Question

An ideal gas is found to obey \( PV^{\frac{3}{2}} = \text{constant} \) during an adiabatic process. If such a gas initially at a temperature \( T \) is adiabatically compressed to \( \frac{1}{4} \)th of its volume, then its final temperature is:

• A. \( \sqrt{3T} \)
• B. \( \sqrt{2T} \)
• C. \( 2T \)
• D. \( 3T \)

Hint:

In an adiabatic process, the final temperature depends on the polytropic index and volume ratio. The temperature increases when volume is compressed.

Answer 1

The Correct Option is C

Step 1: Applying the Adiabatic Relation For an adiabatic process, the relation between temperature and volume is given by: \[ T_1 V_1^{\gamma -1} = T_2 V_2^{\gamma -1} \] where: - \( \gamma \) is the polytropic index, given as \( \gamma = \frac{3}{2} \)
- \( T_1 \) is the initial temperature
- \( V_1 \) is the initial volume
- \( T_2 \) is the final temperature
- \( V_2 \) is the final volume, given as \( V_2 = \frac{V_1}{4} \) Step 2: Substituting Values Rewriting the equation: \[ T \cdot V_1^{\frac{3}{2} - 1} = T_2 \cdot V_2^{\frac{3}{2} - 1} \] Since \( \gamma -1 = \frac{1}{2} \), we get: \[ T \cdot V_1^{\frac{1}{2}} = T_2 \cdot \left(\frac{V_1}{4}\right)^{\frac{1}{2}} \] \[ T \cdot V_1^{\frac{1}{2}} = T_2 \cdot V_1^{\frac{1}{2}} \times \frac{1}{2} \] Canceling \( V_1^{\frac{1}{2}} \) from both sides: \[ T = T_2 \times \frac{1}{2} \] Step 3: Solving for \( T_2 \) \[ T_2 = 2T \] Thus, the correct answer is option (3).

Answer 2

Given:
- The gas follows the adiabatic condition:
\[ PV^{\frac{3}{2}} = \text{constant} \] - Initial temperature = \( T \)
- Final volume \( V_f = \frac{1}{4} V_i \)

Step 1: Identify the adiabatic index \( \gamma \):
For an adiabatic process:
\[ PV^\gamma = \text{constant} \] Comparing with the given:
\[ \gamma = \frac{3}{2} = 1.5 \]

Step 2: Use the ideal gas law \( PV = nRT \) and the adiabatic relations:
\[ TV^{\gamma - 1} = \text{constant} \] This means:
\[ T_i V_i^{\gamma - 1} = T_f V_f^{\gamma - 1} \] where \( T_i \) and \( T_f \) are initial and final temperatures.

Step 3: Substitute values:
\[ T \times V_i^{0.5} = T_f \times \left(\frac{V_i}{4}\right)^{0.5} \] \[ T \times V_i^{0.5} = T_f \times \frac{V_i^{0.5}}{2} \] Cancel \( V_i^{0.5} \) from both sides:
\[ T = \frac{T_f}{2} \Rightarrow T_f = 2T \]

Therefore, the final temperature after adiabatic compression is:
\[ \boxed{2T} \]