For a process to be spontaneous at all conditions of temperature which of the following is true?

Options

A. `Delta S` positive and `Delta H` positive
B. `Delta S` positive and `Delta H` negative
C. `Delta S` negative and `Delta H` positive
D. `Delta S` negative and `Delta H` negative

Answer

To determine which process is spontaneous at all conditions of temperature, we need to analyze the Gibbs free energy equation:

### Step-by-Step Solution:

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

2. **Condition for Spontaneity**:
A process is spontaneous if \(\Delta G < 0\). This means we want to find conditions under which \(\Delta G\) is always negative, regardless of the temperature.

3. **Analyzing the Terms**:
- If \(\Delta H < 0\) (the process is exothermic), this term contributes negatively to \(\Delta G\).
- If \(\Delta S > 0\) (the process increases entropy), this term contributes positively to \(-T \Delta S\), which also helps in making \(\Delta G\) negative.

4. **Combining the Conditions**:
- If both conditions are satisfied:
- \(\Delta H < 0\) (exothermic reaction)
- \(\Delta S > 0\) (increase in entropy)
- Then, regardless of the temperature \(T\), the term \(-T \Delta S\) will always be positive, and since \(\Delta H\) is negative, the overall \(\Delta G\) will be negative.

5. **Conclusion**:
Therefore, the condition for a process to be spontaneous at all temperatures is:
\[ \Delta H < 0 \quad \text{and} \quad \Delta S > 0 \]
This corresponds to option B.

### Final Answer:
The correct answer is **B: \(\Delta H < 0\) and \(\Delta S > 0\)**.