Chapter 18, Problem 140RQ

Interpretation Introduction

Interpretation:

Whether the project to manufacture methanol and hydrogen gas from methane and water is a worthwhile investment or not is to be analyzed.

Concept Introduction:

Equilibrium constant, ${\text{K}}_C$ gives the relation between the amounts or concentrations of the reactant and product, and for a general reaction like $\text{aA}+\text{bB}\rightleftharpoons \text{cC}+\text{dD}$, the equilibrium constant is given as

${\text{K}}_C=\frac{{\left[\text{C}\right]}^{\text{c}}{\left[\text{D}\right]}^{\text{d}}}{{\left[\text{A}\right]}^{\text{a}}{\left[\text{B}\right]}^{\text{b}}}$

Here $\left[\text{A}\right]$, $\left[\text{B}\right]$, $\left[\text{C}\right]$, and $\left[\text{D}\right]$ represent the concentrations of the reactants and the products.

The standard Gibb’s free energy change for a reaction $\left(\Delta G^{°}\right)$ is the difference of the free energies of formation of products $\left(\Delta G^{°}{{}_f}_{\text{product}}\right)$ and reactants $\left(\Delta G^{°}{{}_f}_{\text{reactant}}\right)$, and it is given by the following expression: $\Delta G^{°}={\displaystyle \sum \Delta G^{°}{{}_f}_{\text{product}}}-{\displaystyle \sum \Delta G^{°}{{}_f}_{\text{reactant}}}$

For a spontaneous reaction, Gibb’s free energy change is negative, i.e., $\Delta G<0$

The standard Gibb’s free energy change and equilibrium constant, ${\text{K}}_{\text{C}}$ are related according to the following expression:

$\Delta G^{°}=-\text{R}T\ln {\text{K}}_{\text{C}}$

Where $T$ is the standard temperature, i.e. $25°\text{C}$.