For a certain reaction, $\Delta G^0 = -45$ kJ/mol and $\Delta H^0 = -9

For a certain reaction, $\Delta G^0 = -45$ kJ/mol and $\Delta H^0 = -90$ kJ/mol at 0 °C. What is the minimum temperature at which the reaction will become spontaneous, assuming that $\Delta H^0$ and $\Delta S^0$ are independent of temperature?

273 K

298 K

546 K

596 K

This question was previously asked in

UPSC CDS-1 – 2019

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The spontaneity of a reaction is determined by the change in Gibbs Free Energy, $\Delta G$. A reaction is spontaneous if $\Delta G < 0$. The relationship between $\Delta G$, $\Delta H$, and $\Delta S$ is given by $\Delta G = \Delta H - T\Delta S$. We are given $\Delta G^0 = -45$ kJ/mol and $\Delta H^0 = -90$ kJ/mol at 0 °C (273 K). Assuming $\Delta H^0$ and $\Delta S^0$ are independent of temperature, we can calculate $\Delta S^0$ at 273 K: $\Delta G^0_{273} = \Delta H^0 - (273 \text{ K})\Delta S^0$ $-45 \text{ kJ/mol} = -90 \text{ kJ/mol} - 273\Delta S^0$ $273\Delta S^0 = -90 + 45 = -45 \text{ kJ/mol}$ $\Delta S^0 = \frac{-45}{273} \text{ kJ/(mol·K)}$ The reaction is spontaneous when $\Delta G < 0$: $\Delta H^0 - T\Delta S^0 < 0$ $-90 \text{ kJ/mol} - T \left(\frac{-45}{273} \text{ kJ/(mol·K)}\right) < 0$ $-90 + T \left(\frac{45}{273}\right) < 0$ $T \left(\frac{45}{273}\right) < 90$ $T < 90 \times \frac{273}{45}$ $T < 2 \times 273$ $T < 546 \text{ K}$ So, the reaction is spontaneous at temperatures below 546 K. The temperature at which $\Delta G$ becomes zero (equilibrium) is $T = 546$ K. The question asks for the "minimum temperature at which the reaction will become spontaneous". While the phrasing is awkward for a reaction that is spontaneous below a certain temperature, it likely refers to the boundary temperature (546 K) where spontaneity starts or ends depending on the temperature direction, or potentially the lowest temperature among options where it is spontaneous. However, the calculation directly yields 546 K as the key temperature threshold.

– Spontaneity requires $\Delta G < 0$. - $\Delta G = \Delta H - T\Delta S$. - Calculate $\Delta S$ from the given data at 273 K. - Find the temperature range where $\Delta G < 0$. - The transition temperature where $\Delta G = 0$ is $T_{eq} = \Delta H / \Delta S$.

For reactions with $\Delta H < 0$ and $\Delta S < 0$, the reaction is spontaneous at low temperatures ($T < T_{eq}$) and non-spontaneous at high temperatures ($T > T_{eq}$). The boundary temperature is $T_{eq}$.

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