**2. Modeling**

The Gibbs–Thomson coefficient describes for pure elements the melting temperature depression <sup>Δ</sup>*Tm*½ � *<sup>K</sup>* , based on the solid–liquid interface energy *<sup>γ</sup>sl <sup>N</sup>:m*�<sup>2</sup> ½ � and on the bulk melting entropy by unit volume Δ*S*<sup>∀</sup> *J:K*�<sup>1</sup> *:m*�<sup>3</sup> . Let us consider an isolated solid particle of radius *r* in the liquid phase; the Gibbs–Thomson equation for the structural melting point depression can be expressed by [12, 13]:

$$
\Gamma = \frac{\chi\_{sl}}{\Delta \text{S}\_{\text{V}}} \tag{1}
$$

*<sup>D</sup>*ð Þ¼ *<sup>ω</sup>* <sup>∀</sup>*ω*<sup>2</sup>

*On the Determination of Molar Heat Capacity of Transition Elements: From the Absolute…*

can be expressed as,

*DOI: http://dx.doi.org/10.5772/intechopen.96880*

requires,

tional energy,

to *cv*, provide,

*cRot*

**271**

where *ω* is the frequency, *ν* is the speed of sound in the solid. For a total number of atoms *N* in the volume ∀ and a correspondent density of atoms *n*, these variables

The first Brillouin zone is exchanged by an integral over a sphere of radius *kD*, containing precisely *N* wave vectors allowed. As a volume of space *k* by wave vector

> *<sup>N</sup>* <sup>¼</sup> <sup>4</sup>*πk*<sup>3</sup> *D*

As observed in Eq. (9), the element fundamental frequency is expressed as

*<sup>ω</sup><sup>D</sup>* <sup>¼</sup> *kB* � <sup>Θ</sup>*<sup>D</sup>*

where, Θ*<sup>D</sup>* is the Debye's temperature of the element, *kB* and are the constants

<sup>24</sup> *<sup>π</sup>*<sup>3</sup> *<sup>Z</sup>* <sup>Θ</sup><sup>3</sup>

In 2019, Ferreira et al. [15, 16] considered the following approach for the rota-

<sup>ℏ</sup><sup>2</sup> *J J*ð Þ <sup>þ</sup> <sup>1</sup>

where, *J* is the rotational level corresponding to integer *J* ¼ 0, 1, 2, 3, … , *r* and *M* are the atomic radius and the molar mass, respectively. The rotational contribution

2

where, *ω<sup>D</sup>* is the maximum admissible frequency known as Debye's frequency

*:K*�<sup>1</sup> . Debye's temperature for transition elements is found in the literature [4]. The additions of Eq. (17) of the electronic and of Eq. (19) of the rotational contributions

*J J*ð Þ þ 1

*<sup>M</sup>* � r2 *<sup>J</sup>:mol*�<sup>1</sup>

*D T*2 *Tbulk m*

The electronic contribution to *cve* is written in terms of the phonon energy *cVib*

<sup>¼</sup> <sup>5</sup>

*kB*Θ*<sup>D</sup>* ℏ *ν* <sup>3</sup>

ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup> ∀

*<sup>n</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>3</sup> *D* <sup>6</sup>*π*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 6*π*<sup>2</sup>

> *cve cVib v*

*ERot* <sup>¼</sup> <sup>5</sup> 4

> *<sup>R</sup>* � <sup>ℏ</sup><sup>3</sup> �

*<sup>B</sup>ωD*ð Þ *T* þ Θ*<sup>D</sup>*

Then, the density of atoms *n* can be obtained as,

of Boltzmann and Planck, respectively.

where, *Z* is the valence of the element, *Tbulk*

element ½ � *K* and *T* is the absolute temperature ½ � *K* .

*<sup>v</sup>* to the molar heat capacity can be derived as,

*k*2

*c Rot <sup>v</sup>* <sup>¼</sup> <sup>5</sup> 4

and, *R* is the universal gas constant *J:mol*�<sup>1</sup>

<sup>2</sup> *<sup>π</sup>*2*ν*<sup>3</sup> (6)

*N* ¼ *n*∀ (7)

<sup>3</sup> (8)

<sup>ℏ</sup> (10)

*<sup>m</sup>* is the melting temperature of the

*<sup>M</sup>* � r2 ½ �*<sup>J</sup>* (12)

*:K*�<sup>1</sup> (13)

(9)

*<sup>v</sup>* as,

(11)

According to Gurtin et al. [14], surface stress gives rise to detectable strains in the interior of the crystal whenever a solid surface is created. A relation of surface tension in terms of *η* and *ζ* parameters is given by:

$$
\Gamma = \eta \frac{\sigma\_{sl}}{\Delta \text{S}\_{\text{V}}} \zeta \tag{2}
$$

and,

$$
\eta \cong \frac{\sigma\_{sl}}{\chi\_{sl}} \tag{3}
$$

By substituting (3) into (2) and making *ζ* ¼ 1 ½ � *m* provides

$$
\Gamma \cong \frac{\sigma\_{sl}}{\Delta \mathbf{S}\_{\mathsf{V}}} \cong \frac{\sigma\_{sl} T\_{m}^{bulk}}{\Delta H\_{\mathsf{V}}} = \frac{2}{r} \frac{\Gamma}{}
$$

where *<sup>σ</sup>sl* is the solid–liquid interface tension *<sup>N</sup>:m*�<sup>1</sup> ½ �,*Tbulk <sup>m</sup>* is the bulk melting temperature ½ � *<sup>K</sup>* , <sup>Δ</sup>*H*<sup>∀</sup> is the latent heat of melting per unit volume *<sup>J</sup>:m*�<sup>3</sup> ½ � and *<sup>r</sup>* is the spherical grain radius½ � *m* , respectively.

For a stable nucleus, the critical radius can be expressed in terms of the temperature drop ΔTð Þ*r* through the correlation between the solid–liquid surface tension *σsl* and the bulk melting entropy by unit volume Δ*S*∀, which can be written in terms of the Gibbs–Thomson coefficient Γ.

$$
\Delta \mathbf{T}(r \ge r\_{\mathbf{C}}) = \frac{2}{r} \Gamma \tag{5}
$$

The density of state *D*ð Þ *ω* for a given grain of volume ∀ regarding the critical nucleation radius, is defined as

*On the Determination of Molar Heat Capacity of Transition Elements: From the Absolute… DOI: http://dx.doi.org/10.5772/intechopen.96880*

$$D(\omega) = \frac{\forall \omega^2}{2\,\,\pi^2 \nu^3} \tag{6}$$

where *ω* is the frequency, *ν* is the speed of sound in the solid. For a total number of atoms *N* in the volume ∀ and a correspondent density of atoms *n*, these variables can be expressed as,

$$N = n\forall \tag{7}$$

The first Brillouin zone is exchanged by an integral over a sphere of radius *kD*, containing precisely *N* wave vectors allowed. As a volume of space *k* by wave vector requires,

$$\frac{\left(2\pi\right)^{3}}{\mathsf{V}}N = \frac{4\pi k\_{D}^{3}}{3} \tag{8}$$

Then, the density of atoms *n* can be obtained as,

$$m = \frac{k\_D^3}{6\pi^2} = \frac{1}{6\pi^2} \left(\frac{k\_B \Theta\_D}{\hbar \ \nu}\right)^3 \tag{9}$$

As observed in Eq. (9), the element fundamental frequency is expressed as

$$
\omega\_D = \frac{k\_B \cdot \Theta\_D}{\hbar} \tag{10}
$$

where, Θ*<sup>D</sup>* is the Debye's temperature of the element, *kB* and are the constants of Boltzmann and Planck, respectively.

The electronic contribution to *cve* is written in terms of the phonon energy *cVib <sup>v</sup>* as,

$$\frac{c\_{ve}}{c\_v^{Vib}} = \frac{5}{24 \text{ } \pi^3} Z \frac{\Theta\_D^3}{T^2 T\_m^{bulk}} \tag{11}$$

where, *Z* is the valence of the element, *Tbulk <sup>m</sup>* is the melting temperature of the element ½ � *K* and *T* is the absolute temperature ½ � *K* .

In 2019, Ferreira et al. [15, 16] considered the following approach for the rotational energy,

$$E\_{\rm Rot} = \frac{5}{4} \hbar^2 \frac{J(J+1)}{\overline{M} \cdot \mathbf{r}^2} \text{ [J]} \tag{12}$$

where, *J* is the rotational level corresponding to integer *J* ¼ 0, 1, 2, 3, … , *r* and *M* are the atomic radius and the molar mass, respectively. The rotational contribution *cRot <sup>v</sup>* to the molar heat capacity can be derived as,

$$\mathcal{L}\_{v}^{Rot} = \frac{\mathbf{5}}{4} \frac{\mathbf{R} \cdot \hbar^{3} \cdot \mathbf{}}{k\_{B}^{2} a o\_{\mathrm{D}} (T + \Theta\_{\mathrm{D}})^{2}} \frac{J(J+1)}{\overline{\mathbf{M}} \cdot \mathbf{r}^{2}} \left[J \, mol^{-1} \, \mathrm{K}^{-1}\right] \tag{13}$$

where, *ω<sup>D</sup>* is the maximum admissible frequency known as Debye's frequency and, *R* is the universal gas constant *J:mol*�<sup>1</sup> *:K*�<sup>1</sup> .

Debye's temperature for transition elements is found in the literature [4]. The additions of Eq. (17) of the electronic and of Eq. (19) of the rotational contributions to *cv*, provide,

elements and compounds, regarding the critical radius expressed in terms of the temperature drop employing the correlation between the solid–liquid surface tension and the bulk melting entropy by unit volume, given in terms of the Gibbs– Thomson coefficient [15, 16]. Consequently, based on the nucleation of solid–liquid or solid–solid phases, the total number of atoms in the volume and a correspondent density of *n* atoms limited by nucleation conditions were proposed to calculate the density of state. Ferreira et al.'s model consists of the phonic, electronic, rotations contributions and predicts magnetic anomalies, such as phase transition temperatures. In this paper, model predictions of the molar heat capacity of transitional elements from absolute zero to the melting point are compared with the Thermo-Calc

The Gibbs–Thomson coefficient describes for pure elements the melting temperature depression <sup>Δ</sup>*Tm*½ � *<sup>K</sup>* , based on the solid–liquid interface energy *<sup>γ</sup>sl <sup>N</sup>:m*�<sup>2</sup> ½ �

isolated solid particle of radius *r* in the liquid phase; the Gibbs–Thomson equation

<sup>Γ</sup> <sup>¼</sup> *<sup>γ</sup>sl* Δ*S*<sup>∀</sup>

According to Gurtin et al. [14], surface stress gives rise to detectable strains in the interior of the crystal whenever a solid surface is created. A relation of surface

> <sup>Γ</sup> <sup>¼</sup> *<sup>η</sup> <sup>σ</sup>sl* Δ*S*<sup>∀</sup>

> > *<sup>η</sup>* ffi *<sup>σ</sup>sl γsl*

ffi *<sup>σ</sup>slTbulk m* Δ*H*<sup>∀</sup>

temperature ½ � *<sup>K</sup>* , <sup>Δ</sup>*H*<sup>∀</sup> is the latent heat of melting per unit volume *<sup>J</sup>:m*�<sup>3</sup> ½ � and *<sup>r</sup>* is

ΔTð Þ¼ *r*≥ *rC*

The density of state *D*ð Þ *ω* for a given grain of volume ∀ regarding the critical

For a stable nucleus, the critical radius can be expressed in terms of the temperature drop ΔTð Þ*r* through the correlation between the solid–liquid surface tension *σsl* and the bulk melting entropy by unit volume Δ*S*∀, which can be written in terms

> 2 Γ *r*

<sup>¼</sup> <sup>2</sup> <sup>Γ</sup> *r*

By substituting (3) into (2) and making *ζ* ¼ 1 ½ � *m* provides

<sup>Γ</sup> ffi *<sup>σ</sup>sl* Δ*S*<sup>∀</sup>

where *<sup>σ</sup>sl* is the solid–liquid interface tension *<sup>N</sup>:m*�<sup>1</sup> ½ �,*Tbulk*

the spherical grain radius½ � *m* , respectively.

of the Gibbs–Thomson coefficient Γ.

nucleation radius, is defined as

**270**

for the structural melting point depression can be expressed by [12, 13]:

*:m*�<sup>3</sup> . Let us consider an

*ζ* (2)

(1)

(3)

(4)

(5)

*<sup>m</sup>* is the bulk melting

Software simulations and experimental data.

*Recent Advances in Numerical Simulations*

and on the bulk melting entropy by unit volume Δ*S*<sup>∀</sup> *J:K*�<sup>1</sup>

tension in terms of *η* and *ζ* parameters is given by:

**2. Modeling**

and,

$$\begin{split} c\_{v} &= \left(\mathbf{1}.\mathbf{0} + D(\boldsymbol{\alpha})\right) \, \mathsf{9} \, N\_{\mathsf{u}} k\_{\mathsf{B}} \left(\frac{T}{\Theta\_{\mathsf{D}}}\right)^{3} \int\_{0}^{\frac{T}{\Theta\_{\mathsf{D}}}} \frac{\boldsymbol{\varkappa}^{4} e^{\boldsymbol{\varkappa}}}{\left(e^{\boldsymbol{\varkappa}} - \mathbf{1}\right)^{2}} d\boldsymbol{\varkappa} \, \left(\mathbf{1} + c\_{\mathsf{v}\boldsymbol{\epsilon}}\right) \\ &+ \left(n + \frac{1}{2}\right) \left[\boldsymbol{\mathsf{9}.\mathbf{0}} \, c\_{v}^{\mathrm{Rot}} + \left(\mathbf{1} - \sqrt{\frac{\mathbf{E} \cdot \rho\_{\mathrm{Dia}}}{E\_{\mathrm{Dia}} \cdot \rho}}\right) \frac{\boldsymbol{\mathsf{R}} T^{3}}{\Theta\_{\mathsf{D}} T\_{\mathrm{m}}^{2}}\right] \end{split} \tag{14}$$

to 2180 K. The proposed model agrees, for low and high temperatures, with the

*On the Determination of Molar Heat Capacity of Transition Elements: From the Absolute…*

**Figure 2** shows model calculations for Niobium compared with calculations performed with Thermo-Calc, and four experimental data sets [18–21]. In the literature, experimental values of molar heat capacity at high temperatures (for which measurements are complicated) generally overestimate the heat capacity. On the other side, at low temperatures, where measurements are difficult to control, the experimental values underestimate this property [22]. Furthermore, observations of the thermophysical properties of Nb applied in the model predictions, such as surface tension, Debye's temperature, atomic radius, the density of solid at the melting point, latent heat of fusion, among others, should be carefully compared with those from different authors, as values for the thermophysical properties found in the literature differ from author to author, and they could also be a likely source of the slight deviation observed in the predicted curve. The equivalent

*Comparison of the molar heat capacity of pure Vanadium by applying Debye,Thermo-Calc, and Ferreira et al.*

*Comparison of the molar heat capacity of pure titanium by applying Debye,Thermo-Calc, and Ferreira et al.*

experimental data of Touloukian et al. [17].

*DOI: http://dx.doi.org/10.5772/intechopen.96880*

**Figure 3.**

**Figure 4.**

**273**

*[15], and experimental data Desai [23].*

*[15, 16], and experimental data Desai [23] and Chase [24].*

### **3. Results and discussion**

**Figure 1** presents the model predictions of molar heat capacities for pure Chromium and experimental data from absolute zero to the melting point. Debye's model predictions are presented as a reference for Ferreira's model calculations [15, 16]. Thermo-Calc equilibrium calculations were performed in the range 176 K

#### **Figure 1.**

*Comparison of the molar heat capacity of pure chromium by applying Debye,Thermo-Calc, and Ferreira et al. [15, 16], and Touloukian et al. [17].*

#### **Figure 2.**

*Comparison of the molar heat capacity of pure niobium by applying Debye,Thermo-Calc, and Ferreira et al. [15, 16], and Kirillin et al. [18], Novikov et al. [19], Righini et al. [20] and Scheindlin et al. [21].*

*On the Determination of Molar Heat Capacity of Transition Elements: From the Absolute… DOI: http://dx.doi.org/10.5772/intechopen.96880*

to 2180 K. The proposed model agrees, for low and high temperatures, with the experimental data of Touloukian et al. [17].

**Figure 2** shows model calculations for Niobium compared with calculations performed with Thermo-Calc, and four experimental data sets [18–21]. In the literature, experimental values of molar heat capacity at high temperatures (for which measurements are complicated) generally overestimate the heat capacity. On the other side, at low temperatures, where measurements are difficult to control, the experimental values underestimate this property [22]. Furthermore, observations of the thermophysical properties of Nb applied in the model predictions, such as surface tension, Debye's temperature, atomic radius, the density of solid at the melting point, latent heat of fusion, among others, should be carefully compared with those from different authors, as values for the thermophysical properties found in the literature differ from author to author, and they could also be a likely source of the slight deviation observed in the predicted curve. The equivalent

#### **Figure 3.**

*cv* ¼ ð Þ 1*:*0 þ *D*ð Þ *ω* 9 *NakB*

1 2 � �

9*:*0 *c Rot <sup>v</sup>* þ 1 �

þ *n* þ

*Recent Advances in Numerical Simulations*

**3. Results and discussion**

**Figure 1.**

**Figure 2.**

**272**

*[15, 16], and Touloukian et al. [17].*

*T* Θ*<sup>D</sup>* � �<sup>3</sup> ð

**Figure 1** presents the model predictions of molar heat capacities for pure Chro-

*Comparison of the molar heat capacity of pure chromium by applying Debye,Thermo-Calc, and Ferreira et al.*

*Comparison of the molar heat capacity of pure niobium by applying Debye,Thermo-Calc, and Ferreira et al. [15, 16], and Kirillin et al. [18], Novikov et al. [19], Righini et al. [20] and Scheindlin et al. [21].*

mium and experimental data from absolute zero to the melting point. Debye's model predictions are presented as a reference for Ferreira's model calculations [15, 16]. Thermo-Calc equilibrium calculations were performed in the range 176 K

*T* Θ*D*

*x*4*e<sup>x</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi E � *ρDia EDia* � *ρ*

*<sup>e</sup>*ð Þ *<sup>x</sup>* � <sup>1</sup> <sup>2</sup> *dx* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>c</sup>*v*<sup>e</sup>*

*RT*<sup>3</sup> Θ*DT*<sup>2</sup> *m*

(14)

0

" #

! s

*Comparison of the molar heat capacity of pure Vanadium by applying Debye,Thermo-Calc, and Ferreira et al. [15], and experimental data Desai [23].*

#### **Figure 4.**

*Comparison of the molar heat capacity of pure titanium by applying Debye,Thermo-Calc, and Ferreira et al. [15, 16], and experimental data Desai [23] and Chase [24].*

wavevectors simulated are *n* ¼ 0, 1, and 2. The experimental data are close to the theoretical calculations for *n* ¼ 0.

**Figure 3** shows the experimental scatter for Vanadium from the absolute zero to the melting point, Thermo-Calc and Ferreira et al. model's calculations [15, 16]. The predictions for *n* ¼ 1 agrees, for the whole temperature range, with the experimental data, and Thermo-Calc.

**Figure 4** shows the molar specific heat for Titanium, the experimental data from Chase [24] and found in Desai [23]. Chase experimental data, in green, follow *n* ¼ 2 for the whole temperature range. In this case, Chase's experiment's thermodynamic conditions allow concluding that no phase transition at *T* ¼ 1156*K* takes place, which configures a non-fundamental state specific heat. On the other hand, after the transition temperature, Desai's [23] experimental data and Thermo-Calc agree with the theoretical model for *n* ¼ 0 from 1156 to 1941 K, configuring a fundamental state specific heat.

## **4. Conclusions**

The model previously proposed by Ferreira et al. [15, 16] based on the critical radius of phase nucleation to determine the total numbers of modes, and consequently, the Density of State successfully predicted the molar specific heat capacity of transitional elements. In Cr and V, the experimental data follow the theoretical prediction curves with *n* ¼ 2 and *n* ¼ 1, respectively. Furthermore, the model's calculation for Nb agrees with the experimental data except for the set found in Kirillin et al. [18]. The thermophysical properties of Niobium at high temperatures and experimental difficulties might be the reasons responsible for the slight deviation observed between the predictions and experimental data at high temperatures. For Titanium, non-fundamental states and fundamental state molar heat capacity were predicted experimentally and theoretically, as Chase's experiments follow the model's theoretical predictions for *n* ¼ 2.
