Gibbs Helmholtz equation relates the enthalpy, entropy and free energy change of the process at constant pressure and temperature as
`DeltaG=DeltaH-TDeltaS " (at constant P, T)"`
In General the magnitude of `DeltaH` does not change much with the change in temperature but the terms `TDeltaS` changes appreciably. Hence in some process spontaneity is very much dependent on temperature and such processes are generally known as entropy driven process.
When `CaCO_(3)` is heated to a high temperature it decomposes into CaO and `CO_(2)`, however it is quite stable at room temperature. It can be explained by the fact that

Options

A. `Delta_(r )H` dominates the term `TDeltaS` at high temperature
B. the term `TDeltaS` dominates the `Delta_(r )H` at high temperature
C. at high temperature both `Delta_(r )S` and `Delta_(r )H` becomes negative
D. thermodynamics can not say anything about spontaneity

Explanation

To explain why calcium carbonate (CaCO₃) is stable at room temperature but decomposes into calcium oxide (CaO) and carbon dioxide (CO₂) when heated, we can analyze the situation using the Gibbs-Helmholtz equation:

### Step-by-Step Solution:

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

2. **Analyzing Stability at Room Temperature**:
At room temperature, the decomposition of CaCO₃ into CaO and CO₂ is not spontaneous. This means that:
\[ \Delta G > 0 \]
This indicates that the reaction does not proceed under these conditions.

3. **Considering the Changes in Enthalpy and Entropy**:
- The decomposition reaction can be represented as:
\[ \text{CaCO}_3 (s) \rightarrow \text{CaO} (s) + \text{CO}_2 (g) \]
- Generally, the change in enthalpy (\(\Delta H\)) for this reaction is positive (endothermic), meaning energy is required to break the bonds in CaCO₃.
- The change in entropy (\(\Delta S\)) is positive because the products (CaO and CO₂) have more disorder than the reactant (solid CaCO₃).

4. **Effect of Temperature on Spontaneity**:
As the temperature increases, the term \(T\Delta S\) becomes larger. Since \(\Delta S\) is positive, \(T\Delta S\) will also be positive and can outweigh the positive \(\Delta H\):
\[ \Delta G = \Delta H - T\Delta S \]
At high temperatures, if \(T\Delta S\) becomes greater than \(\Delta H\), then:
\[ \Delta G < 0 \]
This indicates that the reaction becomes spontaneous at high temperatures.

5. **Conclusion**:
Therefore, the stability of CaCO₃ at room temperature can be attributed to the fact that the positive \(\Delta H\) and the relatively lower temperature do not favor the decomposition reaction. However, at high temperatures, the entropy term dominates, making the reaction spontaneous.