If for a certain reaction $\Delta_r H$ is $30 ext{ kJ mol}^{-1}$ at $450 ext{ K}$, the value of $\Delta_r S$

Question

If for a certain reaction $\Delta_r H$ is $30	ext{ kJ mol}^{-1}$ at $450	ext{ K}$, the value of $\Delta_r S$ (in $	ext{J K}^{-1}	ext{ mol}^{-1}$) for which the same reaction will be spontaneous at the same temperature is:

Accepted Answer

For a reaction to be spontaneous, the change in Gibbs free energy ($\Delta G$) must be negative ($\Delta G < 0$). According to the Gibbs equation, $\Delta G = \Delta H - T\Delta S$. Therefore, for spontaneity: $\Delta H - T\Delta S < 0$, which implies $T\Delta S > \Delta H$ or $\Delta S > \frac{\Delta H}{T}$. Given: $\Delta H = 30 ext{ kJ mol}^{-1} = 30000 ext{ J mol}^{-1}$ $T = 450 ext{ K}$ Substituting the given values: $\Delta S > \frac{30000 ext{ J mol}^{-1}}{450 ext{ K}}$ $\Delta S > 66.67 ext{ J K}^{-1} ext{ mol}^{-1}$ Among the given options, $70$ is the only value strictly greater than $66.67$. Therefore, the value of $\Delta_r S$ must be $70 ext{ J K}^{-1} ext{ mol}^{-1}$ for the reaction to be spontaneous.

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