An ideal gas follows a process described by `PV^(2)=C` from `(P_(1), V_(1),T_(1)) " to " (P_(2), V_(2), T_(2))` (C is a constant). Then

Options

A. if `P_(1) gt P_(2) " then " T_(2) gt T_(1)`
B. if `V_(2) gt V_(1) " then " T_(2) lt T_(1)`
C. if `V_(2) gt V_(1) " then " T_(2) gt T_(1)`
D. if `P_(1) gt P_(2) " then " V_(1) gt V_(2)`

Answer

To solve the problem, we need to analyze the given relationship for an ideal gas undergoing a process defined by the equation \( PV^2 = C \), where \( C \) is a constant. We will derive the relationships between pressure \( P \), volume \( V \), and temperature \( T \) of the gas at two states, \( (P_1, V_1, T_1) \) and \( (P_2, V_2, T_2) \).

### Step-by-Step Solution:

1. **Understand the given process**: The process is defined by the equation \( PV^2 = C \). This implies that the product of pressure \( P \) and the square of volume \( V \) remains constant throughout the process.

2. **Relate the two states**: From the equation, we can write:
\[ P_1 V_1^2 = P_2 V_2^2 = C \]
This means:
\[ P_1 V_1^2 = P_2 V_2^2 \]

3. **Express pressure in terms of volume**: Rearranging the above equation gives us:
\[ \frac{P_1}{P_2} = \frac{V_2^2}{V_1^2} \]

4. **Analyze the relationship**: If we assume \( V_2 > V_1 \), then \( \frac{V_2}{V_1} > 1 \) implies:
\[ \frac{P_1}{P_2} > 1 \implies P_1 > P_2 \]

5. **Use the ideal gas law**: The ideal gas law states:
\[ PV = nRT \]
From this, we can express pressure \( P \) as:
\[ P = \frac{nRT}{V} \]

6. **Substitute into the process equation**: Substitute \( P \) into the process equation:
\[ \frac{nRT}{V} V^2 = C \implies nRT V = C \]
This can be rearranged to:
\[ TV = \frac{C}{nR} \]
Let \( C_1 = \frac{C}{nR} \), which is a constant.

7. **Relate temperatures at the two states**: From the above equation, we have:
\[ T_1 V_1 = T_2 V_2 \]
Rearranging gives:
\[ \frac{T_1}{T_2} = \frac{V_2}{V_1} \]

8. **Analyze temperature relationship**: Since we assumed \( V_2 > V_1 \), it follows that:
\[ \frac{T_1}{T_2} > 1 \implies T_1 > T_2 \]

9. **Conclusion**: Based on our analysis, if \( V_2 > V_1 \), then \( T_1 > T_2 \). Therefore, the correct option is that \( T_1 \) is greater than \( T_2 \).

### Final Answer:
The correct option is **B**: \( T_1 > T_2 \).