Question

For a spontaneous process, if entropy and volume are constant, the internal energy of the system must .
A) increase
B) decrease
C) remain constant
D) be zero.

Answer 1

Hint:A process is spontaneous when it occurs on its own without external help in the form of a force or energy. A process is nonspontaneous when it does not occur on its own and you need to provide some external help in the form of a force or energy, for the process to occur.

Complete answer:
A process is spontaneous when its Gibbs free energy change is negative.
\[\Delta G < 0\]
But \[\Delta G = \Delta H - T\Delta S\]
Here, \[\Delta H\] is the change in the enthalpy and \[\Delta S\] is the entropy change.
Hence, a process is spontaneous when \[\Delta H - T\Delta S < 0\]
But \[\Delta H = \Delta U + P\Delta V\]
Here, \[\Delta U\] is the change in the internal energy and \[\Delta V\] is the volume change.
Since the volume of the process is constant, \[\Delta V = 0\] as there is no change in the volume.
Hence, \[\Delta H = \Delta U + P\Delta V = \Delta U + P\left( 0 \right) = \Delta U\]
So, a process is spontaneous when \[\Delta U - T\Delta S < 0\]
But the entropy change for the process is zero as the entropy is constant.
\[\Delta U - T\Delta S = \Delta U - T\left( 0 \right) = \Delta U\]
So, a process is spontaneous when \[\Delta U < 0\]
For a spontaneous process, if entropy and volume are constant, the internal energy of the system must decrease.

Hence, the correct answer is the option (B) decrease.

Note:A process will be spontaneous when the randomness of the molecules present in the system increases. For a spontaneous process entropy change of the system plus surrounding must be positive.