Temperature of an ideal gas is 300 K . The final temperature of the gas when its volume changes from `V " to " 2V` in the process `p=alphaV` (here `alpha` is a positive constant) is

Options

A. 900 K
B. 1200 K
C. 300 K
D. 500 K

Answer

To solve the problem, we need to determine the final temperature \( T_2 \) of an ideal gas when its volume changes from \( V \) to \( 2V \) under the condition that the pressure \( P \) varies as \( P = \alpha V \).

### Step-by-Step Solution:

1. **Understanding the Process**:
The process is defined by the equation \( P = \alpha V \). This implies that pressure is directly proportional to volume, where \( \alpha \) is a positive constant.

2. **Using the Ideal Gas Law**:
According to the ideal gas law, we have:
\[ PV = nRT \]
where \( n \) is the number of moles and \( R \) is the ideal gas constant.

3. **Relating Pressure and Temperature**:
From the ideal gas law, we can express pressure \( P \) in terms of temperature \( T \):
\[ P = \frac{nRT}{V} \]
Substituting \( P = \alpha V \) into this equation gives:
\[ \alpha V = \frac{nRT}{V} \]
Rearranging this, we find:
\[ T = \frac{\alpha V^2}{nR} \]
This shows that \( T \) is proportional to \( V^2 \).

4. **Establishing the Relationship**:
Since \( T \) is proportional to \( V^2 \), we can write:
\[ T_1 V_1^2 = T_2 V_2^2 \]
where \( T_1 \) and \( T_2 \) are the initial and final temperatures, and \( V_1 \) and \( V_2 \) are the initial and final volumes.

5. **Substituting Known Values**:
Given:
- Initial temperature \( T_1 = 300 \, K \)
- Initial volume \( V_1 = V \)
- Final volume \( V_2 = 2V \)

Substituting these values into the equation gives:
\[ 300 \cdot V^2 = T_2 \cdot (2V)^2 \]

6. **Simplifying the Equation**:
This simplifies to:
\[ 300 \cdot V^2 = T_2 \cdot 4V^2 \]
Dividing both sides by \( V^2 \) (assuming \( V \neq 0 \)):
\[ 300 = 4T_2 \]

7. **Solving for \( T_2 \)**:
Now, we can solve for \( T_2 \):
\[ T_2 = \frac{300}{4} = 75 \, K \]

### Final Answer:
The final temperature \( T_2 \) of the gas when its volume changes from \( V \) to \( 2V \) is \( 75 \, K \).