For the reaction at 298 K, $2A + B \rightarrow C$, $\Delta H = 400$ kJ mol$^{-1}$ and $\Delta S = 0.2$ kJ K$^{-1}$ mol$^{-1}$. At what temperature will the reaction become spontaneous considering $\Delta H$ and $\Delta S$ to be constant over the temperature range?

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> [!TLDR] > Spontaneity is governed by $\Delta G = \Delta H - T\Delta S$. For this reaction, $\Delta G$ becomes negative when $T > 2000$ K, so the reaction is spontaneous above **2000 K**. The key to this problem is Gibbs Free Energy — the single most reliable criterion for spontaneity at constant pressure and temperature. Many students jump to thinking about enthalpy alone, but a reaction can be endothermic ($\Delta H > 0$) and still spontaneous if entropy drives it. Here, $\Delta H$ is large and positive, but $\Delta S$ is also positive. That means at high enough temperature, the $-T\Delta S$ term will outweigh the positive $\Delta H$, making $\Delta G$ negative. The condition for spontaneity is $\Delta G < 0$. Since $\Delta H$ and $\Delta S$ are given as constant, we can find the temperature where $\Delta G$ crosses from positive to negative. 1. **Write the Gibbs Free Energy equation** The fundamental relation is: $$\Delta G = \Delta H - T\Delta S$$ For the reaction to be spontaneous, we need $\Delta G < 0$, i.e.: $$\Delta H - T\Delta S < 0$$ 2. **Rearrange to find the threshold temperature** Bring the entropy term to the other side: $$\Delta H < T\Delta S$$ Since $\Delta S > 0$, we can divide without flipping the inequality: $$T > \frac{\Delta H}{\Delta S}$$ This is the temperature above which the reaction becomes spontaneous. At exactly $T = \frac{\Delta H}{\Delta S}$, $\Delta G = 0$ — the reaction is at equilibrium. 3. **Plug in the values** $\Delta H = 400$ kJ mol$^{-1}$ and $\Delta S = 0.2$ kJ K$^{-1}$ mol$^{-1}$. Notice the units: $\Delta S$ is given in kJ, so no conversion is needed. $$T > \frac{400}{0.2} = 2000 ext{ K}$$ So the reaction becomes spontaneous for any temperature above 2000 K. …

$\Delta G$ sign	What it means
Negative ( $\Delta G < 0$ )	Spontaneous — reaction happens on its own
Positive ( $\Delta G > 0$ )	Non-spontaneous — needs constant energy input
Zero ( $\Delta G = 0$ )	Equilibrium — no net change

Q.For the reaction at 298 K, $2A + B \rightarrow C$ , $\Delta H = 400$ kJ mol $^{-1}$ and $\Delta S = 0.2$ kJ K $^{-1}$ mol $^{-1}$ . At what temperature will the reaction become spontaneous considering $\Delta H$ and $\Delta S$ to be constant over the temperature range?

Concept understanding — Gibbs Free Energy

Gibbs Free Energy: The "Why" Behind Spontaneous Reactions

The Two Competing Forces

The Resolution: Gibbs Free Energy

Why Temperature Matters

The Precise Statement

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