**7. Appendix**

### **7.1 Mathematical processing of saturation vapor pressure**

The vapor pressures in dependence on temperature are treated by a relationship:

$$\begin{aligned} \text{I} &-RT\ln\left(p\right) = \Delta H \begin{Bmatrix} T \end{Bmatrix} - \alpha\_1 \cdot T + \alpha\_2 \cdot \begin{Bmatrix} T - \begin{Bmatrix} T \end{Bmatrix} - T \cdot \left(\ln T - \ln\left\{T\right\}\right) \end{Bmatrix} \cdot \tag{A1} \\\\ &+ \alpha\_3 \cdot \left[T^2 \cdot 2 - \begin{Bmatrix} T \end{Bmatrix}^2 / 2 - T \cdot \left\{T\right\} \cdot \left(\ln T - \ln\left\{T\right\}\right)\right] \end{aligned} \tag{A1}$$

where )ln(/( <sup>1</sup> *<sup>T</sup> pRT <sup>T</sup> H* .

Coefficients of equation (A1) are calculated by LSM using orthogonal functions (Kornilov & Vidavski, 1969).

Two functions *<sup>k</sup>* and *s* are orthogonal if the following condition is satisfied:

$$\left[\varphi\_{\not k}\,\varphi\_{\not s}\right] = 0 \quad \text{at} \quad k \neq s \tag{A2}$$

According to Gauss, the *<sup>i</sup>* sign was replaced by . If there are *m* functions <sup>0</sup>, <sup>1</sup>,... *<sup>m</sup>* they may be replaced by linear combination of the orthogonal functions *p* ,...<sup>1</sup> , <sup>0</sup> using Shmidt method:

$$\varphi\_0 = \Phi\_0; \varphi\_1 = \Phi\_1 - \varepsilon\_{10}\varphi\_0; \varphi\_2 = \Phi\_2 - \varepsilon\_{20}\varphi\_0 - \varepsilon\_{21}\varphi\_1; \dots \\ \varphi\_p = \Phi\_p - \sum\_{q=0}^{q=p-1} \varepsilon\_{pq}\varphi\_q \dots \text{ (A3)}$$

where coefficients of orthogonality, *pq* , are defined from equation (A3) and the orthogonal condition as:

$$
\varepsilon\_{pq} = \left[ \Phi\_p \, \varphi\_q \right] \left[ \begin{matrix} 2 \\ \end{matrix} \right] \tag{A4}
$$

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 635

The coefficients *pb* of equation (A8) were determined by LSM using a minimum of a sum of the deviations square of the experimental *i w i pRT i y* )}ln({ values from those ones

> )] <sup>33221100</sup> ([ *<sup>i</sup> <sup>i</sup> <sup>b</sup> <sup>i</sup> <sup>b</sup> <sup>i</sup> bb <sup>i</sup> <sup>y</sup>*

> > <sup>3</sup> [ <sup>3</sup> ] <sup>3</sup> [

] <sup>2</sup>

<sup>0</sup> )( <sup>2</sup>

uncertainties, but one spare digit being excepted. The errors of *pb* and *pq*

2

of the function calculated from approximated equation.

<sup>0</sup> /[] <sup>0</sup> [ <sup>0</sup> *<sup>i</sup> <sup>w</sup> <sup>i</sup> <sup>w</sup> <sup>i</sup> yb* 

*i wb i w i y*

, ] <sup>2</sup>

<sup>3</sup> /[] <sup>3</sup> [ <sup>3</sup> *<sup>i</sup> <sup>w</sup> <sup>i</sup> <sup>w</sup> <sup>i</sup> yb* 

] <sup>2</sup> /[][ *<sup>i</sup> <sup>w</sup> <sup>i</sup> <sup>p</sup> <sup>w</sup> <sup>i</sup> <sup>p</sup> <sup>y</sup> pb* 

*pbs* dispersions of the coefficients *pb* were defined in accordance with the formula

2

<sup>0</sup> is dispersion of individual measurement, *lnr* denotes a

)/( <sup>2</sup>

The constants *i a* of equation (A1) were calculated after the coefficients of *pb* and *pq*

*i wp sy pbs pbs*

number of a freedom degree, *n* is a number of measurements, *l* is the number of terms of approximation equation, *i y* are experimental values of the *xy* )( function, *i Y* are the values

*i wb i w i y*

*i wb i w i y*

*i wb i w i y*

<sup>2</sup> [ <sup>2</sup> ] <sup>2</sup> [

<sup>1</sup> [ <sup>1</sup> ] <sup>1</sup> [

<sup>0</sup> [ <sup>0</sup> ] <sup>0</sup> [

equation being contained only one unknown coefficient:

By differentiation of the expression (A10) upon *pb* and followed by transformation of the obtained equations using a ratio (A2), a system of normal equations were obtained, every

min

, ] <sup>2</sup>

<sup>2</sup> /[] <sup>2</sup> [ <sup>2</sup> *<sup>i</sup> <sup>w</sup> <sup>i</sup> <sup>w</sup> <sup>i</sup> yb* ,

(A12)

(A10)

(A11)

(A13)

have

coefficients were

2

0] <sup>2</sup>

<sup>1</sup> /[] <sup>1</sup> [ <sup>1</sup> *<sup>i</sup> <sup>w</sup> <sup>i</sup> <sup>w</sup> <sup>i</sup> yb* 

] <sup>2</sup>

] <sup>2</sup> /[ <sup>2</sup> 0

coefficients were rounded up to the values of their

Thus, the system of the normal equations is a diagonal matrix. The formulae intended for calculation of *pb* coefficients were obtained on the basis of the normal system of equations

0] <sup>2</sup>

0] <sup>2</sup>

0] <sup>2</sup>

calculated by (A8):

(A11):

The )( <sup>2</sup>

where *<sup>i</sup> <sup>i</sup> <sup>Y</sup> <sup>i</sup> yrs*

calculated by formulae:

)( <sup>12</sup>

been rounded off. The *pb* and *pq*

(A12), as:

A description of the technique of vapor pressures approximation by equation (A1) is given bellow.

The data on the saturation vapor pressure in dependence on the temperature have unequal accuracy; therefore treating those by the LSM were carried out using statistical weights *i w* :

$$\{-RT\ln(p\_{\dot{I}})\}\cdot\mathbf{w}\_{\dot{I}}=\Phi(T\_{\dot{I}})\,.\tag{A5}$$

The value of *i w* is introduced by inversely proportional to the dispersion:

$$\mathbf{w}\_{\dot{l}} = \mathbf{l} / \{ 2 \cdot \mathbf{s} [-RT\_{\dot{l}} \ln(p\_{\dot{l}})] \}^2 \tag{A6}$$

In conformity with the law of an accumulation of the random errors, the dispersions were calculated from the formula:

$$s^{\frac{\mathcal{D}}{2}} \{-RT\_{\dot{l}} \ln(p\_{\dot{l}})\} = \{-R \ln(p\_{\dot{l}})\}^{\mathcal{D}} s\_T^{\mathcal{D}} + \{-RT\_{\dot{l}} / \left. p\_{\dot{l}}\right\}^{\mathcal{D}} s\_P^{\mathcal{D}}$$

For simplification of the calculation, equation (A1) is reduced to the linear form:

$$Y = a\_0 + a\_1 x\_1 + a\_2 x\_2 + a\_3 x\_3 \, \text{} \tag{A7}$$

where *pRTY* )ln( , *<sup>T</sup> Ha* <sup>0</sup> , )}ln(/{ <sup>1</sup> *<sup>T</sup> pRT <sup>T</sup> Ha* , 22*<sup>a</sup>* , <sup>33</sup>*<sup>a</sup>* , *Tx*<sup>1</sup> ,

$$\mathbf{x\_2} = \langle T - \left\langle T \right\rangle - T \left[ \ln(T) - \ln(\left\langle T \right\rangle) \right] \rangle, \qquad \mathbf{x\_3} = \langle T \overset{2}{\prime} \wedge 2 - \left\langle T \right\rangle^2 \wedge 2 - T \left\langle T \right\rangle \ln(T) - \ln(\left\langle T \right\rangle) \rangle \cdot \mathbf{x\_1}$$

Owing to transition to the orthogonal functions, equation (A7) takes a form:

$$Y = b\_0 \wp\_0 + b\_1 \wp\_1 + b\_2 \wp\_2 + b\_3 \wp\_3 \tag{A8}$$

where <sup>1</sup> <sup>0</sup> , <sup>01011</sup> *<sup>x</sup>* , <sup>12102022</sup> *x* , <sup>23213103033</sup> *<sup>x</sup>* .

The coefficients of orthogonality, *pq* (A4), expressed with the use of statistical weights, are:

$$\varepsilon\_{10} = \left[\mathbf{x}\_1 \boldsymbol{\rho}\_0 \mathbf{w}\right] / \left[\boldsymbol{\varrho}\_0^2 \mathbf{w}\right], \quad \varepsilon\_{20} = \left[\mathbf{x}\_2 \boldsymbol{\rho}\_0 \mathbf{w}\right] / \left[\boldsymbol{\varrho}\_0^2 \mathbf{w}\right], \quad \varepsilon\_{30} = \left[\mathbf{x}\_3 \boldsymbol{\rho}\_0 \mathbf{w}\right] / \left[\boldsymbol{\varrho}\_0^2 \mathbf{w}\right],$$

$$\varepsilon\_{21} = \left[\mathbf{x}\_2 \boldsymbol{\rho}\_1 \mathbf{w}\right] / \left[\boldsymbol{\varrho}\_1^2 \mathbf{w}\right],$$

$$\varepsilon\_{31} = \left[\mathbf{x}\_3 \boldsymbol{\rho}\_1 \mathbf{w}\right] / \left[\boldsymbol{\varrho}\_1^2 \mathbf{w}\right], \quad \varepsilon\_{32} = \left[\mathbf{x}\_3 \boldsymbol{\rho}\_2 \mathbf{w}\right] / \left[\boldsymbol{\varrho}\_2^2 \mathbf{w}\right]. \tag{A9}$$

<sup>2</sup> / *qqppq*

A description of the technique of vapor pressures approximation by equation (A1) is given

The data on the saturation vapor pressure in dependence on the temperature have unequal accuracy; therefore treating those by the LSM were carried out using statistical weights *i w* :

In conformity with the law of an accumulation of the random errors, the dispersions were

22}/{ <sup>22</sup> )}ln({)}ln({2

*s i pR i p i RTs*

{ )]}ln()[ln( <sup>2</sup> *TTTTTx* , 2/ )]}ln()[ln(

<sup>33221100</sup> *bbbbY* 

*x*

<sup>1</sup> /[] <sup>12</sup> [ <sup>21</sup> *wwx* ,

*wwx* , ] <sup>2</sup>

] <sup>2</sup>

<sup>0</sup> /[] <sup>02</sup> [ <sup>20</sup> 

*Ha* <sup>0</sup> , )}ln(/{ <sup>1</sup> *<sup>T</sup> pRT <sup>T</sup>*

,

] <sup>2</sup>

<sup>2</sup> /[] <sup>23</sup> [ <sup>32</sup> 

For simplification of the calculation, equation (A1) is reduced to the linear form:

Owing to transition to the orthogonal functions, equation (A7) takes a form:

 , <sup>12102022</sup> 

*wwx* , ] <sup>2</sup>

] 2

<sup>1</sup> /[] <sup>13</sup> [ <sup>31</sup> 

)()}ln({ *<sup>i</sup> <sup>T</sup> <sup>i</sup> <sup>w</sup> <sup>i</sup> pRT* . (A5)

<sup>2</sup> )]}ln([2/{1 *<sup>i</sup> <sup>p</sup> <sup>i</sup> RTs <sup>i</sup> <sup>w</sup>* (A6)

*ps i p i RT T*

<sup>3322110</sup> *xaxaxaaY* , (A7)

2 2/ <sup>2</sup> { <sup>3</sup> *TTTTTTx* .

, (A8)

(A4), expressed with the use of statistical weights, are:

*wwx* , ] <sup>2</sup>

<sup>0</sup> /[] <sup>03</sup> [ <sup>30</sup> *wwx* ,

*wwx* . (A9)

*Ha* , 22*<sup>a</sup>*

(A4)

,

The value of *i w* is introduced by inversely proportional to the dispersion:

bellow.

<sup>33</sup>*<sup>a</sup>* 

where <sup>1</sup> <sup>0</sup>

calculated from the formula:

where *pRTY* )ln( , *<sup>T</sup>*

 , <sup>01011</sup> *<sup>x</sup>* 

<sup>23213103033</sup> *<sup>x</sup>* .

The coefficients of orthogonality, *pq*

<sup>0</sup> /[] <sup>01</sup> [ <sup>10</sup> 

, *Tx*<sup>1</sup> ,

The coefficients *pb* of equation (A8) were determined by LSM using a minimum of a sum of the deviations square of the experimental *i w i pRT i y* )}ln({ values from those ones calculated by (A8):

$$\sum\_{i} \left[ \mathbf{y}\_{i} - (b\_{0}\boldsymbol{\rho}\_{0} + b\_{1}\boldsymbol{\rho}\_{1i} + b\_{2}\boldsymbol{\rho}\_{2i} + b\_{3}\boldsymbol{\rho}\_{3i}) \right]^{2} = \min \tag{A10}$$

By differentiation of the expression (A10) upon *pb* and followed by transformation of the obtained equations using a ratio (A2), a system of normal equations were obtained, every equation being contained only one unknown coefficient:

$$\begin{aligned} \lfloor \boldsymbol{\nu}\_{\dot{I}} \boldsymbol{\rho}\_{\mathbf{0}} \boldsymbol{w}\_{\dot{I}} \rfloor - b\_{\mathbf{0}} \lceil \boldsymbol{\rho}\_{\mathbf{0}}^{2} \boldsymbol{w}\_{\dot{I}} \rceil &= 0 \\ \lfloor \boldsymbol{\nu}\_{\dot{I}} \boldsymbol{\rho}\_{\mathbf{0}} \boldsymbol{w}\_{\dot{I}} \rfloor - b\_{\mathbf{1}} \lceil \boldsymbol{\rho}\_{\mathbf{1}}^{2} \boldsymbol{w}\_{\dot{I}} \rceil &= 0 \\ \lfloor \boldsymbol{\nu}\_{\dot{I}} \boldsymbol{\rho}\_{\mathbf{2}} \boldsymbol{w}\_{\dot{I}} \rfloor - b\_{\mathbf{2}} \lceil \boldsymbol{\rho}\_{\mathbf{2}}^{2} \boldsymbol{w}\_{\dot{I}} \rceil &= 0 \\ \lfloor \boldsymbol{\nu}\_{\dot{I}} \boldsymbol{\rho}\_{\mathbf{3}} \boldsymbol{w}\_{\dot{I}} \rfloor - b\_{\mathbf{3}} \lceil \boldsymbol{\rho}\_{\mathbf{3}}^{2} \boldsymbol{w}\_{\dot{I}} \rfloor &= 0 \end{aligned} \tag{A11}$$

Thus, the system of the normal equations is a diagonal matrix. The formulae intended for calculation of *pb* coefficients were obtained on the basis of the normal system of equations (A11):

$$b\_0 = \lfloor \boldsymbol{\wp}\_{\bar{i}} \boldsymbol{\wp}\_0 \boldsymbol{\wp}\_{\bar{i}} \rfloor \lceil \log\_0^2 \boldsymbol{\wp}\_{\bar{i}} \rceil, \quad b\_1 = \lfloor \boldsymbol{\wp}\_{\bar{i}} \boldsymbol{\wp}\_1 \boldsymbol{\wp}\_{\bar{i}} \rceil \lceil \log\_1^2 \boldsymbol{\wp}\_{\bar{i}} \rceil, \ b\_2 = \lfloor \boldsymbol{\wp}\_{\bar{i}} \boldsymbol{\wp}\_2 \boldsymbol{\wp}\_{\bar{i}} \rfloor \lceil \log\_2^2 \boldsymbol{\wp}\_{\bar{i}} \rceil,$$

$$b\_3 = \lfloor \boldsymbol{\wp}\_{\bar{i}} \boldsymbol{\wp}\_3 \boldsymbol{\wp}\_{\bar{i}} \rfloor \lceil \log\_3^2 \boldsymbol{\wp}\_{\bar{i}} \rceil \tag{A12}$$

$$b\_P = \lfloor \boldsymbol{\wp}\_{\bar{i}} \boldsymbol{\wp}\_P \boldsymbol{\wp}\_{\bar{i}} \rfloor \lceil \log\_P^2 \boldsymbol{\wp}\_{\bar{i}} \rceil$$

The )( <sup>2</sup> *pbs* dispersions of the coefficients *pb* were defined in accordance with the formula (A12), as:

$$s^2(b\_p) = s\_0^2(\partial b\_p / \partial \boldsymbol{\wp})^2 = s\_0^2 \nmid \![\boldsymbol{\wp}\_p^2 \boldsymbol{w}\_{\boldsymbol{\jmath}}] \tag{A13}$$

where *<sup>i</sup> <sup>i</sup> <sup>Y</sup> <sup>i</sup> yrs* 2 )( <sup>12</sup> <sup>0</sup> is dispersion of individual measurement, *lnr* denotes a number of a freedom degree, *n* is a number of measurements, *l* is the number of terms of approximation equation, *i y* are experimental values of the *xy* )( function, *i Y* are the values of the function calculated from approximated equation.

The constants *i a* of equation (A1) were calculated after the coefficients of *pb* and *pq* have been rounded off. The *pb* and *pq* coefficients were rounded up to the values of their uncertainties, but one spare digit being excepted. The errors of *pb* and *pq* coefficients were calculated by formulae:

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 637

Mathematical processing the saturated vapor pressure in dependence on temperature by LSM with orthogonal functions makes it possible to calculate both the enthalpy of

*F H <sup>T</sup>* is designated based on the law of an accumulation of the random errors by the

An equation for computing the temperature dependence of the enthalpy of vaporization was obtained by means of formula (A1) and Clapeyron equation (2) (part 2.1) taking account

2/ <sup>2</sup>

][ 1 {2 0

*i*

<sup>0</sup> )( <sup>2</sup> 

*<sup>w</sup> <sup>s</sup> <sup>i</sup>*

*<sup>w</sup> xTT <sup>w</sup> xb <sup>w</sup> xb <sup>w</sup> TTTT <sup>y</sup> <sup>T</sup>*

0 , <sup>2</sup> , <sup>3</sup> 

*b* by (A14), respectively:

*TT <sup>w</sup> <sup>T</sup> <sup>x</sup> <sup>w</sup> xT <sup>w</sup> <sup>b</sup> <sup>x</sup>*

*T*

where 

, equals to product of the )( *<sup>T</sup>*

*T*

*vap*

The coefficients of equation (A20) are connected with correlations:

tabulated coefficients, the equation (A19) was transformed to the form:

*s H* ( ) *vap* . The latter for the thermodynamic values

22 2 () ( / ) <sup>0</sup> *sF s F yi* (A17)

constants of (A7) and (A15) and connections of

(A18)

,

(A19)

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

] <sup>2</sup> 3 [

*w TT*

<sup>32</sup> (

,}2/ <sup>2</sup>

(A20)

)}2/ <sup>2</sup>

3 32

] <sup>2</sup> 1 [

*i w w x*

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

*<sup>w</sup> <sup>T</sup> <sup>x</sup> <sup>w</sup> xT <sup>w</sup> <sup>x</sup>* <sup>3</sup> 2/ <sup>2</sup> ) <sup>2</sup> ( <sup>1</sup> <sup>32</sup> ) <sup>213231</sup> (

According to approximation accepted by deriving of the equation (A6), the enthalpy of

where *Z* is the difference of the compressibility factors, taking into account vapor deviation from ideality and volume changes of the vapor and liquid. For decreasing a number of

*ZDTCTBRH*

21 2 *Tx T*

)( <sup>3</sup> )( <sup>2</sup> { *TTTT <sup>Z</sup>*

2 1

) <sup>21</sup> <sup>1</sup> <sup>2</sup> ( <sup>1</sup> <sup>1</sup> <sup>2</sup> 2/) <sup>2</sup> <sup>2</sup> (

*Hs* , caused by errors of the *pT* -data, was carried out by

] <sup>2</sup> 2 [

*T*

*<sup>w</sup>* ,

*H* and *Z* values:

) <sup>2</sup> ( (A21)

*i w*

2 )(

**7.2 Calculation of the enthalpies of vaporization** 

*p*

<sup>2</sup> ( <sup>1</sup> <sup>32</sup> ) <sup>213231</sup> {(

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

<sup>2</sup> )}/(){( <sup>2</sup>

*HH*

*Hs*

*<sup>y</sup> <sup>T</sup>*

 vaporization and its dispersion <sup>2</sup>

on interconnections between the

*H T H*

Computing the dispersion, )( <sup>2</sup>

the formula, deduced based on (A17):

*T Hs*

vaporization, *H*

*vap*

*vap* 

these coefficients with

3

formula:

$$
\Delta b\_{P} = t\_{0.05} \left\{ s\_0^2 / \left[ \rho \sigma\_P^2 w \right] \right\}^{1/2} \text{ and } \quad \Delta s\_{pq} = \Delta x\_p / \left\{ \left[ \rho \sigma\_P^2 w \right] \right\}^{1/2} \text{ .}
$$

where *<sup>p</sup> <sup>x</sup>* is uncertainty of rounding of initial *px* value. Dispersion of <sup>2</sup> <sup>0</sup>*<sup>s</sup>* value and sums ] <sup>2</sup> [ *wp* , which were used by computing the *p b* , *pq* , and *i b* constants (A12) were rounded off up to three significant figures. On the basis of formula (A7) and uncertainty of the temperature measurements of ±0.006 K, the *<sup>i</sup> <sup>x</sup>* errors were evaluated equal to: <sup>1</sup> *<sup>x</sup>* <sup>=</sup> 0.006 K, <sup>2</sup> *<sup>x</sup>* =0.024 K and <sup>3</sup> *<sup>x</sup>* =0.048 K2.

The constants <sup>1</sup>*<sup>a</sup>* , 2*<sup>a</sup>* , and 3*<sup>a</sup>* are easily determined by means of coefficients of *pb* and *pq* as -functions are linear combination of the <sup>1</sup> *<sup>x</sup>* , 2*<sup>x</sup>* , and 3*<sup>x</sup>* variable values:

$$a\_3 = b\_3, \quad a\_2 = b\_2 - \varepsilon\_{32} b\_3, \quad a\_1 = b\_1 - \varepsilon\_{21} b\_2 - \varepsilon\_{31} b\_3 + \varepsilon\_{32} \varepsilon\_{21} b\_3 \tag{A14}$$

The constant 0*<sup>a</sup>* of equation (A7) are determined when replacing its terms by appropriate mean ones, namely *iw <sup>x</sup>* and *<sup>w</sup> <sup>y</sup>* :

$$a\_0 = \left\langle \mathbf{y}\_{\mathcal{W}} \right\rangle - a\_1 \left\langle \mathbf{x}\_{1\mathcal{W}} \right\rangle - a\_2 \left\langle \mathbf{x}\_{2\mathcal{W}} \right\rangle - a\_3 \left\langle \mathbf{x}\_{3\mathcal{W}} \right\rangle \tag{A15}$$

A calculation of the *p a* coefficients by the method described above is simpler than straight treating the *pT* -parameter by equation (A7) using LSM. For decreasing a number of tabulated coefficients, the final equation (A7) was transformed to the form:

$$
\ln(p) = A + B \left/ T + C \cdot \ln(T) + D \cdot T \tag{A16}
$$

Here *A*, *B*, *C*, and *D* are correlated constants, which were obtained by reducing the similar terms of equation (A7). A number of significant digits in the coefficients were chosen based on a condition: deviations of calculated vapor pressures from the experimental ones should not exceed the uncertainties of determination of the *p* values (13÷26 Pa). Then, the accuracy of each coefficient of equation (A16) was designated depending on the accuracy of *Tfp* )()ln( function using derivatives /)ln( *Kp* , where *K ABC* , , or *D* coefficients. If ln( ) / ln( ) / *p K p KX* , then *K pX* ln( ) / . This formula was used for evaluation of *K* values on the basis maximum values of *p* = 101.6 kPa and *T* = 530 K, determined in modified ebulliometer.

$$
\Delta A = \Delta \ln(p) = \frac{\Delta p}{p} = \frac{13 \cdot 10^{-3}}{101.6} = 0.00010$$

$$
\Delta B = T \cdot \Delta \ln(p) = \frac{T \cdot \Delta p}{p} = \frac{530 \cdot 13 \cdot 10^{-3}}{101.6} = 0.07$$

$$
\Delta C = \frac{\Delta \ln(p)}{\ln(T)} = \frac{\Delta p}{p \cdot \ln(T)} = \frac{13 \cdot 10^{-3}}{101.6 \cdot 6.3} = 0.00002$$

$$
\Delta D = \frac{\Delta \ln(p)}{T} = \frac{\Delta p}{p \cdot T} = \frac{13 \cdot 10^{-3}}{101.6 \cdot 530} = 0.0000002$$

### **7.2 Calculation of the enthalpies of vaporization**

636 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

rounded off up to three significant figures. On the basis of formula (A7) and uncertainty of the temperature measurements of ±0.006 K, the *<sup>i</sup> <sup>x</sup>* errors were evaluated equal to: <sup>1</sup> *<sup>x</sup>* <sup>=</sup>

The constants <sup>1</sup>*<sup>a</sup>* , 2*<sup>a</sup>* , and 3*<sup>a</sup>* are easily determined by means of coefficients of *pb* and

The constant 0*<sup>a</sup>* of equation (A7) are determined when replacing its terms by appropriate

A calculation of the *p a* coefficients by the method described above is simpler than straight treating the *pT* -parameter by equation (A7) using LSM. For decreasing a number of

Here *A*, *B*, *C*, and *D* are correlated constants, which were obtained by reducing the similar terms of equation (A7). A number of significant digits in the coefficients were chosen based on a condition: deviations of calculated vapor pressures from the experimental ones should not exceed the uncertainties of determination of the *p* values (13÷26 Pa). Then, the accuracy of each coefficient of equation (A16) was designated depending on the accuracy of *Tfp* )()ln( function using derivatives /)ln( *Kp* , where *K ABC* , , or *D* coefficients. If ln( ) / ln( ) / *p K p KX* , then *K pX* ln( ) / . This formula was used for evaluation of *K* values on the basis maximum values of *p* = 101.6 kPa and *T* = 530 K, determined in

> 6.101 <sup>3</sup> <sup>1013</sup>

*p*

*bba* , <sup>3213233122111</sup> *bbba*

and 2/1 ]} <sup>2</sup> /{[ *wppx pq*

*<sup>w</sup> xa <sup>w</sup> xa <sup>w</sup> xa wya*<sup>0</sup> <sup>332211</sup> (A15)

)ln(/)ln( *TDTCTBAp* (A16)

0000002.0

00002.0

07.0

5306.101

3.66.101 <sup>3</sup> <sup>1013</sup>

6.101 <sup>3</sup> <sup>1013530</sup>

00010.0

<sup>3</sup> )ln( <sup>1013</sup>

*Tp p*

*Tp p*

)ln()ln(

*p*

)ln(

*T <sup>p</sup> <sup>C</sup>*

*T <sup>p</sup> <sup>D</sup>*

)ln(

*<sup>p</sup> pA*

)ln(

*pT pTB*

,

*<sup>x</sup>* , 2*<sup>x</sup>* , and 3*<sup>x</sup>* variable values:

, and *i b* constants (A12) were

*b* (A14)

<sup>0</sup>*<sup>s</sup>* value and

2/1 ]} <sup>2</sup> /[ <sup>2</sup>

where *<sup>p</sup> <sup>x</sup>* is uncertainty of rounding of initial *px* value. Dispersion of <sup>2</sup>

, which were used by computing the *p b* , *pq*

tabulated coefficients, the final equation (A7) was transformed to the form:

<sup>0</sup> { 05.0 *wp st pb*


<sup>33</sup> *ba* , <sup>33222</sup>

0.006 K, <sup>2</sup> *<sup>x</sup>* =0.024 K and <sup>3</sup> *<sup>x</sup>* =0.048 K2.

mean ones, namely *iw <sup>x</sup>* and *<sup>w</sup> <sup>y</sup>* :

modified ebulliometer.

sums ] <sup>2</sup> [ *wp* 

*pq* as  Mathematical processing the saturated vapor pressure in dependence on temperature by LSM with orthogonal functions makes it possible to calculate both the enthalpy of vaporization and its dispersion <sup>2</sup> *s H* ( ) *vap* . The latter for the thermodynamic values *F H <sup>T</sup>* is designated based on the law of an accumulation of the random errors by the formula:

$$s^2(F) = s\_0^2 \Sigma(\partial F / \partial y\_j)^2 \tag{A17}$$

An equation for computing the temperature dependence of the enthalpy of vaporization was obtained by means of formula (A1) and Clapeyron equation (2) (part 2.1) taking account on interconnections between the 0 , <sup>2</sup> , <sup>3</sup> constants of (A7) and (A15) and connections of these coefficients with *p b* by (A14), respectively:

$$\begin{aligned} \Delta H^{\circ}\_{T} &= \Delta H^{\circ}\_{\circ} + \alpha\_{2} \left( T - \left\{ \mathbf{\bar{r}} \right\} \right) + \alpha\_{3} \left( T^{2} - \left\{ \mathbf{\bar{r}} \right\}^{2} \right) / 2 = \left\{ \mathbf{\underline{v}}\_{\circ \circ} \right\} - b\_{1} \left\{ \mathbf{x}\_{1 \text{w} } \right\} + b\_{2} \left\{ \mathbf{z}\_{2 1} \left\{ \mathbf{x}\_{1 \text{w} } \right\} + T - \left\{ T \right\} - \left\{ \mathbf{\underline{x}}\_{2 \text{w} } \right\} \right\} \\ &+ b\_{3} \left\{ \left( \mathbf{z}\_{3 1} - \mathbf{z}\_{3 2} \left\{ \mathbf{z}\_{2 1} \right\} \right) + \mathbf{z}\_{3 2} \left\{ \mathbf{\bar{r}} \right\} + \left\{ \mathbf{\underline{x}}\_{2 \text{w} } \right\} - \left\{ T \right\}^{2} / 2 - \left\{ \mathbf{x}\_{3 \text{w} } \right\} - \mathbf{z}\_{3 2} \left\{ T + T^{2} / 2 \right\} \end{aligned} \tag{A18}$$

Computing the dispersion, )( <sup>2</sup> *T Hs* , caused by errors of the *pT* -data, was carried out by the formula, deduced based on (A17):

$$s\_s^2 \left(\Delta H\_{j}^{\prime}\right) = s\_0^2 \sum\_{i} \left< (\partial \Delta H\_{j}^{\prime}) \left(\partial \boldsymbol{\uprho}\_{j}\right) \right>^2 = s\_0^2 \left< \frac{1}{\left[\boldsymbol{\uprho}\_{1}\right]^2} + \frac{\left<\boldsymbol{\uprho}\_{1w}\right>^2}{\left[\boldsymbol{\uprho}\_{1}^2 \boldsymbol{\uprho}\_{j}\right]} + \frac{(\boldsymbol{\uprho}\_{1w})^2}{\left[\boldsymbol{\uprho}\_{2}^2 \boldsymbol{\uprho}\_{j}\right]} + \frac{(\boldsymbol{\uprho}\_{2} \boldsymbol{\uprho}\_{1} + \boldsymbol{\uprho}^2 / 2)^2}{\left[\boldsymbol{\uprho}\_{3}^2 \boldsymbol{\uprho}\_{j}\right]}, \quad \text{(A19)}$$

$$\text{where } \boldsymbol{\uprho} = \boldsymbol{\uprho}\_{21}\left\{\boldsymbol{\upGamma}\right\} - \left<\boldsymbol{\uprho}\_{2w}\right\rangle - \left<\boldsymbol{\uprho}\right>,$$

$$\boldsymbol{\uplambda} = \left(\boldsymbol{\upsigma}\_{31} - \boldsymbol{\upsigma}\_{32}\boldsymbol{\upsigma}\_{21}\right) \left<\boldsymbol{\upmu}\_{1w}\right> + \boldsymbol{\upsigma}\_{32}\left(\left<\boldsymbol{\upmu}\right> + \left<\boldsymbol{\upmu}\_{2w}\right>\right) - \left<\boldsymbol{\upmu}\right>^2 / 2 - \left<\boldsymbol{\upmu}\_{3w}\right>$$

According to approximation accepted by deriving of the equation (A6), the enthalpy of vaporization, *H vap* , equals to product of the )( *<sup>T</sup> H* and *Z* values:

$$
\Delta\_{\text{vap}}H = \left\langle \Delta H'\_{\left\{T\right\}} + a\_2 \left(T - \left\{T\right\} \right) + a\_3 \left(T - \left\{T\right\} \right)^2 / 2 \right\rangle \cdot \Delta Z,\tag{A20}
$$

where *Z* is the difference of the compressibility factors, taking into account vapor deviation from ideality and volume changes of the vapor and liquid. For decreasing a number of tabulated coefficients, the equation (A19) was transformed to the form:

$$
\Delta \sup\_{\text{vap}} H = R(-B + CT + DT^2) \cdot \Delta Z \tag{A21}
$$

The coefficients of equation (A20) are connected with correlations:

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 639

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$$B = (\alpha \frac{\lambda}{2} \left\langle T \right\rangle + \alpha \frac{\lambda}{3} \left\langle T \right\rangle^2 / 2 - \Delta H'\_{\left\langle T \right\rangle} \Big) / / R, \quad C = \alpha \frac{\lambda}{2} / R, \quad \text{and} \qquad D = \alpha \frac{\lambda}{3} / 2R.$$

As an example, Table A1 lists the parameters of equation for calculating the uncertainties of the enthalpies of vaporization caused by errors of the vapor pressures for some freons.


Table A1. Parameters of equation for calculation of the *Hs m T* )}({ values by

$$s\left(\Delta H\_{\text{III}}^{\prime}(T)\right)\left(\left.\left(J\cdot\text{mol}\right)^{-1}\right) = \pm t\_{0.05}\cdot\left[\left.\left.\left(a+b\cdot\left\{\left(T\,\left/\,K+\varepsilon\right)^{2}\right.\\\left.+d\cdot\left\{\text{e}+\text{g}\cdot\left(T\,\left/\,K\right)-0.5\cdot\left(T\,\left/\,K\right)^{2}\right.\right\}^{2}\right\}^{1/2}\right)\right|\right.\right]^{2}\right] = \pm\left(\pm t\_{0.05}\cdot\left[\left.\left(a+b\cdot\left\{\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\right)\right)^{\right)}\right)\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right)}\right) \right.}\right)}\right) \right.}\right.}\right.}\right.}\right.}\right.}\right.$$

for freons CFCl2CFCl2, CF2ClCFCl2 and CF2BrCF2Br at *<sup>T</sup>* = 298.15 K, where 05.0*<sup>t</sup>* denotes the Student criterion.

This equation was obtained by summing up dispersions of the orthogonal coefficients of equation (A19). Errors of the *Hm T* )( *vap* values caused by uncertainties of *pT* data equal to those of the direct calorimetric methods (Table 2).
