Chapter 17, Problem 17.56E

Use the Sackur-Tetrode equation to derive the relationship $\Delta S=R\ln \left(V_2/V_1\right)$ for an isothermal change and $\Delta S=C_{\text{v}}\ln \left(T_2/T_1\right)$ for an isochoric change.

Interpretation Introduction

Interpretation:

The relationship $\Delta S=R\ln \left(V_2/V_1\right)$ for an isothermal change and $\Delta S=C_{\text{v}}\ln \left(T_2/T_1\right)$ for an isochoric change is to be derived using Sackur-Tetrode equation.

Concept introduction:

The Sackur-Tetrode equation is given below.

$S=Nk\left\{\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p}\right]+\frac{5}{2}\right\}$

Where,

• $T$ is the temperature.

• $m$ is the mass.

• $N$ is the number of particles in a system.

This is the best way to calculate the absolute entropies using the statistical thermodynamics applied to gaseous systems.

Answer to Problem 17.56E

The relationship $\Delta S=R\ln \left(V_2/V_1\right)$ for an isothermal change and $\Delta S=C_{\text{v}}\ln \left(T_2/T_1\right)$ for an isochoric change has been derived using Sackur-Tetrode equation.

Explanation of Solution

The Sackur-Tetrode equation is given below.

$S=Nk\left\{\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p}\right]+\frac{5}{2}\right\}$

Where,

• $T$ is the temperature.

• $m$ is the mass.

• $N$ is the number of particles in a system.

For an isothermal change $\Delta T=0$. The change in entropy can be represented as,

$\begin{array}{rll}\Delta S& =& S_2-S_1\\ & =& Nk\left\{\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p_2}\right]+\frac{5}{2}\right\}-Nk\left\{\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p_1}\right]+\frac{5}{2}\right\}\\ & =& Nk\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p_2}\right]-Nk\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p_1}\right]\end{array}$

The pressure can be substituted in terms of $P=\frac{NkT}{V}$ using ideal gas equation as shown below.

$\begin{array}{l}\Delta S=Nk\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT{V}_2}{NkT}\right]-Nk\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT{V}_1}{NkT}\right]\\ =Nk\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{V_2}{N}\right]-Nk\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{V_1}{N}\right]\end{array}$

The given equation is rearranged as shown below.

$\Delta S=Nk\left[\ln {\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{1}{N}+\ln V_2\right]-Nk\left[\ln {\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{1}{N}+\ln V_1\right]$

The common constant terms are cancelled as shown below.

$\begin{array}{l}\Delta S=Nk\ln {\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{1}{N}+Nk\ln V_2-Nk\ln {\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{1}{N}-Nk\ln V_1\\ \Delta S=Nk\ln V_2-Nk\ln V_1\end{array}$

The term $Nk\equiv R$, that is, the product of number of particles and Boltzmann constant is equivalent to gas constant. The identity of logarithm $\left(\ln \frac{A}{B}=\ln A-\ln B\right)$ is used to modify the given equation. The equation becomes,

$\begin{array}{l}\Delta S=R\ln V_2-R\ln V_1\\ \Delta S=R\ln \left(\frac{V_2}{V_1}\right)\end{array}$

Thus, the given equation represents the relationship between entropy and volume at isothermal conditions.

The Sackur-Tetrode equation is given below.

$S=Nk\left\{\ln \left[{\left(\text{}\frac{2\pi mkT}{h^2}\right)}^{3/2}\cdot \frac{kT}{p}\right]+\frac{5}{2}\right\}$

For an isochoric change $\Delta V=0$. The pressure can be substituted in terms of $P=\frac{NkT}{V}$ using ideal gas equation as shown below.

The change in entropy can be represented as,

$\begin{array}{rll}\Delta S& =& S_2-S_1\\ \Delta S& =& Nk\ln \left[{\left(\text{}\frac{2\pi mk{T}_2}{h^2}\right)}^{3/2}\cdot \frac{k{T}_{2}V}{Nk{T}_2}\right]-Nk\ln \left[{\left(\text{}\frac{2\pi mk{T}_1}{h^2}\right)}^{3/2}\cdot \frac{k{T}_{1}V}{Nk{T}_1}\right]\\ \Delta S& =& Nk\ln \left[{\left(\text{}\frac{2\pi mk{T}_2}{h^2}\right)}^{3/2}\cdot \frac{V}{N}\right]-Nk\ln \left[{\left(\text{}\frac{2\pi mk{T}_1}{h^2}\right)}^{3/2}\cdot \frac{V}{N}\right]\end{array}$

The given equation is rearranged as shown below.

$\Delta S=Nk\left[\ln \left[\left\{{\left(\text{}\frac{2\pi mk}{h^2}\right)}^{3/2}\cdot \frac{V}{N}\right\}\right]+\ln {\left(T_2\right)}^{3/2}\right]-Nk\left[\ln \left\{{\left(\text{}\frac{2\pi mk}{h^2}\right)}^{3/2}\cdot \frac{V}{N}\right\}+\ln {\left(T_1\right)}^{3/2}\right]$

The common constant terms are cancelled as shown below.

$\begin{array}{l}\Delta S=Nk\ln \left[{\left(\text{}\frac{2\pi mk}{h^2}\right)}^{3/2}\cdot \frac{V}{N}\right]+Nk\ln {\left(T_2\right)}^{3/2}-Nk\ln \left[{\left(\text{}\frac{2\pi mk}{h^2}\right)}^{3/2}\cdot \frac{V}{N}\right]-Nk\ln {\left(T_1\right)}^{3/2}\\ \Delta S=Nk\ln {\left(T_2\right)}^{3/2}-Nk\ln {\left(T_1\right)}^{3/2}\end{array}$

The term $Nk\equiv R$, that is, the product of number of particles and Boltzmann constant is equivalent to gas constant. The identity of logarithm $\left(\ln \frac{A}{B}=\ln A-\ln B\right)$ is used to modify the given equation. The equation becomes,

$\begin{array}{l}\Delta S=R\ln {\left(T_2\right)}^{3/2}-R\ln {\left(T_1\right)}^{3/2}\\ \Delta S=R\ln {\left(\frac{T_2}{T_1}\right)}^{3/2}\\ \Delta S=\frac{3}{2}R\ln \left(\frac{T_2}{T_1}\right)\end{array}$

For a monoatomic gas $C_{\text{v}}=\frac{3}{2}R$, is substituted in the given equation.

$\Delta S=C_{\text{v}}\ln \left(\frac{T_2}{T_1}\right)$

Thus, given equation represents the relationship between entropy and temperature at isochoric conditions.

Conclusion

The relationship $\Delta S=R\ln \left(V_2/V_1\right)$ for an isothermal change and $\Delta S=C_{\text{v}}\ln \left(T_2/T_1\right)$ for an isochoric change has been derived using Sackur-Tetrode equation.