Chapter 4, Problem 4.76E

Interpretation Introduction

Interpretation:

The value of ${\Delta }_{\text{mix}}G$ as the mixture goes from $x_{\text{B}}=0$ to $x_{\text{B}}=1.00$ at $\text{25}\text{.0}°\text{C}$ is to be calculated. The graph between ${\Delta }_{\text{mix}}G$ verses $x_{\text{B}}$ is to be plotted. The relative concentrations at which the value of ${\Delta }_{\text{mix}}G$ is most negative is to be predicted.

Concept introduction:

The Gibb’s free energy of mixing determines whether the process of mixing is spontaneous or not at a constant pressure and temperature. The expression to calculate the Gibb’s free energy of mixing is shown below.

${\Delta }_{\text{mix}}G=nRT\left(x_{\text{A}}\ln x_{\text{A}}+x_{\text{B}}\ln x_{\text{B}}\right)$

Answer to Problem 4.76E

The graph between ${\Delta }_{\text{mix}}G$ verses $x_{\text{B}}$ is shown below.

The value of ${\Delta }_{\text{mix}}G$ is most negative when the relative concentration of $\text{B}$ and $\text{A}$ in terms of mole fraction is $0.5$.

Explanation of Solution

The formula to calculate Gibb’s free energy of mixing is shown below.-

${\Delta }_{\text{mix}}G=nRT\left(x_{\text{A}}\ln x_{\text{A}}+x_{\text{B}}\ln x_{\text{B}}\right)$…(1)

Where,

• $n$ is the total number of moles.

• $R$ is the gas constant.

• $T$ is the temperature.

• $x_{\text{A}}$ and $x_{\text{B}}$ are the mole fractions of component $\text{A}\text{and}\text{B}$, respectively.

• ${\Delta }_{\text{mix}}{G}^{\text{ideal}}$ is the Gibbs energy of mixing for ideal mixture.

Conversion of temperature from Celsius to Kelvin is shown below.

$T\text{(K)}=T\text{(}°\text{C)}+273\text{K}$…(2)

Substitute the temperature $\left(\text{25}\text{.0}°\text{C}\right)$ in the equation (2).

$\begin{array}{l}T=\text{25}\text{.0}°\text{C}+273\\ T=298\text{K}\end{array}$

The temperature is $298\text{K}$.

The mole fraction of gas A and gas B is equal to one as shown below.

$x_{\text{A}}+x_{\text{B}}=1$

Substitute the value of $x_{\text{B}}=0.1$ in the above formula.

$\begin{array}{rll}{x}_{\text{A}}+0.1& =& 1\\ x_{\text{A}}& =& 0.9\end{array}$

The value of $x_{\text{A}}$ is $0.9$.

Substitute the values of $x_{\text{A}}$, $x_{\text{B}}$$R$, $T$, and $n$ in equation (1).

$\begin{array}{rll}{\Delta }_{\text{mix}}G& =& \left(1\text{mol}\right)\left(8.314\text{J}/\text{K×mol}\right)\left(298\text{K}\right)\left[0.9\ln \left(0.9\right)+0.1\ln \left(0.1\right)\right]\\ & =& 2477.572\left[-0.09482-0.23026\right]\\ & =& -805.409\text{J}\end{array}$

The value of ${\Delta }_{\text{mix}}G$ at $x_{\text{B}}=0.1$ is $-805.409\text{J}$.

Similarly, the value of ${\Delta }_{\text{mix}}G$ as the mixture goes from $x_{\text{B}}=0$ to $x_{\text{B}}=1.00$ is calculated as follows:

$\begin{array}{l}{x}_{\text{B}}{x}_{\text{A}}\ln x_{\text{A}}\ln x_{\text{B}}{x}_{\text{A}}\ln x_{\text{A}}{\Delta }_{\text{mix}}G\\ +x_{\text{B}}\ln x_{\text{B}}\end{array}$

$\begin{array}{rrrrrr}\hfill 0& \hfill 1& \hfill 0& \hfill -& \hfill 0& \hfill 0\\ \hfill 0.1& \hfill 0.9& \hfill -0.10536& \hfill -2.30259& \hfill -0.32508& \hfill -805.409\\ \hfill 0.2& \hfill 0.8& \hfill -0.22314& \hfill -1.60944& \hfill -0.5004& \hfill -1239.78\\ \hfill 0.3& \hfill 0.7& \hfill -0.35667& \hfill -1.20397& \hfill -0.61086& \hfill -1513.46\end{array}$

$\begin{array}{rrrrrr}\hfill 0.4& \hfill 0.6& \hfill -0.51083& \hfill -0.91629& \hfill -0.67301& \hfill -1667.43\\ \hfill 0.5& \hfill 0.5& \hfill -0.69315& \hfill -0.69315& \hfill -0.69315& \hfill -1717.32\\ \hfill 0.6& \hfill 0.4& \hfill -0.91629& \hfill -0.51083& \hfill -0.67301& \hfill -1667.43\\ \hfill 0.7& \hfill 0.3& \hfill -1.20397& \hfill -0.35667& \hfill -0.61086& \hfill -1513.46\end{array}$

$\begin{array}{rrrrrr}\hfill 0.8& \hfill 0.2& \hfill -1.60944& \hfill -0.22314& \hfill -0.5004& \hfill -1239.78\\ \hfill 0.9& \hfill 0.1& \hfill -2.30259& \hfill -0.10536& \hfill -0.32508& \hfill -805.409\\ \hfill 1& \hfill 0& \hfill -& \hfill 0& \hfill 0& \hfill 0\end{array}$

Therefore, the value of ${\Delta }_{\text{mix}}G$ is most negative when the relative concentration of $\text{B}$ and $\text{A}$ in terms of mole fraction is $0.5$.

The graph between ${\Delta }_{\text{mix}}G$ verses $x_{\text{B}}$ is shown below.

Figure 1

Conclusion

The graph between ${\Delta }_{\text{mix}}G$ verses $x_{\text{B}}$ is shown in Figure 1. The value of ${\Delta }_{\text{mix}}G$ is most negative when the relative concentration of $\text{B}$ and $\text{A}$ in terms of mole fraction is $0.5$.