In an adiabatic change, the pressure and temperature of a monoatomic gas are related with relation as `P prop T^(C )`, Where `C` is equal to:

Options

A. `5/4`
B. `5/3`
C. `5/2`
D. `3/5`

Explanation

To solve the problem, we need to find the value of \( C \) in the relationship \( P \propto T^C \) for a monoatomic gas undergoing an adiabatic change.

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. For an ideal gas, the relationship between pressure \( P \), volume \( V \), and temperature \( T \) can be expressed using the adiabatic condition:
\[ PV^\gamma = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

2. **Use the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
Rearranging this gives:
\[ P = \frac{nRT}{V} \]

3. **Substituting into the Adiabatic Condition**:
We can substitute \( P \) from the ideal gas law into the adiabatic condition:
\[ \left(\frac{nRT}{V}\right) V^\gamma = \text{constant} \]
Simplifying this gives:
\[ nRT V^{\gamma - 1} = \text{constant} \]

4. **Relate Pressure and Temperature**:
From the above equation, we can express \( P \) in terms of \( T \):
\[ P \propto T^{\frac{\gamma}{\gamma - 1}} \]
This means:
\[ P \propto T^C \]
where \( C = \frac{\gamma}{\gamma - 1} \).

5. **Determine \( \gamma \) for Monoatomic Gas**:
For a monoatomic gas, \( \gamma = \frac{C_p}{C_v} = \frac{5}{3} \).

6. **Calculate \( C \)**:
Substitute \( \gamma \) into the equation for \( C \):
\[ C = \frac{\frac{5}{3}}{\frac{5}{3} - 1} = \frac{\frac{5}{3}}{\frac{2}{3}} = \frac{5}{2} \]

### Final Answer:
Thus, the value of \( C \) is:
\[ C = \frac{5}{2} \]