**Quantum Perturbation Theory in Fluid Mixtures**

S. M. Motevalli and M. Azimi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54056

### **1. Introduction**

22 Advances in Quantum Mechanics

268 Advances in Quantum Mechanics

wall. in preparation.

84, 2012, pp. 139-154

*Ann. Phys.*, 85:514–545, 1974.

compound media. *J. Math. Phys*, 46:042305, 2005.

[9] G. A. Hagedorn and A. Joye. Semiclassical dynamics with exponentially small error

[10] M. Reed and B. Simon. *Methods of Modern Mathematical Physics II: Fourier Analysis,*

[14] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon. *Schrödinger Operators with Application*

[15] F. D. Mera, S. A. Fulling, J. D. Bouas, and K. Thapa. WKB approximation to the power

[16] J. D. Bouas, S. A. Fulling, F. D. Mera, K. Thapa, C. S. Trendafilova, and J. Wagner. Investigating the spectral geometry of a soft wall. In A. Barnett, C. Gordon, P. Perry, and A. Uribe, editors, *Spectral Geometry, Proceedings of Symposia in Pure Mathematics*, Vol.

[17] R. Balian and C. Bloch. Solution of the Schrödinger equation in terms of classical paths.

[18] I. Pirozhenko, V. V. Nesterenko, and M. Bordag. Integral equations for heat kernel in

[11] W. V. Lovitt. *Linear Integral Equation*. Dover Publications, New York, 1950.

[13] B. L. Moiseiwitsch. *Integral Equations*. Dover Publications, New York, 2005.

[12] F.G. Tricomi. *Integral Equations*. Dover Publications, New York, 1985.

*to Quantum Mechanics and Global Geometry*. Springer, Berlin, 1987.

estimates. *Commun. Math. Phys.*, 207:439–465, 1999.

*Self-Adjointness*. Academic Press, New York, 1975.

Experimental assessment of macroscopic thermo-dynamical parameters under extreme conditions is almost impossible and very expensive. Therefore, theoretical EOS for further experiments or evaluation is inevitable. In spite of other efficient methods of calculation such as integral equations and computer simulations, we have used perturbation theory because of its extensive qualities. Moreover, other methods are more time consuming than perturbation theories. When one wants to deal with realistic intermolecular interactions, the problem of deriving the thermodynamic and structural properties of the system becomes rather formida‐ ble. Thus, perturbation theories of liquid have been devised since the mid-20th century. Thermodynamic perturbation theory offers a molecular, as opposed to continuum approach to the prediction of fluid thermodynamic properties. Although, perturbation predictions are not expected to rival those of advanced integral-equations or large scale computer simulations methods, they are far more numerically efficient than the latter approaches and often produced comparably accurate results.

Dealing with light species such as *He* and *H*2 at low temperature and high densities makes it necessary taking into account quantum mechanical effects. Quantum rules and shapes related with the electronic orbital change completely the macroscopic properties.

Furthermore, for this fluid mixture, the quantum effect has been exerted in terms of first order quantum mechanical correction term in the Wigner-Kirkwood expansion. This term by generalizing the Wigner-Kirkwood correction for one component fluid to binary mixture produce acceptable results in comparison with simulation and other experimental data. Since utilizing Wigner-Kirkwood expansion in temperatures below 50 K bears diverges, we preferred to restrict our investigations in ranges above those temperatures from 50 to 4000 degrees. In these regions our calculations provide more acceptable results in comparison with other studies.

© 2013 Motevalli and Azimi; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Motevalli and Azimi; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This term make a negligible contribution under high temperatures conditions. Taking into account various contributions, we have utilized an improved version of the equation of state to study the Helmholtz free energy *F*, to investigate the effects of *P* and *T* on thermodynamic properties of helium and hydrogen isotopes mixtures over a wide range of densities. We also have studied effects of concentrations of each component on macroscopic parameters. In addition, comparisons among various perturbation and ideal parts have been presented in logarithmic diagrams for different densities and concentrations for evaluation of perturbation terms validity in respect to variables ranges.

The first section is dedicated to a brief description of Wigner expansion which leads to derivation of first quantum correction term in free energy. With the intention of describing effects of quantum correction term we have explained theoretical method of our calculations in the frame work of statistical perturbation theory of free energy in section two. In section three we have depicted diagrams resulted from our theoretical evaluations and gave a brief explanation for them. In section four we have focused on the description of our calculations and its usages in different areas. Finally, some applications of this study have been introduced in the last section.

#### **2. Quantum correction term**

Considering quantum system of *N* identical particles of mass *m* confined to the region of *Λ* with the interacting potential of*U* . This structure is considered in *υ*-dimension space (*R <sup>υ</sup>*). In the absence of external fields the Hamiltonian of particles is given as

$$H = \frac{1}{2m} \left( -i\hbar \vec{\nabla} \right)^2 + \mathcal{U}\left(\vec{r}\right) \tag{1}$$

**2.1. Wigner-Kirkwood expansion**

regime can be expanded in powers of *h* <sup>2</sup>

b

sional integral defined in an infinite domain *R <sup>υ</sup>*:

Via integrating equation 2 in respect to *β* we have

b

0

b

b

Let us introduce following definition

That *Q* and *D* respectively represent

b

ur r r

0

b

b

 b

ur r r

in configuration space *r*

Where *p*

To have an analytical equation for quantum effects in fluid we must derive partition function of it. In approximating partition function we need to evaluate Boltzmann density. Conse‐ quently having an expansion of quantum correction terms it is necessary to expand Boltzmann density. Considering system of *N* particles in the infinite space in standard Wigner-Kirkwood expansion [1, 2] fermions or boson exchange effects between quantum particles have been neglected. In the "bulk" regime, equilibrium quantities of this system in the nearly classical

this expansion for utilizing it in statistical perturbation framework. The Boltzmann density *B<sup>β</sup>*

*H H i pr i pr N*

<sup>→</sup> =(*p*1, *<sup>p</sup>*2, *<sup>p</sup>*3, ...) is the *υ<sup>N</sup>* -dimensional momentum vector. Instead of considering we

*H Z*

*H z i pr i pr N*

u

u p


( )

take the Laplace transform of this operator with respect to the inverse temperature *β*,

*H z* <sup>1</sup> *de e*

( )

p

2

( )

= -Ñ -

*Qi P m m D P Ur z m*

1 1 2 2 1 2

2

= ++

( )

2 2

<sup>r</sup> ur <sup>h</sup>


*dP d re re <sup>e</sup> <sup>e</sup>*

b b

2

*dP B re r e e e*

. In this section, we review briefly the derivation of

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

271

<sup>h</sup> (2)

<sup>→</sup> can be formally written in the basis of plane waves as a *υN* -dimen‐

( ) . . ( )

 b


( ) . . ( )

1

*H Z*

*H z DQ* += + (5)

ur <sup>r</sup> (6)

<sup>h</sup> (4)

Where, ℏ is the Plank constant. The equilibrium statistical mechanics of the particle system is studied in the canonical ensemble at the temperature *T* (or, alternatively, the inverse temper‐ ature *β* =1 / *kBT* with *kB* being Boltzmann's constant). Quantum effects will be considered via de Broglie wavelength *λ* =ℏ *β* / *m*. For a typical microscopic length of particles *l*, for suffi‐ ciently small dimensionless parameter *λ* / *l* semi-classical regime is dominant. In such system Boltzmann density in configuration space *r* <sup>→</sup> can be expanded in powers of *λ* <sup>2</sup> within the wellknown Wigner-Kirkwood expansion [1, 2]. In the case of an inverse-power-law repulsive potential *V* (*r*)=*V*<sup>0</sup> (*a* /*r*)*n* from the range 1<*n* <*∞*, the Wigner-Kirkwood expansion turns out to be analytic in *λ* <sup>2</sup> [3]. In the hard-core limit *n* →*∞*, this expansion is not further correct and one has the non-analyticity of type (*λ* 2)1/2 , as was shown in numerous analytic studies [4-7]. In contrast to the bulk case, the resulting Boltzmann density involves also position dependent terms which are non-analytic in *λ*. Under some condition about the classical density profile, the analyticity in *λ* is restored by integrating the Boltzmann density over configuration space.

#### **2.1. Wigner-Kirkwood expansion**

This term make a negligible contribution under high temperatures conditions. Taking into account various contributions, we have utilized an improved version of the equation of state to study the Helmholtz free energy *F*, to investigate the effects of *P* and *T* on thermodynamic properties of helium and hydrogen isotopes mixtures over a wide range of densities. We also have studied effects of concentrations of each component on macroscopic parameters. In addition, comparisons among various perturbation and ideal parts have been presented in logarithmic diagrams for different densities and concentrations for evaluation of perturbation

The first section is dedicated to a brief description of Wigner expansion which leads to derivation of first quantum correction term in free energy. With the intention of describing effects of quantum correction term we have explained theoretical method of our calculations in the frame work of statistical perturbation theory of free energy in section two. In section three we have depicted diagrams resulted from our theoretical evaluations and gave a brief explanation for them. In section four we have focused on the description of our calculations and its usages in different areas. Finally, some applications of this study have been introduced

Considering quantum system of *N* identical particles of mass *m* confined to the region of *Λ* with the interacting potential of*U* . This structure is considered in *υ*-dimension space (*R <sup>υ</sup>*). In

( ) ( ) <sup>1</sup> <sup>2</sup>

Where, ℏ is the Plank constant. The equilibrium statistical mechanics of the particle system is studied in the canonical ensemble at the temperature *T* (or, alternatively, the inverse temper‐ ature *β* =1 / *kBT* with *kB* being Boltzmann's constant). Quantum effects will be considered via de Broglie wavelength *λ* =ℏ *β* / *m*. For a typical microscopic length of particles *l*, for suffi‐ ciently small dimensionless parameter *λ* / *l* semi-classical regime is dominant. In such system

known Wigner-Kirkwood expansion [1, 2]. In the case of an inverse-power-law repulsive

In contrast to the bulk case, the resulting Boltzmann density involves also position dependent terms which are non-analytic in *λ*. Under some condition about the classical density profile, the analyticity in *λ* is restored by integrating the Boltzmann density over configuration space.

(*a* /*r*)*n* from the range 1<*n* <*∞*, the Wigner-Kirkwood expansion turns out

[3]. In the hard-core limit *n* →*∞*, this expansion is not further correct and

= -Ñ + <sup>r</sup> <sup>r</sup> <sup>h</sup> (1)

<sup>→</sup> can be expanded in powers of *λ* <sup>2</sup> within the well-

, as was shown in numerous analytic studies [4-7].

*H i Ur m*

the absence of external fields the Hamiltonian of particles is given as

2

terms validity in respect to variables ranges.

in the last section.

270 Advances in Quantum Mechanics

potential *V* (*r*)=*V*<sup>0</sup>

to be analytic in *λ* <sup>2</sup>

**2. Quantum correction term**

Boltzmann density in configuration space *r*

one has the non-analyticity of type (*λ* 2)1/2

To have an analytical equation for quantum effects in fluid we must derive partition function of it. In approximating partition function we need to evaluate Boltzmann density. Conse‐ quently having an expansion of quantum correction terms it is necessary to expand Boltzmann density. Considering system of *N* particles in the infinite space in standard Wigner-Kirkwood expansion [1, 2] fermions or boson exchange effects between quantum particles have been neglected. In the "bulk" regime, equilibrium quantities of this system in the nearly classical regime can be expanded in powers of *h* <sup>2</sup> . In this section, we review briefly the derivation of this expansion for utilizing it in statistical perturbation framework. The Boltzmann density *B<sup>β</sup>* in configuration space *r* <sup>→</sup> can be formally written in the basis of plane waves as a *υN* -dimen‐ sional integral defined in an infinite domain *R <sup>υ</sup>*:

$$B\_{\beta} = \left\langle \vec{r} \; \middle| \; e^{-\beta H} \; \middle| \; \vec{r} \right\rangle = \int \frac{d\vec{P}}{\left(2\pi\hbar\right)^{\nu N}} e^{-(i\hbar)\vec{p}\cdot\vec{r}} \; e^{-\beta H} e^{(i\hbar)\vec{p}\cdot\vec{r}} \tag{2}$$

Where *p* <sup>→</sup> =(*p*1, *<sup>p</sup>*2, *<sup>p</sup>*3, ...) is the *υ<sup>N</sup>* -dimensional momentum vector. Instead of considering we take the Laplace transform of this operator with respect to the inverse temperature *β*,

$$\int\_0^\beta d\beta e^{-\beta H} e^{-\beta z} = \frac{1}{H+Z} \tag{3}$$

Via integrating equation 2 in respect to *β* we have

$$\int\_0^\beta d\beta \left\langle \vec{r} \left| e^{-\beta H} \right| \, \vec{r} \right\rangle e^{-\beta z} = \int \frac{d\vec{P}}{\left(2\pi\hbar\right)^{\alpha N}} e^{-(\beta\hbar)\vec{p}\cdot\vec{r}} \xrightarrow[H+Z]{} \tag{4}$$

Let us introduce following definition

$$H + z = D + Q \tag{5}$$

That *Q* and *D* respectively represent

$$\begin{aligned} Q &= \frac{1}{2m} \left( -i\hbar \vec{\nabla} \right)^2 - \frac{1}{2m} \vec{P}^2 \\ D &= \frac{1}{2m} \vec{P}^2 + \mathcal{U} \left( \vec{r} \right) + z \end{aligned} \tag{6}$$

One can expand

$$\frac{1}{H+Z} = \frac{1}{D} - \frac{1}{D} \mathbb{Q}\frac{1}{D} + \frac{1}{D} \mathbb{Q}\frac{1}{D} \mathbb{Q}\frac{1}{D} - \dots \tag{7}$$

( ) ( )

r

0

b

b

b

( ) ( )

( ) ( )

to o

rde

b

( )

p

l

2

*f* (*r*

<sup>→</sup> ) as follows

( )

p l

2

*Br e*

=

1

*N*

u

*N*

u

*N*

*H U*

in partition function of fluids mixture.

partition function takes the expansion form

L

*N*

( )

p l

*qu*

At the one-particle level, one introduces the particle density

u


p l

l

u

1

p l

2

, b

2

l

r y

<sup>1</sup> <sup>1</sup>

b

*N*

u

p l

( ) ( )

é ù - <sup>=</sup> ê ú Ñ+ Ñ

<sup>1</sup> , 4 6 <sup>2</sup>

b

2 2 2 2

ï ï ì ü é ù í ý ê ú - Ñ +Ñ + ï ï î þ ê ú ë û

ur ur

24 24

*U U*

*e U eO*

ì ü ï ï é ù <sup>=</sup> í ý ê ú Ñ- Ñ +

b

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup>

ur ur r

l

*U*

*Br e U U*

b


*U*

ur ur r

b


*U*

b


( ) ( )

*Br e U U O*

l

<sup>1</sup> <sup>1</sup>


 b

ur ur r r

*N*

u

b

We conclude that the quantum Boltzmann density in configuration space is g

( ) ( )

l

12 24 <sup>2</sup>

1 !

*qu V*

*N*

*re r e U UO*

 l

ï ï î þ ë û

 b

*Z dr r e r*

<sup>=</sup> <sup>ò</sup> rr r

b

ì ü ï ï é ù - <sup>=</sup> í ý + Ñ+ Ñ + = ê ú ï ï î þ ë û

 b

 l

<sup>2</sup> <sup>2</sup> <sup>2</sup> 2 4

 l

> b

l

 b

4

*H*

*U U*


 l

b

Substituting the *λ*-expansion of the Boltzmann density (12A) into formula (13), the quantum

2 2 2 2 1 1 <sup>4</sup> 1 1

ì ü ï ï é ù <sup>=</sup> í ý ê ú - Ñ +Ñ + ï ï î þ ê ú ë û <sup>ò</sup> <sup>r</sup> ur ur

lb

For expressing macroscopic physical quantities, one defines the quantum average of a function

( ) <sup>1</sup>

b

*<sup>f</sup> dr r e r f r Z N*

<sup>L</sup> <sup>=</sup> <sup>ò</sup> rr r r

*H*

! 24 24 <sup>2</sup>

b

!

*qu*

*qu <sup>N</sup> Z dr e U eO*

l

Integrating Boltzmann density ignoring exchange effects over configuration space will result

 b

ë û

iven,

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

(12)

273

(13)

(14)

(15)

6 8

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> 2 4

*Q*, operates in the following manner

$$\mathcal{Q}\left[f\left(\overline{r}\right)e^{\left(i\not\hbar\right)\overline{p}\cdot\overline{r}}\right] = e^{-\left(i\not\hbar\right)\overline{p}\cdot\overline{r}}\left[\frac{i\hbar}{m}\overline{p}\cdot\overline{\nabla} + \frac{\hbar^2}{2m}\overline{\nabla}^2\right]f\left(\overline{r}\right) \tag{8}$$

And then we can find that

$$e^{-(\not H)\vec{p}\cdot\vec{r}}\frac{1}{H+z}e^{(\not H)\vec{p}\cdot\vec{r}}=\frac{1}{D}\sum\_{n=0}^{\infty}\left\{\left[\frac{i\hbar}{m}\vec{p}\cdot\vec{\nabla}+\frac{\hbar^{2}}{2m}\vec{\nabla}^{2}\right]\frac{1}{D}\right\}^{n}$$

$$\int \frac{d\vec{P}}{(2\pi\hbar)^{\alpha N}}e^{-(\not H)\vec{p}\cdot\vec{r}}\frac{1}{H+z}e^{(\not H)\vec{p}\cdot\vec{r}}=\int \frac{d\vec{P}}{(2\pi\hbar)^{\alpha N}}\frac{1}{D}\sum\_{n=0}^{\infty}\left\{\left[\frac{i\hbar}{m}\vec{p}\cdot\vec{\nabla}+\frac{\hbar^{2}}{2m}\vec{\nabla}^{2}\right]\frac{1}{D}\right\}^{n}=\tag{9}$$

$$\int\_{0}^{\theta}d\beta\left\{\vec{r}\cdot\Big[e^{-\beta H}\Big|\ \vec{r}\right]e^{-\beta z}=\sum\_{n=0}^{\infty}\left\{\frac{d\vec{P}}{2\pi\hbar}\frac{1}{D}\frac{1}{D}\left[\left[\frac{i\hbar}{m}\vec{p}\cdot\vec{\nabla}+\frac{\hbar^{2}}{2m}\vec{\nabla}^{2}\right]\frac{1}{D}\right]^{n}\right\}$$

So we have expanded series in ℏ2*n* which enable us power series of ℏ*n*. It remains to define 1 / *D <sup>j</sup>*

$$\frac{1}{D^j} = \int\_0^\infty d\beta \frac{1}{\left(j-1\right)!} \beta^{j-1} e^{-\beta D} = \int\_0^\infty d\beta e^{-\beta z} \frac{1}{\left(j-1\right)!} \beta^{j-1} e^{-\beta \left[\frac{-^2}{p} \not\to m + \mathcal{U}\left(\bar{r}\right)\right]} \tag{10}$$

and finally integrating on the momentum variables *p* <sup>→</sup> , the Boltzmann density in configuration space is obtained as the series

$$\left\langle \vec{r} \mid e^{-\beta H} \mid \vec{r} \right\rangle = \sum\_{n=0}^{\infty} \mathcal{B}\_{\beta}^{(n)} \left( \vec{r} \right),\tag{11}$$

where

$$\begin{aligned} B\_{\beta}^{(0)}\left(\bar{r}\right) &= \frac{1}{\left(\sqrt{2\pi}\,\bar{\lambda}\right)^{\alpha N}} e^{-\beta\mathcal{U}}\\ B\_{\beta}^{(1)}\left(\bar{r}\right) &= \frac{1}{\left(\sqrt{2\pi}\,\bar{\lambda}\right)^{\alpha N}} e^{-\beta\mathcal{U}}\bar{\lambda}^{2} \left[\frac{-\beta}{4}\overline{\nabla}^{2}\mathcal{U} + \frac{\beta^{2}}{6} \left(\overline{\nabla}\mathcal{U}\right)^{2}\right] \\ B\_{\beta}^{(2)}\left(\bar{r}\right) &= \frac{1}{\left(\sqrt{2\pi}\,\bar{\lambda}\right)^{\alpha N}} e^{-\beta\mathcal{U}} \left\{\lambda^{2} \left[\frac{\beta}{6}\overline{\nabla}^{2}\mathcal{U} - \frac{\beta^{2}}{8} \left(\overline{\nabla}\mathcal{U}\right)^{2}\right] + O\lambda^{4}\right\} \end{aligned} \tag{12}$$

We conclude that the quantum Boltzmann density in configuration space is g iven, to o 2 rde , b r y l

$$\begin{aligned} \left\langle \bar{r} \mid e^{-\beta H} \mid \bar{r} \right\rangle &= \frac{1}{\left(\sqrt{2\pi}\,\lambda\right)^{\upsilon N}} e^{-\beta\mathcal{U}} \left\{ 1 + \lambda^2 \left[ \frac{-\beta}{12} \overline{\nabla}^2 \mathcal{U} + \frac{\beta^2}{24} \left(\overline{\nabla}\mathcal{U}\right)^2 \right] + \mathcal{O}\lambda^4 \right\} = 0\\ \frac{1}{\left(\sqrt{2\pi}\,\lambda\right)^{\upsilon N}} & \left\{ e^{-\beta\mathcal{U}} \left[ 1 - \frac{\lambda^2 \beta}{24} \overline{\nabla}^2 \mathcal{U} \right] + \frac{\lambda^2}{24} \overline{\nabla}^2 e^{-\beta\mathcal{U}} + \mathcal{O}\lambda^4 \right\} \end{aligned}$$

One can expand

272 Advances in Quantum Mechanics

1 111111 *Q QQ* ... *HZ DD DD D D* =- + -

( ) ( ) ( ) ( ) <sup>2</sup> <sup>2</sup> . . . <sup>2</sup> *i pr i pr <sup>i</sup> Q f re e p f r*

( )

 p

*N*

= Ñ +

u

<sup>111</sup> . <sup>2</sup> 2 2

ur ur r ur ur h h

ì ü ï ï é ù <sup>=</sup> í ý ê ú Ñ+ Ñ

r ur ur h h

<sup>2</sup> <sup>2</sup> . .

 u

*e e <sup>p</sup> H z D m mD*

<sup>1</sup> . <sup>2</sup> <sup>2</sup>

So we have expanded series in ℏ2*n* which enable us power series of ℏ*n*. It remains to define

*<sup>n</sup> m D*

ï ï î þ ê ú ë û <sup>å</sup> ur

<sup>2</sup> 1 1

 b

*p m Ur j j D z*

( ) ( )

b

0 , *<sup>H</sup> <sup>n</sup> n re r B r*

=

 b

0

=

<sup>+</sup> ï ï î þ ê ú ë û

*n*

2

ì ü ï ï é ù í ý ê ú Ñ

2

b

<sup>=</sup> å rr r (11)

1 *n*

( ) <sup>2</sup>

<sup>→</sup> , the Boltzmann density in configuration

r r

ì ü ï ï é ù <sup>=</sup> í ý ê ú Ñ+ Ñ =

*n*

( )

( ) ( )

*<sup>j</sup> d e de e*

b

¥ -

b

1 ! 1 !

 b

é ù ¥ ¥ - + ê ú - - - - ë û = = - - ò ò

h

p


*m m*

(7)

(8)

(9)

(10)

+

( ) ( )

*H z*


ò ò

*i pr i pr*

r r r r h h

h h

 b

<sup>2</sup> <sup>2</sup> . .

*ee p Hz D m m D*

ò ò å

*n*

0

=

å

*dP dP i*

¥ -

ur r ur r r h h

*N N <sup>n</sup>*

¥

0

=

*dP i d r e re <sup>p</sup> D m*

0 0 1 1 1

bb

and finally integrating on the momentum variables *p*

*D j j*

11 1 . <sup>2</sup>

<sup>+</sup> ï ï î þ ê ú ë û

*i*

*Q*, operates in the following manner

And then we can find that

( )

b

p

u

space is obtained as the series

0

1 / *D <sup>j</sup>*

where

b

( ) ( )

*i pr i pr*

r r r r h h

¥ -

b

Integrating Boltzmann density ignoring exchange effects over configuration space will result in partition function of fluids mixture.

$$Z\_{qu} = \frac{1}{N!} \int\_{V} \vec{dr} \left\langle \stackrel{\rightarrow}{r} \left| e^{-\beta H} \right| \stackrel{\rightarrow}{r} \right\rangle \tag{13}$$

Substituting the *λ*-expansion of the Boltzmann density (12A) into formula (13), the quantum partition function takes the expansion form

$$Z\_{qu} = \frac{1}{N!} \int\_{\Lambda} d\vec{r} \frac{1}{\left(\sqrt{2\pi}\lambda\right)^{\upsilon N}} \left\{ e^{-\beta\mathcal{U}} \left[1 - \frac{\lambda^2 \mathcal{B}}{24} \overrightarrow{\nabla}^2 \mathcal{U}\right] \mathbf{1} + \frac{\lambda^2}{24} \overrightarrow{\nabla}^2 e^{-\beta\mathcal{U}} + \mathcal{O}\lambda^4 \right\} \tag{14}$$

For expressing macroscopic physical quantities, one defines the quantum average of a function *f* (*r* <sup>→</sup> ) as follows

$$\left\{ f \right\}\_{qu} = \frac{1}{Z\_{qu}N!} \int\_{\Lambda} d\vec{r} \left\langle \stackrel{\rightarrow}{r} \left| e^{-\beta H} \right| \stackrel{\rightarrow}{r} \right\rangle f\left(\stackrel{\rightarrow}{r}\right) \tag{15}$$

At the one-particle level, one introduces the particle density

$$m\_{qu}\left(r\right) = \left\langle \sum\_{j=1}^{N} \mathcal{S}\left(r - r\_j\right) \right\rangle\_{qu} \tag{16}$$

Since we have ln(1− *x*)= − *x* − *x* <sup>2</sup> / 2−... we can expand the second term in the right side. By

which indicates the first term of Wigner-Kirkwood correction part that is consist of the second derivative of potential function that leads to below equation for quantum correction term with

0

¥

*F U r g r dr m* s

0

is the ith particle's mass. *Vij* ¯ is the average molecular volume. Distribution function defines

probability of finding particle at particular point *r*. In many literatures that have studied distribution function found it more versatile to use Laplace transform of this function *G*(*s*).

In this chapter the two formula which use RDF, we will encounter below integral equation

( ) ( ) ( ) ( ) { ( ) ( ) ( ) ( ) } <sup>0</sup> 1 3 3 2 2

On the right side of above equation from the right in the first equation we approximate distribution function with its values at contact points. This choice has been resulted from the behavior of molecules of which their repulsive interactions dominate their attractive potential.

<sup>=</sup> <sup>=</sup> <sup>=</sup> - ò ò òò (27)

 f

 s

0 0 1 00 *I r r rg r dr x x xg x dx x x g x dx x x g x dx*

¥¥ ¥

( ) <sup>0</sup> ( ) *sr G s rg r e dr ij ij*

*F u r g r r V dr*

¥

*qu ij ij ij*

(1) 2 2

b

<sup>2</sup> ()()

*g*(*r*) represents radial distribution function, which is a measure of the spatial structure of the particles in reference system, is the expected number of particles at a distance *r*. *NA* is Avogadro

is the distance in which potential function effectively tend to zero.

( ) ( )4

p

<sup>=</sup> å <sup>Ñ</sup> ò (25)

¥ - <sup>=</sup> ò (26)

f

2 (1) 2

*hN n*

p

96 *A*

*qu*

Generalizing to multi-component system we have [8]

2

<sup>2</sup> ,

2

p

*hN n c c*

b

96 *ij A i j*

*i j ij*

*m* s

<sup>→</sup> <sup>2</sup>*<sup>U</sup>* we can have explicit formula for the second term of

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

275

<sup>=</sup> <sup>Ñ</sup> ò (24)

means of equation 18 in deriving ∇

the number density of *n* we have

*m*<sup>11</sup> =*m*1, *m*<sup>22</sup> =*m*2, *m*<sup>12</sup> =*c*1*m*<sup>1</sup> + *c*2*m*

that need expansion.

 sf

s f

constant and *σ* <sup>0</sup>

**2.2. Free energy**

*mi*

At the two-particle level, the two-body density is given by

$$m\_{qu}^{(2)}\left(r,r'\right) = \left\langle \sum\_{\substack{j,k=1\\j\neq k}}^N \delta\left(r-r\_j\right) \delta\left(r'-r\_j\right) \right\rangle\_{qu} \tag{17}$$

And the pair distribution function

$$\log\_{qu}\left(r, r'\right) = \frac{n\_{qu}^{(2)}\left(r, r'\right)}{n\_{qu}\left(r\right)n\_{qu}\left(r'\right)}\tag{18}$$

The classical partition function and the classical average of a function *f* (*r* <sup>→</sup> ) are defined as follows

$$Z = \frac{1}{N!} \int\_{\Lambda} \frac{d\vec{r}}{\left(\sqrt{2\pi}\lambda\right)^{\nu N}} e^{-\beta\mathcal{U}(\vec{r})} \tag{19}$$

$$\left\langle f \right\rangle = \frac{1}{\text{ZN}!} \int\_{\Lambda} \frac{d\vec{r}}{\left(\sqrt{2\pi}\,\lambda\right)^{\iota\aleph}} e^{-\beta\mathcal{U}\left(\vec{r}\right)} f\left(\vec{r}\right) \tag{20}$$

Consequently with the definition of equation 19 one can derive below equation for *Zqu*

$$Z\_{qu} = Z \left\{ 1 - \lambda^2 \left. \frac{\beta}{24} \sqrt{\overline{\nabla}^2 U} \right| + O\lambda^4 \right\} \tag{21}$$

$$
\beta F\_{qu} = -\ln\left(Z\_{qu}\right) \tag{22}
$$

$$\ln\left(Z\_{q\mu}\right) = \ln(Z) + \ln(1 - \lambda^2 \frac{\beta}{24} \left\langle \overline{\nabla}^2 U \right\rangle + O\lambda^4 + \dots \tag{23}$$

Since we have ln(1− *x*)= − *x* − *x* <sup>2</sup> / 2−... we can expand the second term in the right side. By means of equation 18 in deriving ∇ <sup>→</sup> <sup>2</sup>*<sup>U</sup>* we can have explicit formula for the second term of which indicates the first term of Wigner-Kirkwood correction part that is consist of the second derivative of potential function that leads to below equation for quantum correction term with the number density of *n* we have

$$F\_{qu}^{(1)} = \frac{\hbar^2 N\_A n \beta}{96\pi^2 m} \int\_{\sigma^0}^{\infty} \nabla^2 \mathcal{U}(r) g(r) dr \tag{24}$$

*g*(*r*) represents radial distribution function, which is a measure of the spatial structure of the particles in reference system, is the expected number of particles at a distance *r*. *NA* is Avogadro constant and *σ* <sup>0</sup> is the distance in which potential function effectively tend to zero.

#### **2.2. Free energy**

( ) ( ) 1

*<sup>j</sup> qu*

*j k qu*

( ) ( ) ( ) ( )

*n rr*

*qu qu*

*n rn r*

 d

= - å (16)

¢ ¢ = -- å (17)

¢ ¢ <sup>=</sup> ¢ (18)

<sup>→</sup> ) are defined as

(19)

(20)

(21)

(23)

*N qu j*

=

( ) ( ) ( ) ( ) <sup>2</sup> , 1

<sup>2</sup> , , *qu*

( )

( )

p l

Consequently with the definition of equation 19 one can derive below equation for *Zqu*

<sup>L</sup> <sup>=</sup> <sup>ò</sup> <sup>r</sup> <sup>r</sup> <sup>r</sup>

p l

<sup>L</sup> <sup>=</sup> <sup>ò</sup> <sup>r</sup> <sup>r</sup>

! <sup>2</sup>

! <sup>2</sup>

b

*N*

*dr Z e*

1 ( )

*N*

u

( ) ( ) <sup>1</sup>

u

<sup>2</sup> 2 4 <sup>1</sup> 24 *Z Z UO qu* b l

 ì ü = - Ñ+ í ý î þ ur

( ) <sup>2</sup> 2 4 ln ln( ) ln(1 ...) <sup>24</sup> *Z Z qu U O*

b = + - Ñ+ + l

*N dr <sup>f</sup> e fr ZN*

*U r*

*U r*

 l

*F Z qu* = - ln( *qu* ) (22)

 l

ur

b


b


*n rr r r r r* d

> = ¹

( )

The classical partition function and the classical average of a function *f* (*r*

*qu*

*g rr*

*N qu j j j k*

*n r rr* d

At the two-particle level, the two-body density is given by

,

And the pair distribution function

274 Advances in Quantum Mechanics

follows

Generalizing to multi-component system we have [8]

$$F^{(1)}\_{q\mu} = \frac{\hbar^2 N\_A m \mathcal{B}}{96\pi^2} \sum\_{i,j} \frac{c\_i c\_j}{m\_{ij}} \int\_{\sigma\_{\vec{\eta}}^0}^{\infty} \nabla^2 u\_{ij}(r) g\_{i\vec{\eta}}(r) 4\pi r^2 \overline{V\_{i\vec{\eta}}} dr \tag{25}$$

*m*<sup>11</sup> =*m*1, *m*<sup>22</sup> =*m*2, *m*<sup>12</sup> =*c*1*m*<sup>1</sup> + *c*2*m* 2

*mi* is the ith particle's mass. *Vij* ¯ is the average molecular volume. Distribution function defines probability of finding particle at particular point *r*. In many literatures that have studied distribution function found it more versatile to use Laplace transform of this function *G*(*s*).

$$\mathcal{G}\_{ij}\begin{pmatrix}\mathbf{s}\end{pmatrix} = \int\_0^\infty r g\_{ij}(r) e^{-sr} dr\tag{26}$$

In this chapter the two formula which use RDF, we will encounter below integral equation that need expansion.

$$I = \int\_{\sigma\_0}^{\sigma} r\phi(r)\operatorname{rg}(r)dr = \sigma\_0^3 \int\_1^{\sigma} \mathbf{x}\phi(\mathbf{x})\operatorname{xg}(\mathbf{x})d\mathbf{x} = \sigma\_0^3 \left\{ \int\_0^{\sigma} \mathbf{x}^2 \phi(\mathbf{x})\operatorname{g}(\mathbf{x})d\mathbf{x} - \int\_0^1 \mathbf{x}^2 \phi(\mathbf{x})\operatorname{g}(\mathbf{x})d\mathbf{x} \right\} \tag{27}$$

On the right side of above equation from the right in the first equation we approximate distribution function with its values at contact points. This choice has been resulted from the behavior of molecules of which their repulsive interactions dominate their attractive potential. However, for the second term (*I* ′ ) we will use change in integrals to employ Laplace transform of RDF instead of RDF directly.

$$\log\left(\mathbf{x}\right) = \frac{1}{2\pi i} \int\_{\gamma - i\infty}^{\gamma - i\infty} G\left(\mathbf{s}\right) e^{-s\mathbf{x}} d\mathbf{s} \tag{28}$$

expressed as of the "unperturbed" system) plus a correction term. This in turn implies that the thermodynamic and structural properties of the real system may be expressed in terms of those of the reference system which, of course, should be known. In the case of two component fluids, a natural choice for the reference system is the hard-sphere fluid, even for this simple system the thermodynamic and structural properties are known only approximately. Let us now consider a system defined by a pair interaction potential *u*(*r*). The usual perturbation expan‐ sion for the Helmholtz free energy, *F* , to first order in *β* =1 / *kβT* , with *T* being the absolute temperature and *kB* being the Boltzmann constant, leads to *F* . Common starting point of many thermodynamic perturbation theories is an expansion of the Helmholtz free energy, the resulting first-order prediction for a fluid composed of particles helium and hydrogen is given

The terms respectively are perturbation, Quantum, hard convex body and ideal terms.

*i j ij ij ij ij*

l

s

¥

rs

Perturbation term due to long range attraction of potential is given by [10]

0

to long-ranged attraction for DY potential we can employ below equation:

0

( 1) 3 13 <sup>1</sup> sin

e s

2 22

*l l*

*<sup>i</sup>* define the number of element in a molecule, *l*

 s

molecule. *δ<sup>F</sup> <sup>t</sup>* corresponds to the interval of *σij*

. By this approach we can express this term as:

*Vij* ¯ the average molecular volume defined as:

s

s

2 ( ) ( , , )4 *ij t HS*

*F n c c u r g r r V dr*

0 0 ,

<sup>=</sup> ç ÷ ç÷ ç÷ - - ç÷ ç÷ è ø èø èø

*i i ij ii ij i i ii jj i ij ii jj ii*

s s*l*

é ù ¢ - é ù ê ú é ù =+ + - - + - ê ú ê ú ë û ê ú <sup>+</sup> ë û ë û

*n l*

l

*t t ij ij ij i j ij ij ij i j ij ij F kT c c A V e G e G F*

,

*i j*

p *t Q HB id FF F F F* =+ + + (32)

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

277

2

 u

 s

1

å (34)

*HS* ) in calculation of first order perturbation contribution due

d

s s


s

is distance of centre to centre for each

<sup>0</sup> which long range attractive range is not

(35)

<sup>=</sup> å ò (33)

 p

*ij ij*

 u

æ ö æö æö

( ) ( ) ( ) ( )

, *σij*

further applicable. Consequently, we prefer to use the contact value of hard sphere RDF at

<sup>3</sup> <sup>2</sup> <sup>2</sup> 2 2 2 1 <sup>2</sup>

*i*

via the following equation.

Via Laplace transform of RDF (*rgij*

3

s

*V*

Where *n* ′

*r* =*σij*

Substituting above equation in *I* ′ we have

$$I'=\int\_0^\wp \varphi\left(s\right)G\left(s\right)ds\tag{29}$$

Where *φ*(*s*) represents

$$\log \left( s \right) = \frac{1}{2\pi i} \int\_{\gamma - i\infty}^{\gamma - i\infty} \mathbf{x} \phi \left( \mathbf{x} \right) e^{-s\mathbf{x}} d\mathbf{x} \tag{30}$$

That indicates inverse Laplace of *xϕ*(*x*). So it suffices to just define inverse Laplace of potential function multiplied by *x*.

Therefore, Using Laplace transform of RDF *G*(*s*) [9] quantum correction term for DY potential turn out to be

$$F^{\mathcal{Q}} = \frac{\hbar^2 N\_A n \mathcal{J}}{24\pi} \sum\_{i,j} \frac{c\_i c\_j \varepsilon\_{ij} A\_{\bar{\eta}} \overline{V\_{\bar{\eta}}}}{m\_{\bar{\eta}} \sigma\_{\bar{\eta}}^0} \left\{ \lambda\_{\bar{\eta}}^2 e^{\lambda\_{\bar{\eta}}} G \left( \frac{\lambda\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^0} \right) - \nu\_{\bar{\eta}}^2 e^{\nu\_{\bar{\eta}}} G \left( \frac{\nu\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^0} \right) \right\} \tag{31}$$

*ci* is the *i* particle's concentration and *n* represents number density. *Aij* , *λij* and *υij* are controlling parameters of double Yukawa(DY). *εij* is the attractive well depth of mutual interacting potential.

#### **3. Framework**

The derivation of the thermodynamic and structural properties of a fluid system becomes a rather difficult problem when one wants to deal with realistic intermolecular interactions. For that reason, since the mid-20th century, simplifying attempts to (approximately) solve this problem have been devised, among which the perturbation theories of liquids have played a prominent role [10]. In this instance, the key idea is to express the actual potential in terms of a reference potential (that in terms of Ross perturbation theory Helmholtz free energy is expressed as of the "unperturbed" system) plus a correction term. This in turn implies that the thermodynamic and structural properties of the real system may be expressed in terms of those of the reference system which, of course, should be known. In the case of two component fluids, a natural choice for the reference system is the hard-sphere fluid, even for this simple system the thermodynamic and structural properties are known only approximately. Let us now consider a system defined by a pair interaction potential *u*(*r*). The usual perturbation expan‐ sion for the Helmholtz free energy, *F* , to first order in *β* =1 / *kβT* , with *T* being the absolute temperature and *kB* being the Boltzmann constant, leads to *F*. Common starting point of many thermodynamic perturbation theories is an expansion of the Helmholtz free energy, the resulting first-order prediction for a fluid composed of particles helium and hydrogen is given via the following equation.

$$F = F^t + F^Q + F^{HB} + F^{id} \tag{32}$$

The terms respectively are perturbation, Quantum, hard convex body and ideal terms. Perturbation term due to long range attraction of potential is given by [10]

$$F^{t} = 2\pi n \sum\_{i,j} c\_i c\_j \int\_{\sigma\_{ij}^0}^{\sigma} u\_{ij}(r) g\_{ij}^{HS}(r, \rho, \sigma\_{ij}) 4\pi r^2 \overline{V\_{ij}} dr \tag{33}$$

Via Laplace transform of RDF (*rgij HS* ) in calculation of first order perturbation contribution due to long-ranged attraction for DY potential we can employ below equation:

$$F^t = kT \sum\_{i,j} c\_i c\_j \varepsilon\_{ij} \sigma\_{ij}^0 A\_{ij} \overline{V}\_{ij} \left( e^{\lambda\_{\overline{\gamma}}} G \left( \frac{\lambda\_{\overline{\gamma}}}{\sigma\_{i\overline{\gamma}}^0} \right) - e^{\nu\_{\overline{\gamma}}} G \left( \frac{\nu\_{\overline{\gamma}}}{\sigma\_{i\overline{\gamma}}^0} \right) \right) - \delta F^t \tag{34}$$

*Vij* ¯ the average molecular volume defined as:

However, for the second term (*I* ′

276 Advances in Quantum Mechanics

of RDF instead of RDF directly.

Substituting above equation in *I* ′

Where *φ*(*s*) represents

function multiplied by *x*.

2

parameters of double Yukawa(DY). *εij*

turn out to be

*ci*

potential.

**3. Framework**

) we will use change in integrals to employ Laplace transform


¥ ¢ <sup>=</sup> ò (29)


( ) ( ) <sup>1</sup> 2

p g

we have

*<sup>i</sup> xg x G s e ds i* g

> ( ) ( ) <sup>0</sup> *I s G s ds* j

( ) ( ) <sup>1</sup> 2

g

*i* g

*ij ij Q A i j ij ij ij*

s

*m*

is the *i* particle's concentration and *n* represents number density. *Aij*

e

*F eG eG*

p

j

*hN n cc A V*

b

p

*i s x x e dx*

*i sx*


 f

That indicates inverse Laplace of *xϕ*(*x*). So it suffices to just define inverse Laplace of potential

Therefore, Using Laplace transform of RDF *G*(*s*) [9] quantum correction term for DY potential

0 00 , 24

The derivation of the thermodynamic and structural properties of a fluid system becomes a rather difficult problem when one wants to deal with realistic intermolecular interactions. For that reason, since the mid-20th century, simplifying attempts to (approximately) solve this problem have been devised, among which the perturbation theories of liquids have played a prominent role [10]. In this instance, the key idea is to express the actual potential in terms of a reference potential (that in terms of Ross perturbation theory Helmholtz free energy is

l

= ç ÷ ç÷ ç÷ -

l

2 2

l

*ij ij i j ij ij ij ij*

*ij ij*

å (31)

ss

 u

æ ö æö æö

è ø èø èø

 u  u

is the attractive well depth of mutual interacting

, *λij*

and *υij*

are controlling

*i sx*


$$\overline{V\_{\vec{\eta}}} = 1 + \frac{\langle n\_i'-1|}{\sigma\_{\vec{\imath}i}^3} \left[ \frac{3}{2} \left( \sigma\_{\vec{\imath}i}^2 + \sigma\_{\vec{\imath}\vec{\jmath}}^2 \right) l\_i - \frac{1}{2} \left( l\_i \right)^3 - \frac{3}{2} \left[ \left( \sigma\_{\vec{\imath}i} + \sigma\_{\vec{\jmath}\vec{\jmath}} \right)^2 - l\_i^2 \right]^{\frac{1}{2}} \sin^{-1} \left[ \frac{l\_i}{\sigma\_{\vec{\imath}i} + \sigma\_{\vec{\jmath}\vec{\jmath}}} \right] \left( \sigma\_{\vec{\imath}\vec{\jmath}}^2 \right) \right] \tag{35}$$

Where *n* ′ *<sup>i</sup>* define the number of element in a molecule, *l i* is distance of centre to centre for each molecule. *δ<sup>F</sup> <sup>t</sup>* corresponds to the interval of *σij* , *σij* <sup>0</sup> which long range attractive range is not further applicable. Consequently, we prefer to use the contact value of hard sphere RDF at *r* =*σij* . By this approach we can express this term as:

$$
\delta F^t \approx \frac{n}{2kT} \sum\_{i,j} c\_i c\_j \int\_{\sigma\_{\vec{\eta}}}^{\sigma\_{\vec{\eta}}^0} u\_{ij}(r) g\_{i\vec{\eta}}^{HS}(\sigma\_{ij}) 4\pi r^2 \overline{V\_{ij}} dr \tag{36}
$$

( )

 s

> s

*HS* (*σ*12) refer to as hard sphere radial distribution function at *<sup>r</sup>* <sup>=</sup>*σ*12 contact point

 s

<sup>2</sup> <sup>2</sup>

æ ö

s s

 ss

 d

*ij ij*

 d s

s

is arithmetic mean of hard-core

*g* (41)

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

11 22

s s

s s

 s

The correction term due to nonadditivity of the hard sphere diameter is the first order

by inclusion of Barrio and Solana correction on equation of state of BMCSL. Undoubtedly, the availability of the analytical HS RDF obtained from the solution to the corresponding Percus– Yevick (PY) equation represented a major step toward the successful application of the perturbation theory of liquids to more realistic inter-particle potentials. However, the lack of thermodynamic consistency between the virial and compressibility routes to the equation of state present in the PY approximation (as well as in other integral equation theories) is a drawback that may question the results derived from its use within a perturbation treatment. Fortunately, for our purposes, another analytical approximation for the RDF of the HS fluid, which avoids the thermodynamic consistency problem, has been more recently derived [15, 16]. We used improved RDF that yields exact asymptotic expression for the thermodynamic properties. However, we have used improved version of RDF that yields exact asymptotic expression for the thermodynamic properties. This have been derived by inclusion of Barrio

( ) ( )

<sup>1</sup> 3 2 ( ) , <sup>1</sup> 1 1

<sup>3</sup> 3 3

s s

*ii jj*

() () ()

= +

 ss

*BMCSL ii jj ii jj*

( ) 1

s s

diameters of each species. Otherwise, the system is said to be non-additive.

s s

12 12 2 3

h

h

*gg g*

*HS BMCSL BS ij ij ij ij ij ij*

12 12 2 11 22

h

is the Kronecker delta function. For additive mixtures *σij*

2 2

=+ + ç ÷ -+ + ç ÷ - - è ø

ss

2


( ) ( ) ( )

1 2 22

ss

<sup>11</sup> <sup>3</sup>

*ii jj ii jj*

 h

 h

11 22 32 2 (40)

279

(42)

+ è ø

1 1 2 11 22

s s

ss

*z cc* 2

*z cc*

p

and Solana correction on EOS of BMCSL at *r* =*σ*12 [9]

s

2 1

*c*

d

*BS ij i ii jj*

h h

h

1

s

*g*

*g*

*<sup>η</sup><sup>i</sup>* <sup>=</sup> *<sup>π</sup>* <sup>6</sup> *<sup>n</sup>*∑ *j cj σ jj i*

*δij*

s

perturbation correction [14]

In Eq. (41), *g*<sup>12</sup>

2 1 2 11 22 11 22

( )( ) 1 2 11 22 11 22 12 12 12 2 () *nd HS F kT nc c* =- + + -

ssss

= -

æ ö - <sup>=</sup> ç ÷

*gij HS* (*σij* ) is the contact value of radial distribution function. *σij* stands for separation distance at contact between the centers of two interacting fluid particles, with species *i* and *j*. Although via minimization of Helmholtz free energy we can achieve value for hard sphere diameter, we preferred to use its analytical form due to its practical approach [17]. Hard sphere diameter will be calculated by means of Barker-Henderson equation as a function of interacting potential and temperature. Using Gauss-Legendre qudrature integration method we are able to evaluate its values numerically.

$$\sigma\_{ij} = \int\_0^{\sigma\_{ij}^0} \left( 1 - \exp\left( -\beta u\_{ij}^{D\mathcal{Y}}(r) \right) \right) dr \tag{37}$$

*F HB*, Helmholtz free energy of hard convex body is given by following equation:

$$F^{HB} = \mathfrak{a}\_{mix} \left( F^{HS} + F^{ud} \right) \tag{38}$$

Non-sphericity parameter *amix* for the scaling theory [11] is defined as

$$a\_{\rm mix} = \frac{1}{3\pi} \frac{\sum\_{i,j} c\_i c\_j V\_{ij}^{\rm eff} \left( V\_{ij}^{\rm eff} \right)^{\prime} \left( V\_{ij}^{\rm eff} \right)^{\prime}}{\sum\_{i,j} c\_i c\_j V\_{ij}^{\rm eff}}, \qquad V\_{ij}^{\rm eff} = \frac{\pi}{6} \sigma\_{ij}^{\, 3} \,\overline{V\_{ij}} \tag{39}$$

(*Vij eff* )′ and (*Vij eff* )″ are the first and second partial derivatives of *Vij eff* with respect to *σii* and *σ jj* . From Boublik, Mansoori, Carnahan, Starling, Leland (BMCSL) [12, 8] with correction term of Barrio [13] on EOS, the Helmholtz free energy, *F HS* for hard sphere term becomes:

$$\begin{aligned} \frac{F}{KT} &= \frac{\eta\_3 \mathbb{E}\xi\_1 + (2 - \eta\_3)\xi\_2}{1 - \eta\_3} + \frac{\eta\_3 \xi\_3}{(1 - \eta\_3)^2} + (\xi\_3 + 2\xi\_2 - 1)\ln(1 - \eta\_3) \lambda \\\\ \xi\_1 &= \frac{3\eta\_1\eta\_2}{\eta\_0\eta\_3}, \quad \xi\_2 = \frac{\eta\_1\eta\_2}{\eta\_3^2}(\eta\_4 z\_1 + \eta\_0 z\_2), \quad \xi\_3 = \frac{\eta\_2\lambda}{\eta\_0\eta\_3^3} \end{aligned}$$

$$\begin{aligned} z\_1 &= 2c\_1 c\_2 \sigma\_{11} \sigma\_{22} \left( \frac{\sigma\_{11} - \sigma\_{22}}{\sigma\_{11} + \sigma\_{22}} \right) \\ z\_2 &= c\_1 c\_2 \sigma\_{11} \sigma\_{22} \left( \sigma\_{11}^{-2} - \sigma\_{22}^{-2} \right) \end{aligned} \tag{40}$$

The correction term due to nonadditivity of the hard sphere diameter is the first order perturbation correction [14]

$$F^{nl} = -kT\pi m c\_1 c\_2 \left(\sigma\_{11} + \sigma\_{22}\right) \left(\sigma\_{11} + \sigma\_{22} - 2\sigma\_{12}\right) \mathbf{g}\_{12}^{HS}(\sigma\_{12}) \tag{41}$$

In Eq. (41), *g*<sup>12</sup> *HS* (*σ*12) refer to as hard sphere radial distribution function at *<sup>r</sup>* <sup>=</sup>*σ*12 contact point by inclusion of Barrio and Solana correction on equation of state of BMCSL. Undoubtedly, the availability of the analytical HS RDF obtained from the solution to the corresponding Percus– Yevick (PY) equation represented a major step toward the successful application of the perturbation theory of liquids to more realistic inter-particle potentials. However, the lack of thermodynamic consistency between the virial and compressibility routes to the equation of state present in the PY approximation (as well as in other integral equation theories) is a drawback that may question the results derived from its use within a perturbation treatment. Fortunately, for our purposes, another analytical approximation for the RDF of the HS fluid, which avoids the thermodynamic consistency problem, has been more recently derived [15, 16]. We used improved RDF that yields exact asymptotic expression for the thermodynamic properties. However, we have used improved version of RDF that yields exact asymptotic expression for the thermodynamic properties. This have been derived by inclusion of Barrio and Solana correction on EOS of BMCSL at *r* =*σ*12 [9]

$$\begin{aligned} g\_{\vec{ij}}^{HS}(\sigma\_{\vec{ij}}) &= g\_{\vec{ij}}^{BMCL}(\sigma\_{\vec{ij}}) + g\_{\vec{ij}}^{BS}(\sigma\_{\vec{ij}})\\ g\_{12}^{BMCL}(\sigma\_{12}) &= \frac{1}{1-\eta\_{3}} + \frac{3\eta\_{2}}{(1-\eta\_{3})^{2}} \frac{\sigma\_{\vec{ii}}\sigma\_{\vec{jj}}}{\sigma\_{\vec{ii}}+\sigma\_{\vec{jj}}} + \frac{2\eta\_{2}^{2}}{(1-\eta\_{3})^{3}} \left(\frac{\sigma\_{\vec{ii}}\sigma\_{\vec{jj}}}{\sigma\_{\vec{ii}}+\sigma\_{\vec{jj}}}\right)^{2},\\ g\_{12}^{BS}(\sigma\_{12}) &= \frac{1-\delta\_{ij}c\_{i}}{2} \frac{\eta\_{i}\eta\_{2}}{(1-\eta\_{3})^{2}} \left(\frac{\sigma\_{\vec{ii}}\sigma\_{\vec{jj}}}{\sigma\_{\vec{ii}}+\sigma\_{\vec{jj}}}\right)^{2} \left(\sigma\_{11}-\sigma\_{22}\right) \left(\delta\_{ij}+\left(1-\delta\_{ij}\right)\frac{\sigma\_{22}}{\sigma\_{11}}\right) \end{aligned} \tag{42}$$

*<sup>η</sup><sup>i</sup>* <sup>=</sup> *<sup>π</sup>* <sup>6</sup> *<sup>n</sup>*∑ *j cj σ jj i*

0

s

( ) ( )4 <sup>2</sup> *ij ij t HS*

*<sup>n</sup> F c c u r g r V dr kT* s

*i j ij ij ij ij*

at contact between the centers of two interacting fluid particles, with species *i* and *j*. Although via minimization of Helmholtz free energy we can achieve value for hard sphere diameter, we preferred to use its analytical form due to its practical approach [17]. Hard sphere diameter will be calculated by means of Barker-Henderson equation as a function of interacting potential and temperature. Using Gauss-Legendre qudrature integration method we are able to evaluate

> ( ( )) <sup>0</sup> <sup>0</sup> 1 exp ( ) *ij DY ij ij u r dr*

*F HB*, Helmholtz free energy of hard convex body is given by following equation:

( ) *HB HS nd*

( ) ( )

*ef eff eff i j ij ij ij i j eff mix ij ij ij eff i j ij*

= =

*ccV V V*

<sup>1</sup> , 3 6

are the first and second partial derivatives of *Vij*

of Barrio [13] on EOS, the Helmholtz free energy, *F HS* for hard sphere term becomes:

(1−*η*3)<sup>2</sup> <sup>+</sup> (*ξ*<sup>3</sup> <sup>+</sup> <sup>2</sup>*ξ*<sup>2</sup> <sup>−</sup>1)ln(1−*η*3),

3 *η*0*η*<sup>3</sup> 3

*a V V ccV*

, 3

. From Boublik, Mansoori, Carnahan, Starling, Leland (BMCSL) [12, 8] with correction term

¢ ²

 b s p

,

) is the contact value of radial distribution function. *σij*

s

Non-sphericity parameter *amix* for the scaling theory [11] is defined as

,

*i j*

p

+

*η*3*ξ*<sup>3</sup>

<sup>2</sup> (*η*4*z*<sup>1</sup> <sup>+</sup> *<sup>η</sup>*0*z*2), *<sup>ξ</sup>*<sup>3</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup>

å

s

*i j*

» å

d

*gij HS* (*σij*

(*Vij eff* )′

*σ jj*

*F HS KT* <sup>=</sup>

*ξ*<sup>1</sup> =

3*η*1*η*<sup>2</sup> *η*0*η*<sup>3</sup>

and (*Vij*

*eff* )″

*η*<sup>3</sup> *ξ*<sup>1</sup> + (2−*η*3)*ξ*<sup>2</sup> 1−*η*<sup>3</sup>

> *η*1*η*<sup>2</sup> *η*3

, *ξ*<sup>2</sup> =

its values numerically.

278 Advances in Quantum Mechanics

2

= -- ò (37)

*mix F aF F* = + (38)

p

s

<sup>å</sup> (39)

*eff* with respect to *σii* and

ò (36)

stands for separation distance

*δij* is the Kronecker delta function. For additive mixtures *σij* is arithmetic mean of hard-core diameters of each species. Otherwise, the system is said to be non-additive.

The ideal free energy with *N* particle for the atomic and molecular components of fluid mixture are given by,

$$F^{id}(n,T) = \frac{3}{2} \ln\left(\frac{h^2}{2\pi k T m\_1^{c\_1} m\_2^{c\_2}}\right) + \ln n + \sum\_i c\_i \ln c\_i - 1 \tag{43}$$

Compressibility factor of ideal term is one and *Z HB* would be estimated with the following derivation of related Helmholtz free energy

$$\mathbf{Z}^{\rm{HB}} = \mathbf{n} \frac{\partial}{\partial \mathbf{n}} \frac{\mathbf{F}^{\rm{HB}}}{kT} \tag{44}$$

*<sup>N</sup> GF P n*

stability. Negative values for this energy describe stable state. This is expressed as

(0) *cc*

*S NkT G*

That *Gi* 0

structure factor is defined as

Compairing this equation with *Scc*

coordination) and when *Scc*(0)> >*Scc*

composition if *Scc*(0)< <*Scc*

in this work, written as:

their reference [20].

*Aij* , *λij* , *υij*

**3.1. Potentials**

<sup>0</sup> ln *xs i i i i i i*

Furthermore, Gibbs excess free energy is an appropriate measure in the definition of phase

represents the Gibbs free energy of pure fluid of species i. Concentration-concentration

<sup>1</sup> <sup>2</sup> 2

*c* - æ ö ¶ <sup>=</sup> ç ÷ ç ÷ ¶è ø

It is convenient to consider interacting potential with short-range sharply repulsive and longer-range attractive tail and treat them within a combined potential. The most practical method for the repulsive term of potential is the hard-sphere model with the benefit of preventing particles overlap. Furthermore, attractive or repulsive tails may be included using a perturbation theory. It is incontrovertible to generalize this potential to multi-component mixtures. This behavior is conveyed in double Yukawa (DY) potential which provides accurate thermodynamic properties of fluid in low temperatures and high density [18, 19]. At first we define DY potential as its effects on pressure of *He* −*H*2 mixture has been studied

> ( ) ( ) 0 0 <sup>0</sup> 1 1

é ù - -

are controlling parameters. These parameters for *He* and *H*2 are listed in table 1 with

 us

( ) *ij ij ij ij r r DY ij ij ij ij ur A e e r* s

e

ls

<sup>=</sup> - ê ú ë û

, ,

*TPN*

= + (47)

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

281

(49)

(50)

*G G c G NkT c c* =- - å å (48)

*id* enable us to define degree of hetero-coordination. In a given

*id* then unlike atoms tend to pair as nearest neighbors (hetero-

*id* then like atoms are preferred as a neighbor.

For the perturbation term due to long rage attraction of potential tail employing (44) we will have

$$\begin{split} \mathbb{Z}\mathbb{Z}^{t} &= \frac{2\pi n}{k\Gamma} \sum\_{i,j} c\_{i} c\_{j} \varepsilon\_{ij} \sigma\_{ij}^{0} A\_{ij} \overline{V}\_{ij} \Big[ e^{\lambda\_{\bar{\eta}}} \Big( \mathcal{G} \Big( \frac{\lambda\_{\bar{\eta}}}{\sigma\_{ij}^{0}} \Big) - n \frac{\partial}{\partial n} \mathcal{G} \Big( \frac{\lambda\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}} \Big) \Big] - e^{\upsilon\_{\bar{\eta}}} \Big( \mathcal{G} \Big( \frac{\nu\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}} \Big) - n \frac{\partial}{\partial n} \mathcal{G} \Big( \frac{\nu\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}} \Big) \Big) \Big] - \delta Z^{-1} \\\\ \delta Z^{-1} &\approx \frac{n}{2kT} \sum\_{i,j} c\_{i} c\_{j} \Big[ \mathcal{g}\_{\bar{\eta}\bar{\eta}}^{HS} \{ \sigma\_{\bar{\eta}\bar{\eta}} \} + n \frac{\partial}{\partial n} \mathcal{g}\_{\bar{\eta}\bar{\eta}}^{HS} \{ \sigma\_{\bar{\eta}} \} \Big] \Big|\_{\mathcal{S}\_{\bar{\eta}}}^{\mathcal{G}^{0}} u\_{\bar{\eta}\bar{\eta}}^{DN} \{ r \} 4\pi r^{2} \overline{V}\_{\bar{\eta}\bar{\eta}} dr \end{split}$$

Numerical integration has been used for calculation of *δZ <sup>t</sup>* in the range of *σij* , *σij* <sup>0</sup> . Expressions for first order perturbation and quantum correction term of compressibility factor are achiev‐ able via applying (44) for the free energy part of the quantum correction term.

$$\boldsymbol{Z}^{Q} = \frac{\boldsymbol{h}^{2} \mathbf{N}\_{A} \boldsymbol{n} \boldsymbol{\rho}^{2}}{24\pi} \sum\_{i,j} \frac{\boldsymbol{c}\_{i} \boldsymbol{c}\_{j} \boldsymbol{c}\_{i\bar{j}} \boldsymbol{A}\_{\bar{\eta}} \overline{\boldsymbol{V}\_{\bar{\eta}}}}{\boldsymbol{m}\_{\bar{\eta}} \sigma\_{\bar{\eta}}^{0}} \left(\boldsymbol{\lambda}\_{\bar{\eta}}^{2} \boldsymbol{e}^{\lambda\_{\bar{\eta}}} \left(\boldsymbol{G} \left(\frac{\boldsymbol{\lambda}\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}}\right) - \boldsymbol{n} \frac{\boldsymbol{\mathcal{O}}}{\bar{\boldsymbol{\mathcal{O}}}} \boldsymbol{G} \left(\frac{\boldsymbol{\lambda}\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}}\right)\right) - \nu\_{\bar{\eta}}^{2} \boldsymbol{e}^{\lambda\_{\bar{\eta}}} \left(\boldsymbol{G} \left(\frac{\boldsymbol{\alpha}\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}}\right) - \boldsymbol{n} \frac{\boldsymbol{\mathcal{O}}}{\bar{\boldsymbol{\mathcal{O}}}} \boldsymbol{G} \left(\frac{\boldsymbol{\alpha}\_{\bar{\eta}}}{\sigma\_{\bar{\eta}}^{0}}\right)\right) \right) \tag{45}$$

Summation over compressibility factors gives the total pressure of mixture

$$P = nkT\left(1 + Z^{HB} + Z^{t} + Z^{Q}\right) \tag{46}$$

Defining Gibbs free energy provides information at critical points of phase stability diagram. Concavity and convexity of Gibbs diagram indicates if mixture is in one phase or not,

$$G = F + \frac{N}{m}P\tag{47}$$

Furthermore, Gibbs excess free energy is an appropriate measure in the definition of phase stability. Negative values for this energy describe stable state. This is expressed as

$$\mathbf{G}\_{\rm xs} = \mathbf{G} - \sum\_{i} c\_{i} \mathbf{G}\_{i}^{0} - \text{NkT} \sum\_{i} c\_{i} \ln c\_{i} \tag{48}$$

That *Gi* 0 represents the Gibbs free energy of pure fluid of species i. Concentration-concentration structure factor is defined as

$$S\_{cc}(0) = NkT \left(\frac{\partial^2}{\partial c^2} G\right)^{-1}\_{T, P, N} \tag{49}$$

Compairing this equation with *Scc id* enable us to define degree of hetero-coordination. In a given composition if *Scc*(0)< <*Scc id* then unlike atoms tend to pair as nearest neighbors (heterocoordination) and when *Scc*(0)> >*Scc id* then like atoms are preferred as a neighbor.

#### **3.1. Potentials**

The ideal free energy with *N* particle for the atomic and molecular components of fluid mixture

*i i c c <sup>i</sup>*

å (43)

¶ <sup>=</sup> ¶ (44)

∂ ∂*n <sup>G</sup>*( *<sup>υ</sup>ij σij*

in the range of *σij*

<sup>0</sup> ))) <sup>−</sup>*δ<sup>Z</sup> <sup>t</sup>*

, *σij*

<sup>0</sup> . Expressions

1 2

æ ö <sup>=</sup> ç ÷ ++ -

Compressibility factor of ideal term is one and *Z HB* would be estimated with the following

*HB*

*n kT*

For the perturbation term due to long rage attraction of potential tail employing (44) we will

*DY* (*r*)4*πr* <sup>2</sup>

for first order perturbation and quantum correction term of compressibility factor are achiev‐

2 2 0 00 00 , 24

ll

*ij ij i j ij ij ij ij ij ij*

æ ö <sup>æ</sup> æö æö æö æö ö æ <sup>ö</sup> ¶ ¶ <sup>=</sup> ç ÷ <sup>ç</sup> ç÷ ç÷ ç÷ ç÷ -- - ÷ ç <sup>÷</sup> ç ÷ <sup>ç</sup> ç÷ ç÷ ç÷ ç÷ ¶ ¶ ÷ ç <sup>÷</sup> ç ÷

ss

Defining Gibbs free energy provides information at critical points of phase stability diagram.

Concavity and convexity of Gibbs diagram indicates if mixture is in one phase or not,

*Vij* ¯*dr*

*ij ij ij ij*

 u

è ø è èø èø èø èø ø è ø å (45)

 u (1 ) *HB t Q P nkT Z Z Z* = + ++ (46)

uu

ss

1 2 <sup>3</sup> ( , ) ln ln ln 1

*HB <sup>F</sup> Z n*

∂ ∂*n <sup>G</sup>*( *<sup>λ</sup>ij σij* <sup>0</sup> )) −*e υij* (*G*( *<sup>υ</sup>ij σij* <sup>0</sup> ) −*n*

*HS* (*σij* ))*∫σij σij* 0 *uij*

able via applying (44) for the free energy part of the quantum correction term.

*<sup>Z</sup> eG nG eG nG <sup>m</sup> n n*

Summation over compressibility factors gives the total pressure of mixture

*ij ij Q A i j ij ij ij*

l

l

2

*<sup>h</sup> F nT n cc*

è ø

2 2

p*kTm m*

*id*

derivation of related Helmholtz free energy

) + *n* ∂ <sup>∂</sup>*<sup>n</sup> gij*

Numerical integration has been used for calculation of *δZ <sup>t</sup>*

are given by,

280 Advances in Quantum Mechanics

have

*<sup>Z</sup> <sup>t</sup>* <sup>=</sup> <sup>2</sup>*π<sup>n</sup> kT* ∑ *i*, *j ci cj εij σij* 0 *Aij V*¯ *ij*(*e λij* (*G*( *<sup>λ</sup>ij σij* <sup>0</sup> ) −*n*

*<sup>δ</sup><sup>Z</sup> <sup>t</sup>* <sup>≈</sup> *<sup>n</sup>*

<sup>2</sup>*kT* ∑ *i*, *j ci cj* (*gij HS* (*σij*

2 2

p b

*hN n cc A V*

e

s

It is convenient to consider interacting potential with short-range sharply repulsive and longer-range attractive tail and treat them within a combined potential. The most practical method for the repulsive term of potential is the hard-sphere model with the benefit of preventing particles overlap. Furthermore, attractive or repulsive tails may be included using a perturbation theory. It is incontrovertible to generalize this potential to multi-component mixtures. This behavior is conveyed in double Yukawa (DY) potential which provides accurate thermodynamic properties of fluid in low temperatures and high density [18, 19]. At first we define DY potential as its effects on pressure of *He* −*H*2 mixture has been studied in this work, written as:

$$\ln\_{\vec{i}\vec{j}}^{D\text{Y}}(\boldsymbol{r}) = \varepsilon\_{i\vec{j}} A\_{i\vec{j}} \frac{\sigma\_{i\vec{j}}^{0}}{r} \left[ e^{\lambda\_{\vec{i}\vec{j}} \left(1 - \frac{r}{\mu} \boldsymbol{\sigma}\_{\vec{i}}^{0}\right)} - e^{\nu\_{\vec{i}} \left(1 - \boldsymbol{\eta} \left\{\boldsymbol{\sigma}\_{\vec{i}}^{0}\right\}\right)} \right] \tag{50}$$

*Aij* , *λij* , *υij* are controlling parameters. These parameters for *He* and *H*2 are listed in table 1 with their reference [20].


affect sensibly. In addition, this figure shows the predicted equimolar surface of the deuterium and tritium mixture for quantum correction term. This part is the most significant contribution at low temperature and varying smoothly in higher temperature. At very high densities, perturbation term contribution increases sharply with reducing density. Also, terms, *P <sup>t</sup>*

> α 13.10 12.7 11.1 ε / *kB* 10.80 15.50 36.40 σmin 0.29673 0.337 0.343

> > 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Reduced density

**Figure 1.** Different contributions of pressure as a function of reduced density for *T* =100*oK* for fluid mixture of deute‐

For helium-hydrogen mixtures different parts of pressure due to correction terms and ideal parts have been showed in figure 2 at *T* =100*<sup>o</sup> K*. Ideal pressure at reduced density of approx‐ imately zero, to about 0.25 rises drastically. However, afterward it soars gently up to 100M (pa). Pressure due to hard sphere is the most significant contribution except that it is less than perturbation part at value of 1.5 for reduced density. Effects of perturbation and Quantum correction are important in high densities. In low densities, these contributions are insignificant and may possibly be ignored. Non-additive part has been caused by dissimilarity of particles

*He* **−** *He He-H2 H2-H2*

*P HS* and *P id* , tend to infinity as *ρ* →*∞*.

**Table 2.** Potential parameters for *He, H2* interactions for exp-6 potential [20]

 PHS Pid Pt PQ T= 100 (K)

107

which surges steadily from the beginning.

rium and tritium

**4. Results**

108

Presure (pa)

109

, *P <sup>Q</sup>*,

283

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

**Table 1.** Potential parameters for *He*, *H*2 interactions for DY potential [20].

For the atomic and molecular fluids studies in this mixture, these particles interact via a exponential six (exp-6) or Double Yukawa (DY) potential energy function [20]. The fluids considered in this work are binary mixtures that their constituents are spherical particles of two species, *i* and *j*, interacting via pair potential *uij* (*r*).

$$u\_{i\bar{j}}^{\text{exp-}6}(r) = \begin{cases} \infty & r < \sigma\_{\infty} \\ \varepsilon\_{i,j} \frac{a\_{\bar{j}j}}{\alpha\_{\bar{j}\bar{j}} - 6} \left( 6 \frac{6}{\alpha\_{\bar{j}\bar{j}}} \exp(a\_{\bar{j}j} (1 - \frac{r}{\sigma\_{\text{min},\bar{j}}})) - (\frac{\sigma\_{\text{min},\bar{j}}}{r})^6 \right) & r > \sigma\_{\infty} \end{cases} \tag{51}$$

So we consider two-component fluid interacting via Buckingham potential *uij* (*r*) between molecules of types *i* and *j*. This potential is more realistic than square-well or Yukawa type potential for hydrogen isotope's mixture [21] at high temperatures. Because of same atomic structure of hydrogen and its isotopes, the three constant of potential are same for hydrogen isotopes. These constants obtained experimentally from molecular scattering [22].

*σ*min indicate the range of interaction and the parameter *α* regulates the stiffness of repulsion. For hydrogen and helium type atoms these parameters have been organized in Table 2. It is well known that the long range attractive part of exp-6 potential is similar to Lenard–Jones potential.

In view of the energy equation (32), one can readily obtain equation for total pressure and different contributions to pressure from standard derivation of respective Helmholtz free energy. By the exp-6 potential, we have computed the Helmholtz free energy. The ten-point Gausses quadrature has been used to calculate integrals in quantum correction and perturba‐ tion contribution. The calculated pressure for *D*<sup>2</sup> + *T*2 fluid mixture with equal mole fraction

and at temperature of *T* =100*<sup>o</sup> K* is showed logarithmically in figure 1. As it is clear from this figure, the effect of hard sphere term of pressure in given rang of temperature is significant and the range of pressure variation is wider than ideal part. As it is mentioned earlier the difference between isotopes is simply related to the neutron number in each nucleus and correction due to difference in mass which involves in non-additive correction term doesn't affect sensibly. In addition, this figure shows the predicted equimolar surface of the deuterium and tritium mixture for quantum correction term. This part is the most significant contribution at low temperature and varying smoothly in higher temperature. At very high densities, perturbation term contribution increases sharply with reducing density. Also, terms, *P <sup>t</sup>* , *P <sup>Q</sup>*, *P HS* and *P id* , tend to infinity as *ρ* →*∞*.


**Table 2.** Potential parameters for *He, H2* interactions for exp-6 potential [20]

**Figure 1.** Different contributions of pressure as a function of reduced density for *T* =100*oK* for fluid mixture of deute‐ rium and tritium

#### **4. Results**

*He* **−** *He He-H2 H2-H2*

(*r*).

 s *r*

s

s

(51)

(*r*) between

σ <sup>0</sup> 2.634 2.970 2.978 *A* 2.548 2.801 3.179 ε / *kB* 10.57 15.50 36.40 υ 3.336 3.386 3.211 λ 12.204 10.954 9.083

For the atomic and molecular fluids studies in this mixture, these particles interact via a exponential six (exp-6) or Double Yukawa (DY) potential energy function [20]. The fluids considered in this work are binary mixtures that their constituents are spherical particles of

> s

molecules of types *i* and *j*. This potential is more realistic than square-well or Yukawa type potential for hydrogen isotope's mixture [21] at high temperatures. Because of same atomic structure of hydrogen and its isotopes, the three constant of potential are same for hydrogen

*σ*min indicate the range of interaction and the parameter *α* regulates the stiffness of repulsion. For hydrogen and helium type atoms these parameters have been organized in Table 2. It is well known that the long range attractive part of exp-6 potential is similar to Lenard–Jones

In view of the energy equation (32), one can readily obtain equation for total pressure and different contributions to pressure from standard derivation of respective Helmholtz free energy. By the exp-6 potential, we have computed the Helmholtz free energy. The ten-point Gausses quadrature has been used to calculate integrals in quantum correction and perturba‐ tion contribution. The calculated pressure for *D*<sup>2</sup> + *T*2 fluid mixture with equal mole fraction and at temperature of *T* =100*<sup>o</sup> K* is showed logarithmically in figure 1. As it is clear from this figure, the effect of hard sphere term of pressure in given rang of temperature is significant and the range of pressure variation is wider than ideal part. As it is mentioned earlier the difference between isotopes is simply related to the neutron number in each nucleus and correction due to difference in mass which involves in non-additive correction term doesn't

<sup>ï</sup> ç ÷ -- > ¥ - ç <sup>÷</sup> îï è ø

ì¥ < ¥ ïï <sup>=</sup> æ ö <sup>í</sup>

exp 6 ( ) <sup>6</sup> min, <sup>6</sup> 6 exp( (1 )) ( ) , <sup>6</sup> min,

*u r ij <sup>r</sup> ij <sup>r</sup> i j ij <sup>r</sup> ij ij ij*

So we consider two-component fluid interacting via Buckingham potential *uij*

isotopes. These constants obtained experimentally from molecular scattering [22].

 a

**Table 1.** Potential parameters for *He*, *H*2 interactions for DY potential [20].

two species, *i* and *j*, interacting via pair potential *uij*

a

aa

e

*ij*

282 Advances in Quantum Mechanics

potential.


For helium-hydrogen mixtures different parts of pressure due to correction terms and ideal parts have been showed in figure 2 at *T* =100*<sup>o</sup> K*. Ideal pressure at reduced density of approx‐ imately zero, to about 0.25 rises drastically. However, afterward it soars gently up to 100M (pa). Pressure due to hard sphere is the most significant contribution except that it is less than perturbation part at value of 1.5 for reduced density. Effects of perturbation and Quantum correction are important in high densities. In low densities, these contributions are insignificant and may possibly be ignored. Non-additive part has been caused by dissimilarity of particles which surges steadily from the beginning.

*T* **(***K***)** *cHe* **ρ \* η** *PMC P***[19]** *P***[23]** *PDY* **[19]**

**Table 3.** Comparison of values of pressure results from our study [19], Monte-Carlo simulation [24] and Isam Ali's

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Helium concentration

**Figure 4.** Comparision of efect of DY and EXP-6 potential on pressure of mixture in che=0.5, T=300 vs. Reduced density

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for helium-hydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6's more steady

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represents that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described *P <sup>Q</sup>* / *P* in figure 7 for helium-hydrogen and in figure 8 for deuterium tritium mixture. For the third picture increase in pressure is similar to what have been elaborated for figures 5 and 6. For figure 8 this manner remains analogous to helium-hydrogen mixture and temperatures next to 100 (K). However, for temperature lower than this it would behave inversely. For this range any increase in

 DY Exp-6 T=1000 (K) cHe=0.5

0.0

behavior makes it adjustable with previous studies and MC simulation.

20.0G

40.0G

Pressure (pa)

tritium concentration bears decrease in pressure.

60.0G

80.0G

study [23].

 0.25 1.101 0.433 2.3090 2.7039 1.9664 2.8678 0.5 1.101 0.400 1.8560 1.7001 1.5729 1.8402 0.75 1.101 0.367 1.4240 1.2816 1.3160 1.3887 0.5 1.223 0.335 4.5100 4.4205 4.1094 4.9406 0.75 1.223 0.307 3.7150 3.5190 3.5904 3.9328 0.5 1.376 0.247 12.4300 12.0832 12.1014 14.154 0.5 1.572 0.282 16.3300 16.4485 16.4720 19.859

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

285

**Figure 2.** Different contribution of correction terms on pressure of helium-hydrogen mixture at T=100, che=0.5 vs. re‐ duce density

Gibbs excess free energy which is a measure for indicating phase stability of matters has been depicted in figure 3. Stability is limited to the areas that Gibbs excess free energy tends to negative values. This figure explains that stability rages for helium-hydrogen mixture at room temperature is confide in the boundaries in which helium concentration is less than 0.1.

**Figure 3.** Gibbs excess free energy for helium-hydrogen mixture

Table 3 presents a comparison between results of pressure from this work using DY potential in place of exp-6, Monte–Carlo simulations and additionally study of reference [23] Obviously, there are appreciable adaption among our investigation results and MC which proves validity of our calculations. As Table 3, exhibits in low temperatures DY potential have more consistent results in comparison with exp-6. However, values of pressure extracted using DY potential cannot adjust with simulation resembling exp-6. Moreover, at higher temperatures after *T* =1000*<sup>o</sup> K*, DY potential is not good choice for evaluating EOS of hydrogen and helium mixture. We clarify our deduction presenting comparison between effects of these two potentials over wide ranges of temperatures.


**Table 3.** Comparison of values of pressure results from our study [19], Monte-Carlo simulation [24] and Isam Ali's study [23].

0.4 0.6 0.8 1.0 1.2 1.4 1.6

 P= 0.5G (pa) P= 1G (pa) T=300 K, rs=1.3

Reduced density

**Figure 2.** Different contribution of correction terms on pressure of helium-hydrogen mixture at T=100, che=0.5 vs. re‐

Gibbs excess free energy which is a measure for indicating phase stability of matters has been depicted in figure 3. Stability is limited to the areas that Gibbs excess free energy tends to negative values. This figure explains that stability rages for helium-hydrogen mixture at room temperature is confide in the boundaries in which helium concentration is less than 0.1.

0.0 0.2 0.4 0.6 0.8

Table 3 presents a comparison between results of pressure from this work using DY potential in place of exp-6, Monte–Carlo simulations and additionally study of reference [23] Obviously, there are appreciable adaption among our investigation results and MC which proves validity of our calculations. As Table 3, exhibits in low temperatures DY potential have more consistent results in comparison with exp-6. However, values of pressure extracted using DY potential cannot adjust with simulation resembling exp-6. Moreover, at higher temperatures after *T* =1000*<sup>o</sup> K*, DY potential is not good choice for evaluating EOS of hydrogen and helium mixture. We clarify our deduction presenting comparison between effects of these two

Helium mole fraction

100M

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

**Figure 3.** Gibbs excess free energy for helium-hydrogen mixture

potentials over wide ranges of temperatures.

Gxs/NkT

1G

Pressure (pa)

duce density

284 Advances in Quantum Mechanics

10G

 Pid Pnonadd PHs PQ Pt T= 100 K

**Figure 4.** Comparision of efect of DY and EXP-6 potential on pressure of mixture in che=0.5, T=300 vs. Reduced density

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for helium-hydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6's more steady behavior makes it adjustable with previous studies and MC simulation.

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represents that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described *P <sup>Q</sup>* / *P* in figure 7 for helium-hydrogen and in figure 8 for deuterium tritium mixture. For the third picture increase in pressure is similar to what have been elaborated for figures 5 and 6. For figure 8 this manner remains analogous to helium-hydrogen mixture and temperatures next to 100 (K). However, for temperature lower than this it would behave inversely. For this range any increase in tritium concentration bears decrease in pressure.

range any increase in tritium concentration bears decrease in pressure.

range any increase in tritium concentration bears decrease in pressure.

adjustable with previous studies and MC simulation.

adjustable with previous studies and MC simulation.

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for heliumhydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6's more steady behavior makes it

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for heliumhydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6's more steady behavior makes it

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represent that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described *QP P* in figure 7 for helium-hydrogen and in figure 8 for deuterium tritium mixture. For the third picture increase in pressure is similar to what have been elaborated for figures 5 and 6. For number 8 this manner remains analogous for deuteriumtritium mixture and temperatures next to 100 (K). However, for temperature lower than this it would behave inversely. For this

0.00 0.05 0.10 0.15 0.20 0.25

Fract oi n of quan ut m ot ot t al pressure

<sup>50</sup> <sup>100</sup> 150 200 250 300 350


> Tritium mole fraction

**5. Conclusion** 

pressures below 100G pa.

Fract oi n of quan ut m et mr ot ot at l pressure

**5. Conclusion**

ourselves to pressures below 100G (pa).

2e-1 4e-1 6e-1 8e-1

**Figure 8.** Fraction of quantum perturbation term to total pressure for Deuterium-Tritium mixture.

An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pres‐ sure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with the code VASP,[25] which combines classical molecular dynamics simula‐ tion for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mix‐ tures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted

Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value, to decrease unnecessary efforts. Likewise, we can speculate

Temprature (K)

2e-1 4e-1 6e-1 8e-1

Helium mole fraction

Figure 7. Fraction of quantum perturbation term to total pressure for helium-hydrogen mixture.

<sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

Temprature (K)

Figure 8. Fraction of quantum perturbation term to total pressure for Deuterium-Tritium mixture.

An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pressure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with the code VASP,[25] which combines classical molecular dynamics simulation for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mixtures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted ourselves to

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

287

Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value to decrease unnecessary

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represent that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described *QP P* in figure 7 for helium-hydrogen and in figure 8 for deuterium tritium mixture. For the third picture increase in pressure is similar to what have been elaborated for figures 5 and 6. For number 8 this manner remains analogous for deuteriumtritium mixture and temperatures next to 100 (K). However, for temperature lower than this it would behave inversely. For this

Figure 5. Pressure of quantum correction term at *s* 1.3 for helium-hydrogen mixture. **Figure 5.** Pressure of quantum correction term at ρ*s* = 1.3 for helium-hydrogen mixture. Figure 5. Pressure of quantum correction term at *s* 1.3 for helium-hydrogen mixture.

 (K)

Figure 6. Total pressure from 50 K at for helium-hydrogen mixture. Figure 6. Total pressure from 50 K at for helium-hydrogen mixture. **Figure 6.** Total pressure from 50 K at ρ*S*=1.3 for helium-hydrogen mixture.

Figure 7. Fraction of quantum perturbation term to total pressure for helium-hydrogen mixture. **Figure 7.** Fraction of quantum perturbation term to total pressure for helium-hydrogen mixture.


> Tritium mole fraction

**5. Conclusion** 

pressures below 100G pa.

Fract oi n of quan ut m et mr ot ot at l pressure

2e-1 4e-1 6e-1 8e-1

<sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

Temprature (K)

Figure 8. Fraction of quantum perturbation term to total pressure for Deuterium-Tritium mixture.

An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pressure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with the code VASP,[25] which combines classical molecular dynamics simulation for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mixtures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted ourselves to

Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value to decrease unnecessary

An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pressure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with

2e-1 4e-1 6e-1 8e-1

Helium mole fraction

Figure 7. Fraction of quantum perturbation term to total pressure for helium-hydrogen mixture.

Temprature (K)

Figure 8. Fraction of quantum perturbation term to total pressure for Deuterium-Tritium mixture. **Figure 8.** Fraction of quantum perturbation term to total pressure for Deuterium-Tritium mixture.

**5. Conclusion** 

0.00 0.05 0.10 0.15 0.20 0.25

Fract oi n of quan ut m ot ot t al pressure

<sup>50</sup> <sup>100</sup> 150 200 250 300 350

#### **5. Conclusion**

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for heliumhydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6's more steady behavior makes it

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for heliumhydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6's more steady behavior makes it

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represent that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described *QP P* in figure 7 for helium-hydrogen and in figure 8 for deuterium tritium mixture. For the third picture increase in pressure is similar to what have been elaborated for figures 5 and 6. For number 8 this manner remains analogous for deuteriumtritium mixture and temperatures next to 100 (K). However, for temperature lower than this it would behave inversely. For this

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represent that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described *QP P* in figure 7 for helium-hydrogen and in figure 8 for deuterium tritium mixture. For the third picture increase in pressure is similar to what have been elaborated for figures 5 and 6. For number 8 this manner remains analogous for deuteriumtritium mixture and temperatures next to 100 (K). However, for temperature lower than this it would behave inversely. For this

adjustable with previous studies and MC simulation.

adjustable with previous studies and MC simulation.

0 1e+10 2e+10 3e+10 4e+10 5e+10 6e+10 7e+10

0 1e+10 2e+10 3e+10 4e+10 5e+10 6e+10 7e+10

**Figure 5.** Pressure of quantum correction term at ρ*s* = 1.3 for helium-hydrogen mixture.

0.0 5.0e+10 1.0e+11 1.5e+11 2.0e+11 2.5e+11 3.0e+11 3.5e+11

0.0 5.0e+10 1.0e+11 1.5e+11 2.0e+11 2.5e+11 3.0e+11 3.5e+11

Pressure (pa)

**Figure 6.** Total pressure from 50 K at ρ*S*=1.3 for helium-hydrogen mixture.

0.00 0.05 0.10 0.15 0.20 0.25

Fract oi n of quan ut m ot ot t al pressure

<sup>50</sup> <sup>100</sup> 150 200 250 300 350


> Tritium mole fraction

**5. Conclusion** 

pressures below 100G pa.

Fract oi n of quan ut m et mr ot ot at l pressure

2e-1 4e-1 6e-1 8e-1

Temprature (K)

**Figure 7.** Fraction of quantum perturbation term to total pressure for helium-hydrogen mixture.

Pressure (pa)

<sup>50100150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

Temprature (K)

<sup>50100150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

Temprature (K)

Figure 6. Total pressure from 50 K at for helium-hydrogen mixture.

Figure 6. Total pressure from 50 K at for helium-hydrogen mixture.

Quan ut m ef ef ct pressure (pa)

286 Advances in Quantum Mechanics

Quan ut m ef ef ct pressure (pa)

<sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

<sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

Figure 5. Pressure of quantum correction term at

Figure 5. Pressure of quantum correction term at

Temprature

Temprature

range any increase in tritium concentration bears decrease in pressure.

range any increase in tritium concentration bears decrease in pressure.

2e-1 4e-1 6e-1 8e-1

2e-1 4e-1 6e-1 8e-1

2e-1 4e-1 6e-1 8e-1

2e-1 4e-1 6e-1 8e-1

Helium mole fraction

Figure 7. Fraction of quantum perturbation term to total pressure for helium-hydrogen mixture.

<sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup>

Temprature (K)

Figure 8. Fraction of quantum perturbation term to total pressure for Deuterium-Tritium mixture.

An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pressure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with the code VASP,[25] which combines classical molecular dynamics simulation for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mixtures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted ourselves to

Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value to decrease unnecessary

2e-1 4e-1 6e-1 8e-1

Helium concentration

Helium concentration

*s* 1.3 for helium-hydrogen mixture.

*s* 1.3 for helium-hydrogen mixture.

Helium mole fraction

Helium mole fraction

the code VASP,[25] which combines classical molecular dynamics simulation for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mixtures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted ourselves to pressures below 100G pa. Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value to decrease unnecessary An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pres‐ sure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with the code VASP,[25] which combines classical molecular dynamics simula‐ tion for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mix‐ tures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted ourselves to pressures below 100G (pa).

Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value, to decrease unnecessary efforts. Likewise, we can speculate from Fig. 1 that in low temperature and high densities, long range perturbation term has the most significant effect in comparison with other parts. On the other hand, hard sphere part can be assumed as the most noticeable part in high temperature ranges. Moreover, comparison of DY and exp-6 potentials effects, on pressure of this mixture has been studied to express benefits of using exp-6 potential for higher temperatures and densities. Additionally, as it is obvious in high temperature and density difference between effects of two potentials are considerable for this equimolar mixture. This discriminating property makes exp-6 potential preferable.

**Author details**

**References**

S. M. Motevalli and M. Azimi

Review 40, 749 (1932).

45, 116 (1934).

54, 1523 (1971).

Department of Physics, Faculty of Science, University of Mazandaran, Babolsar, Iran

[1] Wigner E. P. On the Quantum Correction for Thermodynamic Equilibrium. Physical

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

289

[2] Kirkwood J. G. Quantum Statistics of Almost Classical Assemblies. Physical Review

[3] DeWitt H. E. Analytic Properties of the Quantum Corrections to the Second Virial

[4] Hill R. N. Quantum Corrections to the Second Virial Coefficient at High Tempera‐

[5] Jancovici B. Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High

[6] Pisani C. and McKellar B. H. J, Semiclassical propagators and Wigner-Kirkwood ex‐

[7] Mason E. A Siregar J. and Huang Y. Simplified calculation of quantum corrections to the virial coefficients of hard convex bodies. Molecular Physics 73, 1171 (1991).

[8] Mansoori G. A., Carnahan N. F., Starling K. E., and T. W. Leland. Equilibrium Ther‐ modynamic Properties of the Mixture of Hard Spheres. Journal of Chemical Physics

[9] Tang Y. and Lu B. C.Y. Improved expressions for the radial distribution function of

[10] Barker J. A. and Henderson D., Perturbation Theory and Equation of State for Fluids:

[11] Largo L. and Solana J. R., Equation of state for fluid mixtures of hard spheres and linear homo-nuclear fused hard spheres Physical Review E 58, 2251 (1998).

[12] Boublik T. Hard‐Sphere Equation of State. Journal of Chemical Physics 54, 471 (1970).

[13] Barrio C. and Solana J. Consistency conditions and equation of state for additive hard-sphere fluid mixtures. Journal of Chemical Physics 113, 10180 (2000).

The Square‐Well Potential. Journal of Chemical Physics 47, 2856 (1967).

pansions for hard-core potentials. Physical Review A 44, 1061 (1991).

hard spheres. Journal of Chemical Physics 103, 7463 (1995).

Coefficient. Journal of Mathematical Physics 3, 1003 (1962).

tures. Journal of Mathematical Physics 9, 1534 (1968).

Temperature. Physical Review 178, 295 (1969).

Furthermore, this approach has been used to evaluate EOS of *D*<sup>2</sup> + *T*2 mixture. Also, we have used this method to compare different contribution parts of pressure. These comparisons indicate that in low temperature quantum effects are more important, however in high temperatures, hard sphere part is the most effective. The last two three dimensional diagrams reveals the importance of quantum term in comparison with total pressure. However, for temperatures below 100 (k) for deuterium-tritium mixture negative pressure express that in low tritium concentration, deuterium rich fluid tend to consolidate.

### **6. Applications**

One of the topics which can count on a great deal of interest from both theoretical and experimental physics is research in fluid mixture properties. These interests, not only comprise in the wide abundance of mixtures in our everyday life and in our universe but also the surprising new phenomena which were detected in the laboratories responsible for this increased attention. Mixtures, in general, have a much richer phase diagram than their pure constituents and various effects can be observed only in multi-component systems.

These kinds of studies have allowed a more complete modeling of mixture and consequently a better prediction and a more accurate calculation of thermodynamic quantities of mixture, such as activity coefficient, partial molar volume, phase behavior, local composition in general and have promoted a deeper understanding of the microscopic structure of mixtures.

Furthermore, for astronomical applications it is known that most of giant gas planets are like Jupiter is consisting primarily of hydrogen and helium. Modeling the interior of such planets requires an accurate equation of state for hydrogen-helium mixtures at high pressure and temperature conditions similar to those in planetary interiors [29]. Thus, the characterization of such system by statistical perturbation calculations will help us to answer questions concerning the inner structure of planets, their origin and evolution [29, 30].

In addition, in perturbation consideration of plasma via chemical picture, perturbation corrections will be included by means of additional free energy correction terms. Therefore, in considering transition behavior of molecular fluid to fully ionized plasma these terms are suitable in studying the neutral interaction parts. Consequently this will help us in studying inertial confinement fusion [31] and considering plasma as a fluid mixture in tokomak [32].

### **Author details**

from Fig. 1 that in low temperature and high densities, long range perturbation term has the most significant effect in comparison with other parts. On the other hand, hard sphere part can be assumed as the most noticeable part in high temperature ranges. Moreover, comparison of DY and exp-6 potentials effects, on pressure of this mixture has been studied to express benefits of using exp-6 potential for higher temperatures and densities. Additionally, as it is obvious in high temperature and density difference between effects of two potentials are considerable for this equimolar mixture. This discriminating property makes exp-6 potential

Furthermore, this approach has been used to evaluate EOS of *D*<sup>2</sup> + *T*2 mixture. Also, we have used this method to compare different contribution parts of pressure. These comparisons indicate that in low temperature quantum effects are more important, however in high temperatures, hard sphere part is the most effective. The last two three dimensional diagrams reveals the importance of quantum term in comparison with total pressure. However, for temperatures below 100 (k) for deuterium-tritium mixture negative pressure express that in

One of the topics which can count on a great deal of interest from both theoretical and experimental physics is research in fluid mixture properties. These interests, not only comprise in the wide abundance of mixtures in our everyday life and in our universe but also the surprising new phenomena which were detected in the laboratories responsible for this increased attention. Mixtures, in general, have a much richer phase diagram than their pure

These kinds of studies have allowed a more complete modeling of mixture and consequently a better prediction and a more accurate calculation of thermodynamic quantities of mixture, such as activity coefficient, partial molar volume, phase behavior, local composition in general

Furthermore, for astronomical applications it is known that most of giant gas planets are like Jupiter is consisting primarily of hydrogen and helium. Modeling the interior of such planets requires an accurate equation of state for hydrogen-helium mixtures at high pressure and temperature conditions similar to those in planetary interiors [29]. Thus, the characterization of such system by statistical perturbation calculations will help us to answer questions

In addition, in perturbation consideration of plasma via chemical picture, perturbation corrections will be included by means of additional free energy correction terms. Therefore, in considering transition behavior of molecular fluid to fully ionized plasma these terms are suitable in studying the neutral interaction parts. Consequently this will help us in studying inertial confinement fusion [31] and considering plasma as a fluid mixture in tokomak [32].

constituents and various effects can be observed only in multi-component systems.

and have promoted a deeper understanding of the microscopic structure of mixtures.

concerning the inner structure of planets, their origin and evolution [29, 30].

low tritium concentration, deuterium rich fluid tend to consolidate.

preferable.

288 Advances in Quantum Mechanics

**6. Applications**

S. M. Motevalli and M. Azimi

Department of Physics, Faculty of Science, University of Mazandaran, Babolsar, Iran

### **References**


[31] Collins G. W., Da Silva L. B., Celliers P., Gold D. M., Foord M. E., Wallace R. J., Ng A., Weber S. V., Budil K. S., Cauble R. Measurements of the Equation of State of Deu‐

Quantum Perturbation Theory in Fluid Mixtures

http://dx.doi.org/10.5772/54056

291

[32] Hakel P. and Kilcrease D. P. A New Chemical-Picture-Based Model for Plasma Equa‐ tion-of-State Calculations, 14th APS Topical Conference on Atomic Processes in Plas‐

terium at the Fluid Insulator-Metal Transition. Science 281, 1178 (1998).

ma, (2004).


[31] Collins G. W., Da Silva L. B., Celliers P., Gold D. M., Foord M. E., Wallace R. J., Ng A., Weber S. V., Budil K. S., Cauble R. Measurements of the Equation of State of Deu‐ terium at the Fluid Insulator-Metal Transition. Science 281, 1178 (1998).

[14] Leonard P. J. Henderson D. and Barker J., Molecular Physic 21, 107 (1971).

[17] Tang Y., Jianzhong W. Journal of Chemical Physics 119, 7388 (2003).

[18] Garcia A. and Gonzalez D. J. Physical Chemistry Liquid 18, 91 (1988).

view A 43, 5418 (1991).

290 Advances in Quantum Mechanics

[20] Ree F. H. Mol. Phys. 96, 87 (1983).

Europhys. Lett. 13, 679 (1990).

cal Physics 78, 409 (1983).

Physical Review B 84, 235109 (2011).

Thermodynamics 2, 27 (2006).

view B 47, 558 (1993).

fluids. Physical Review E 53, 4820 (1996).

Journal of Modern Physic B 26, 1250103 (2012).

Journal of Chemical Physics. 124, 154505 (2006).

[15] Yuste S. B. and Santos A., Radial distribution function for hard spheres. Physical Re‐

[16] Yuste S. B. Lopez de Haro M. and Santos A., Structure of hard-sphere meta-stable

[19] Motevalli S. M., Pahlavani and M. R. Azimi M. Theoretical Investigations of Proper‐ ties of Hydrogen and Helium Mixture Based on Perturbation Theory. , International

[21] Paricaud P. A general perturbation approach for equation of state development.

[22] Kuijper A. D. et al., Fluid-Fluid Phase Separation in a Repulsive α-exp-6 Mixture .

[23] Ali I. et al., Thermodynamic properties of He-H2 fluid mixtures over a wide range of

[24] Ree F. H. Simple mixing rule for mixtures with exp‐6 interactions. Journal of Chemi‐

[25] Kresse G. and Hafner J. Ab initio molecular dynamics for liquid metals. Physical Re‐

[26] Lorenzen W. Halts B. and Redmer R. Metallization in hydrogen-helium mixtures

[27] Ross M. Ree F. H. and D. A. Young, The equation of state of molecular hydrogen at

[28] Chen Q. F. and Cai L. C. Equation of State of Helium-Hydrogen and Helium-Deuteri‐ um Fluid Mixture at High Pressures and Tempratures. International of Journal of

[29] Guillot T., D. J. Stevenson, W. B. Hubbard, and D. Saumon, in Jupiter, edited by Ba‐ genal F., Chapter three. University of Arizona Press, Tucson; 2003. p35–57.

[30] Saumon D. and Guillot T. Shock Compression of Deuterium and the Interiors of Jupi‐

temperatures and pressures. Physical Review E 69, 056104 (2004).

very high density. Journal of Chemical Physics 79, 1487 (1983).

ter and Saturn. The Astrophysical Journal 609. 1170 (2004).

[32] Hakel P. and Kilcrease D. P. A New Chemical-Picture-Based Model for Plasma Equa‐ tion-of-State Calculations, 14th APS Topical Conference on Atomic Processes in Plas‐ ma, (2004).

**Chapter 13**

**Quantal Cumulant Mechanics**

Yasuteru Shigeta

**1. Introduction**

and electronic structures.

http://dx.doi.org/10.5772/53703

**as Extended Ehrenfest Theorem**

Additional information is available at the end of the chapter

dinger equation approximately is extensively explored, yet.

Since Schrödinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrödinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrödinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schrö‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrödinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schrö‐

In contrast to the time-dependent wave mechanics, Heisenberg developed the equations of motion (EOM) derived for time-dependent operator rather than wave function [5]. This equation is now referred as the Heisenberg' EOM. This equation is exactly equivalent to the time-dependent Schrödinger equation so that the trials to solve the Heisenberg' EOM rather than Schrödinger one were also done for long time. For example, the Dyson equation, which is the basic equation in the Green's function theory, is also derived from the Heisenberg' EOM. Various approximate methods were deviced to solve the Dyson equation for nuclear

In this chapter, we propose a new approximate methodology to solve dynamical properties of given systems on the basis of quantum mechanics starting from the Heisenberg' EOM. First, theoretical background of the method is given for one-dimensional systems and an ex‐ tension to multi-dimensional cases is derived. Then, we show three applications in molecu‐

> © 2013 Shigeta; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Shigeta; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

### **Chapter 13**
