Animated Solution for Chemistry - Chemical Thermodynamics: For the chemical reaction, $X\rightleftharpoons Y$, the standard reaction Gibbs energy depends on temperature $T$ (in K) as ${\Delta }_r{G}^{\circ }({\text{in kJ mol}}^{-1})=120-\frac{3}{8}T$ The major component of the reaction mixture at $T$ is

Imagine a chemical tug-of-war between two components, $X$ and $Y$. The reaction is given as $X\rightleftharpoons Y$. The ultimate judge of who wins this tug-of-war is the standard Gibbs free energy change, ${\Delta }_r{G}^{\circ }$.

The relationship between ${\Delta }_r{G}^{\circ }$ and the major component is straightforward but crucial: - If ${\Delta }_r{G}^{\circ }<0$, the forward reaction is spontaneous. The system naturally wants to convert $X$ into $Y$, making $Y$ the major component. - If ${\Delta }_r{G}^{\circ }>0$, the forward reaction is non-spontaneous (meaning the backward reaction is spontaneous). The system prefers to stay as $X$, making $X$ the major component.

We are provided with a beautiful linear equation that dictates how ${\Delta }_r{G}^{\circ }$ changes with temperature $T$:

Our mission is to test the temperature given in each option, calculate the corresponding ${\Delta }_r{G}^{\circ }$, and see if the predicted major component matches the option's claim.

Let's test Option (a): $T=280\text{ K}$ Substituting $T=280$ into our master equation:

Since $15>0$, the reaction is non-spontaneous in the forward direction, meaning $X$ is the major component. However, Option (a) claims $Y$ is major. So, Option (a) is incorrect.

Let's test Option (b): $T=350\text{ K}$ Substituting $T=350$:

Since $-11.25<0$, the reaction is spontaneous, making $Y$ the major component. Option (b) claims $X$ is major, which is incorrect.

Let's test Option (c): $T=315\text{ K}$ Substituting $T=315$:

Since $1.875>0$, the reaction is non-spontaneous, meaning $X$ is the major component. Option (c) correctly states that $X$ is major at this temperature! This is our winner.

Let's test Option (d): $T=300\text{ K}$ Just to be absolutely certain, let's check the last option. Substituting $T=300$:

Since $7.5>0$, $X$ should be the major component. Option (d) claims $Y$ is major, so it is incorrect.

As a bonus thought experiment, what happens when the tug-of-war is perfectly tied? This occurs when ${\Delta }_r{G}^{\circ }=0$.

At exactly $320\text{ K}$, the system is in standard equilibrium. For any temperature below $320\text{ K}$, ${\Delta }_r{G}^{\circ }$ is positive, and $X$ dominates. For any temperature above $320\text{ K}$, ${\Delta }_r{G}^{\circ }$ becomes negative, and $Y$ takes over. This elegant linear relationship perfectly maps out the thermodynamic landscape of the reaction!