In which case, process will be spontaneous at all temperature?

Options

A. `Delta H lt 0` and `DeltaS gt 0`
B. `Delta H gt 0` and `Delta S lt 0`
C. `DeltaH lt 0` and `DeltaS lt 0`
D. `DeltaH gt 0` and `DeltaS lt 0`

Explanation

To determine the conditions under which a process will be spontaneous at all temperatures, we can analyze the Gibbs free energy equation:

### Step-by-Step Solution:

1. **Understand the Gibbs Free Energy Equation**:
The Gibbs free energy (G) is defined by the equation:
\[ \Delta G = \Delta H - T \Delta S \]
where:
- \(\Delta G\) = change in Gibbs free energy
- \(\Delta H\) = change in enthalpy
- \(T\) = absolute temperature
- \(\Delta S\) = change in entropy

2. **Determine Conditions for Spontaneity**:
A process is spontaneous if \(\Delta G < 0\). Therefore, we need to analyze the signs of \(\Delta H\) and \(\Delta S\):
- If \(\Delta H < 0\) (exothermic process) and \(\Delta S > 0\) (increase in entropy), then:
\[ \Delta G = \Delta H - T \Delta S \]
will always be negative regardless of the temperature \(T\).

3. **Evaluate Other Cases**:
- If \(\Delta H < 0\) and \(\Delta S < 0\): Spontaneity occurs at low temperatures.
- If \(\Delta H > 0\) and \(\Delta S > 0\): Spontaneity occurs at high temperatures.
- If \(\Delta H > 0\) and \(\Delta S < 0\): The process is never spontaneous.

4. **Conclusion**:
The only scenario where the process is spontaneous at all temperatures is when:
\[ \Delta H < 0 \quad \text{and} \quad \Delta S > 0 \]

### Final Answer:
The process will be spontaneous at all temperatures when \(\Delta H < 0\) and \(\Delta S > 0\).

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