Animated Solution for Chemistry - Chemical Thermodynamics: The reaction, $\text{MgO}(s)+\text{C}(s)⟶\text{Mg}(s)+\text{CO}(g)$, for which ${\Delta }_r{H}^{\circ }=+491.1{\text{ kJ mol}}^{-1}$ and ${\Delta }_r{S}^{\circ }=198.0{\text{ J K}}^{-1}{\text{mol}}^{-1}$, is not feasible at 298 K. Temperature above which reaction will be feasible is

Entropy vs. Enthalpy

Have you ever wondered why some chemical reactions happen spontaneously while others refuse to budge unless you heat them up? The answer lies in the ultimate arbiter of chemical thermodynamics: Gibbs Free Energy ($\Delta G$).

In any chemical reaction, there is a constant tug-of-war between two fundamental forces. On one side, we have Enthalpy ($\Delta H$), which represents the heat energy of the system. Nature generally prefers to minimize energy, so exothermic reactions (negative $\Delta H$) are favored. On the other side, we have Entropy ($\Delta S$), which measures the randomness or chaos of the system. Nature loves chaos, so an increase in entropy (positive $\Delta S$) is also favored.

These two forces are tied together by the famous Gibbs-Helmholtz equation:

For a reaction to be "feasible" or spontaneous, the overall free energy change must be negative, meaning $\Delta G<0$.

In our specific problem, the reaction is highly endothermic, with $\Delta H^{\circ }=+491.1{\text{ kJ mol}}^{-1}$. This means the enthalpy factor is strongly opposing the reaction. However, the entropy change is positive, $\Delta S^{\circ }=198.0{\text{ J K}}^{-1}{\text{mol}}^{-1}$, because we are producing a gas ($\text{CO}$) from solid reactants.

Notice the variable $T$ (temperature) in the equation. Temperature acts like a volume knob for entropy. At low temperatures, the $T\Delta S$ term is too small to overcome the massive positive $\Delta H$. But as we crank up the heat, the entropy term grows larger and larger until it eventually dominates the equation.

Before we calculate the exact tipping point, we must navigate a classic trap that catches many students off guard: Unit Mismatch.

Enthalpy is given in kilo-joules ($\text{kJ}$), while entropy is given in Joules ($\text{J}$). You cannot subtract apples from oranges! We must convert $\Delta H$ into Joules by multiplying it by $10^3$.

We set up our inequality for spontaneity:

Rearranging this to solve for $T$, we get:

Now, we carefully substitute our unit-matched values:

Rounding to one decimal place as per the options, we find that the temperature must be strictly greater than $2480.3\text{ K}$.

At temperatures above this threshold, the chaos of the system (entropy) completely overpowers the energy barrier (enthalpy), driving the reaction forward. This beautiful phenomenon is known as an entropy-driven reaction.