The correct relation between `gamma=(c_(p))/(c_(v))` and temperature `T` is :

Options

A. `gamma prop T^(@)`
B. `gamma prop (1)/(T)`
C. `gamma prop T`
D. `gamma prop(1)/(sqrt(T))`

Answer

To solve the problem regarding the relationship between gamma (γ) and temperature (T), we need to understand the definitions and properties of specific heats.

### Step-by-Step Solution:

1. **Understanding Gamma (γ)**:
- Gamma (γ) is defined as the ratio of specific heat at constant pressure (Cp) to specific heat at constant volume (Cv):
\[ \gamma = \frac{C_p}{C_v} \]

2. **Nature of Specific Heats**:
- Specific heats (Cp and Cv) are properties of a substance that can depend on temperature. However, the ratio of these two specific heats (γ) is a dimensionless quantity.

3. **Dependence on Temperature**:
- While Cp and Cv can vary with temperature, their ratio (γ) is often considered constant for an ideal gas over a limited temperature range. This means that γ does not change with temperature.

4. **Conclusion**:
- Since γ is a dimensionless ratio that does not depend on temperature, we can express this relationship mathematically:
\[ \gamma \propto T^0 \]
- This indicates that γ is independent of temperature, as any quantity raised to the power of 0 is equal to 1.

5. **Final Statement**:
- Therefore, the correct relation is that γ does not depend on temperature T.