Question

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

Answer 1

 

Given

Reaction: 2A + B -> C

\[ \Delta H = 400\ \text{kJ mol}^{-1},\quad \Delta S = 0.2\ \text{kJ K}^{-1}\text{mol}^{-1} \]

Assume ΔH and ΔS are constant with temperature.

Condition for Spontaneity

Gibbs free energy change: \[ \Delta G = \Delta H - T\Delta S \]

For the reaction to become spontaneous: \[ \Delta G < 0 \]

Boundary temperature (just becomes spontaneous) when: \[ \Delta G = 0 \Rightarrow \Delta H = T\Delta S \Rightarrow T = \frac{\Delta H}{\Delta S} \]

Calculation

Use consistent units (kJ):

\[ T = \frac{400\ \text{kJ mol}^{-1}}{0.2\ \text{kJ K}^{-1}\text{mol}^{-1}} = \frac{400}{0.2}\ \text{K} = 2000\ \text{K} \]

The reaction becomes spontaneous at temperatures \[ T > 2000\ \text{K}, \] with the threshold temperature for spontaneity \[ \boxed{T = 2000\ \text{K}.} \]