Ratio of `C_(P)` and `C_(v)` depends upon temperature according to the following relation

Options

A. `gamma alpha` T
B. `gamma alpha 1//T`
C. `gamma alpha sqrt(T)`
D. `gamma alpha T^(o)`

Answer

To solve the problem regarding the ratio of \( C_p \) (molar specific heat capacity at constant pressure) and \( C_v \) (molar heat capacity at constant volume) and how it depends on temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- \( C_p \) is the heat capacity at constant pressure.
- \( C_v \) is the heat capacity at constant volume.
- The ratio \( \gamma \) is defined as:
\[ \gamma = \frac{C_p}{C_v} \]

2. **Identifying the Nature of Gases**:
- The value of \( \gamma \) varies depending on the type of gas:
- For monoatomic gases, \( \gamma = \frac{5}{3} \).
- For diatomic gases, \( \gamma = \frac{7}{5} \).
- For polyatomic gases, \( \gamma = \frac{9}{3} \) (which simplifies to \( 3 \)).

3. **Temperature Independence**:
- It is important to note that \( C_p \) and \( C_v \) are generally considered to be independent of temperature for ideal gases in the context of this problem.
- Thus, \( \gamma \) can be treated as a constant for a given type of gas.

4. **Conclusion about Temperature Dependence**:
- Since \( \gamma \) is constant for different types of gases and does not change with temperature, we can conclude that:
\[ \gamma \text{ is independent of temperature, or } \gamma \propto T^0 \]
- Therefore, the correct answer to the question is that the ratio \( \frac{C_p}{C_v} \) does not depend on temperature.

### Final Answer:
The ratio \( \frac{C_p}{C_v} \) (denoted as \( \gamma \)) is independent of temperature.