The correct thermodynamic conditions for the spontaneous reaction at all temperatures is

Options

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

Answer

To determine the correct thermodynamic conditions for a spontaneous reaction at all temperatures, we can analyze the Gibbs free energy equation, which is given by:

\[ \Delta G = \Delta H - T \Delta S \]

Where:
- \(\Delta G\) = change in Gibbs free energy
- \(\Delta H\) = change in enthalpy
- \(T\) = temperature in Kelvin
- \(\Delta S\) = change in entropy

### Step-by-Step Solution:

1. **Understand Spontaneity**: A reaction is spontaneous when the change in Gibbs free energy (\(\Delta G\)) is negative. Thus, we need to ensure that \(\Delta G < 0\).

2. **Analyze the Gibbs Free Energy Equation**: From the equation \(\Delta G = \Delta H - T \Delta S\), we can see that:
- If \(\Delta H\) is negative (exothermic reaction), it contributes to making \(\Delta G\) negative.
- If \(\Delta S\) is positive (increase in disorder), it also contributes to making \(\Delta G\) negative, especially when multiplied by the positive temperature \(T\).

3. **Conditions for Spontaneity**:
- **Case 1**: If \(\Delta H < 0\) (exothermic) and \(\Delta S > 0\) (increase in entropy), then:
\[ \Delta G = \Delta H - T \Delta S < 0 \]
This condition ensures that \(\Delta G\) is negative at all temperatures.

- **Case 2**: If \(\Delta H > 0\) (endothermic) and \(\Delta S < 0\) (decrease in entropy), then \(\Delta G\) will be positive at all temperatures, which is not spontaneous.

- **Case 3**: If \(\Delta H > 0\) and \(\Delta S > 0\), \(\Delta G\) can be negative at high temperatures, but it is not guaranteed to be negative at all temperatures.

- **Case 4**: If \(\Delta H < 0\) and \(\Delta S < 0\), \(\Delta G\) can be negative at low temperatures, but it is not guaranteed to be negative at all temperatures.

4. **Conclusion**: The only condition that guarantees spontaneity at all temperatures is when \(\Delta H < 0\) and \(\Delta S > 0\). Therefore, the correct thermodynamic conditions for spontaneous reactions at all temperatures are:
- \(\Delta H < 0\) (negative enthalpy change)
- \(\Delta S > 0\) (positive entropy change)

### Final Answer:
The correct thermodynamic conditions for spontaneous reactions at all temperatures are:
- \(\Delta H < 0\) and \(\Delta S > 0\).