Chapter 4, Problem 4A.4E

(a)

Interpretation Introduction

Interpretation:

Change in molar Gibbs energy of carbon dioxide at ${20}^{o}C$ when its pressure is changed isothermally from 1.0 bar to 3.0 bar has to be determined.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by $\text{ΔG}$.

Molal Gibbs free energy can be determined using the following formulas as shown below,

$\Delta G=RT\ln \frac{V_1}{V_2}\boxed{\begin{array}{l}Where,\\ \Delta G:ChangeinGibbsfreeenergy\\ R:Gas\text{constant}\\ T:Temperature\\ V_1:Initialvolume\\ V_2:Finalvolume\end{array}}$

$\Delta G=\Delta H-T\Delta S\boxed{\begin{array}{l}where,\\ \Delta H:Enthalpychange\\ T:Temperature\\ \Delta S:Changeinentropy\end{array}}$

For an isothermal reaction enthalpy change is zero, thus this equation can be written as,

$\begin{array}{c}\Delta G=-T\Delta S\\ =-T\left(C_P\ln \frac{T_2}{T_1}-R\ln \frac{P_2}{P_1}\right)\\ =-T\left(0-R\ln \frac{P_2}{P_1}\right)\end{array}$

Explanation of Solution

Change in molar Gibbs energy of carbon dioxide at ${20}^{o}C$ when its pressure is changed isothermally from 1.0 bar to 3.0 bar can be determined as follows,

Herein,

$\begin{array}{c}\text{Temperature}\text{,}\text{T}\text{=}293\text{K}\\ \text{Initial}\text{pressure}\left(P_{\text{1}}\right)\text{=}\text{1}\text{bar}\\ \text{Final}\text{pressure}\left(P_2\right)\text{=}\text{3}\text{bar}\end{array}$

Substituting the known values in the above following equation,

$\begin{array}{c}\text{ΔG}\text{=}\text{-T}\left(\text{0-Rln}\frac{{\text{P}}_{\text{2}}}{{\text{P}}_{\text{1}}}\right)\\ \text{=}\text{-293}\text{K}\left(\text{0-8}\text{.314}{\text{Jmol}}^{\text{-1}}{\text{K}}^{\text{-1}}\text{×}\text{ln}\frac{\text{3}}{\text{1}}\right)\\ \text{=}2676.22Jmo{l}^{-1}\end{array}$

As per the above calculation, the change in molar Gibbs energy in the given sample is found to be $2676.22Jmo{l}^{-1}$

(b)

Interpretation Introduction

Interpretation:

Change in molar Gibbs energy of carbon dioxide at ${20}^{o}C$ when its pressure is changed isothermally from 1.0 bar to $2.7\times {10}^{-4}atm$ has to be determined.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by $\text{ΔG}$.

Molal Gibbs free energy can be determined using the following formulas as show below,

$\Delta G=RT\ln \frac{V_1}{V_2}\boxed{\begin{array}{l}Where,\\ \Delta G:ChangeinGibbsfreeenergy\\ R:Gascons\tan t\\ T:Temperature\\ V_1:Initialvolume\\ V_2:Finalvolume\end{array}}$

$\Delta G=\Delta H-T\Delta S\boxed{\begin{array}{l}where,\\ \Delta H:Enthalpychange\\ T:Temperature\\ \Delta S:Changeinentropy\end{array}}$

For an isothermal reaction enthalpy change is zero, thus this equation can be written as,

$\begin{array}{c}\Delta G=-T\Delta S\\ =-T\left(C_P\ln \frac{T_2}{T_1}-R\ln \frac{P_2}{P_1}\right)\\ =-T\left(0-R\ln \frac{P_2}{P_1}\right)\end{array}$

Explanation of Solution

Change in molar Gibbs energy of carbon dioxide at ${20}^{o}C$ when its pressure is changed isothermally from 1.0 bar to 3.0 bar can be determined as follows,

Herein,

$\begin{array}{c}\text{Temperature}\text{,}\text{T}\text{=}293\text{K}\\ \text{Initial}\text{pressure}\left(P_{\text{1}}\right)\text{=}\text{1}\text{bar}\\ \text{Final}\text{pressure}\left(P_2\right)\text{=}2.735\times {10}^{-4}bar\left(\because 2.7\times {10}^{-4}atm\times \frac{1.013bar}{1atm}\right)\end{array}$

Substituting the known values in the above following equation,

$\begin{array}{c}\text{ΔG}\text{=}\text{-T}\left(\text{0-Rln}\frac{{\text{P}}_{\text{2}}}{{\text{P}}_{\text{1}}}\right)\\ \text{=}\text{-293}\text{K}\left(\text{0-8}\text{.314}{\text{Jmol}}^{\text{-1}}{\text{K}}^{\text{-1}}\text{×}\text{ln}\frac{2.735\times {10}^{-4}}{\text{1}}\right)\\ \text{=}-19985.47Jmo{l}^{-1}\end{array}$

As per the above calculation, the change in molar Gibbs energy in the given sample is found to be $-19985.47Jmo{l}^{-1}$