One mole of an ideal gas passes through a process where pressure and volume obey the relation `P=P_0 [1-1/2 (V_0/V)^2]` Here `P_0` and `V_0` are constants. Calculate the change in the temperature of the gas if its volume changes from `V_0` to `2V_0`.

Options

A. `3/4 (P_0 V_0)/R`
B. `1/2 (P_0 V_0)/R`
C. `1/4 (P_0 V_0)/R`
D. `5/4 (P_0 V_0)/R`

Explanation

To solve the problem, we need to calculate the change in temperature of the gas as it goes from an initial volume \( V_0 \) to a final volume \( 2V_0 \). We will use the provided relation between pressure \( P \) and volume \( V \) of the gas, as well as the ideal gas law.

### Step-by-Step Solution:

1. **Identify the Initial and Final States:**
- Initial volume \( V_1 = V_0 \)
- Final volume \( V_2 = 2V_0 \)

2. **Calculate Initial Pressure \( P_1 \):**
Using the relation \( P = P_0 \left( 1 - \frac{1}{2} \left( \frac{V_0}{V} \right)^2 \right) \):
\[ P_1 = P_0 \left( 1 - \frac{1}{2} \left( \frac{V_0}{V_0} \right)^2 \right) = P_0 \left( 1 - \frac{1}{2} \right) = \frac{1}{2} P_0 \]

3. **Calculate Final Pressure \( P_2 \):**
For \( V_2 = 2V_0 \):
\[ P_2 = P_0 \left( 1 - \frac{1}{2} \left( \frac{V_0}{2V_0} \right)^2 \right) = P_0 \left( 1 - \frac{1}{2} \cdot \frac{1}{4} \right) = P_0 \left( 1 - \frac{1}{8} \right) = P_0 \cdot \frac{7}{8} \]

4. **Use the Formula for Change in Temperature:**
The change in temperature \( \Delta T \) is given by:
\[ \Delta T = \frac{P_2 V_2 - P_1 V_1}{nR} \]
Since \( n = 1 \) (1 mole of gas), we have:
\[ \Delta T = \frac{P_2 V_2 - P_1 V_1}{R} \]

5. **Substituting the Values:**
Substitute \( P_2 = \frac{7}{8} P_0 \), \( V_2 = 2V_0 \), \( P_1 = \frac{1}{2} P_0 \), and \( V_1 = V_0 \):
\[ \Delta T = \frac{\left( \frac{7}{8} P_0 \cdot 2V_0 \right) - \left( \frac{1}{2} P_0 \cdot V_0 \right)}{R} \]

6. **Calculating the Terms:**
\[ \Delta T = \frac{\left( \frac{14}{8} P_0 V_0 \right) - \left( \frac{1}{2} P_0 V_0 \right)}{R} \]
Convert \( \frac{1}{2} P_0 V_0 \) to a fraction with a common denominator:
\[ \frac{1}{2} P_0 V_0 = \frac{4}{8} P_0 V_0 \]
Thus,
\[ \Delta T = \frac{\left( \frac{14}{8} P_0 V_0 - \frac{4}{8} P_0 V_0 \right)}{R} = \frac{\frac{10}{8} P_0 V_0}{R} = \frac{5}{4} \frac{P_0 V_0}{R} \]

### Final Answer:
The change in temperature \( \Delta T \) is:
\[ \Delta T = \frac{5}{4} \frac{P_0 V_0}{R} \]