To determine which conditions lead to a nonspontaneous process at every temperature, we need to analyze the relationship between the Gibbs free energy change (ΔG), enthalpy change (ΔH), and entropy change (ΔS). The Gibbs free energy equation is given by:
\[
\Delta G = \Delta H - T \Delta S
\]
A process is nonspontaneous if ΔG > 0 at all temperatures. Let's analyze each option step by step:
### Step 1: Analyze Option (i) ΔH > 0, ΔS = 0
- Here, ΔH is positive and ΔS is zero.
- Substituting into the Gibbs equation:
\[
\Delta G = \Delta H - T \cdot 0 = \Delta H
\]
- Since ΔH > 0, it follows that ΔG > 0.
- **Conclusion**: This condition is nonspontaneous at all temperatures.
### Step 2: Analyze Option (ii) ΔH < 0, ΔS > 0
- Here, ΔH is negative and ΔS is positive.
- Substituting into the Gibbs equation:
\[
\Delta G = \Delta H - T \Delta S
\]
- As T increases, the term \(T \Delta S\) becomes larger and negative, which can make ΔG negative.
- **Conclusion**: This condition is spontaneous at high temperatures, so it is not nonspontaneous at every temperature.
### Step 3: Analyze Option (iii) ΔH > 0, ΔS < 0
- Here, ΔH is positive and ΔS is negative.
- Substituting into the Gibbs equation:
\[
\Delta G = \Delta H - T \Delta S
\]
- Since ΔS < 0, the term \(-T \Delta S\) becomes positive, making ΔG positive.
- **Conclusion**: This condition is nonspontaneous at all temperatures.
### Step 4: Analyze Option (iv) ΔH = 0, ΔS < 0
- Here, ΔH is zero and ΔS is negative.
- Substituting into the Gibbs equation:
\[
\Delta G = 0 - T \Delta S
\]
- Since ΔS < 0, the term \(-T \Delta S\) becomes positive, making ΔG positive.
- **Conclusion**: This condition is nonspontaneous at all temperatures.
### Final Conclusion:
The conditions under which the process is nonspontaneous at every temperature are:
- (i) ΔH > 0, ΔS = 0
- (iii) ΔH > 0, ΔS < 0
- (iv) ΔH = 0, ΔS < 0
Thus, the correct options are (i), (iii), and (iv).
