Question

For a certain reaction \(\Delta H = -50\) kJ and \(\Delta S = -100\) J/K find the temperature so that the reaction is spontaneous.

• A. \(650\) K
• B. \(600\) K
• C. \(550\) K
• D. \(450\) K

Hint:

A reaction is spontaneous when the Gibbs energy change is negative, so write the Gibbs equation and solve for the temperature.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A reaction is spontaneous when its Gibbs free energy change is negative. Here both the enthalpy change and the entropy change are negative, so temperature decides the sign of \(\Delta G\).

Step 2: Key Formula or Approach:
\[ \Delta G = \Delta H - T\Delta S \] The condition for spontaneity is \(\Delta G < 0\). Keep the units of \(\Delta H\) and \(\Delta S\) consistent, so convert kJ to J.

Step 3: Detailed Explanation:
\(\Delta H = -50\text{ kJ} = -50000\text{ J}\) and \(\Delta S = -100\text{ J/K}\).
\[ \Delta G = -50000 - T(-100) = -50000 + 100\,T \]
For \(\Delta G < 0\) we need \(100\,T < 50000\), so \(T < 500\text{ K}\).
The reaction is spontaneous only below 500 K. At higher temperature the negative entropy term \(-T\Delta S\) becomes larger than the favourable \(\Delta H\).

Step 4: Check the options.
650 K, 600 K and 550 K are all above 500 K, so \(\Delta G\) is positive there and the reaction is not spontaneous. Only 450 K is below 500 K.

Final Answer:
The reaction is spontaneous only for \(T < 500\text{ K}\), so the required temperature is 450 K, option (D). \[ \boxed{450\text{ K}} \]