Question

An ideal gas undergoes an adiabatic expansion from volume \( V \) to \( 2V \). If the initial temperature is \( T \), what is the final temperature? (Assume the ratio of specific heats \( \gamma = \frac{5}{3} \))

• A. \( T \)
• B. \( \frac{T}{2} \)
• C. \( \frac{T}{2^{2/3}} \)
• D. \( \frac{T}{2^{5/3}} \)

Hint:

\textbf{Tip:} For adiabatic processes, use \( TV^{\gamma - 1} = \text{const} \) to directly relate initial and final temperatures and volumes.

Answer 1

The Correct Option is C

To determine the final temperature of an ideal gas undergoing adiabatic expansion, we use the adiabatic process relation:

\( TV^{\gamma-1} = \text{constant} \)

For an initial state with temperature \( T_1 = T \) and volume \( V_1 = V \), and a final state with temperature \( T_2 \) and volume \( V_2 = 2V \), the equation becomes:

\( TV^{\gamma-1} = T_2(2V)^{\gamma-1} \)

Simplifying gives:

\( T V^{\gamma-1} = T_2 \cdot 2^{\gamma-1} \cdot V^{\gamma-1} \)

Cancel out \( V^{\gamma-1} \) on both sides:

\( T = T_2 \cdot 2^{\gamma-1} \)

Solve for \( T_2 \):

\( T_2 = \frac{T}{2^{\gamma-1}} \)

Given \( \gamma = \frac{5}{3} \), calculate \( \gamma-1 \):

\( \gamma-1 = \frac{5}{3} - 1 = \frac{2}{3} \)

Substitute back to find \( T_2 \):

\( T_2 = \frac{T}{2^{2/3}} \)

The final temperature \( T_2 \) is \( \frac{T}{2^{2/3}} \).

Answer 2

To find the final temperature \( T_f \) of an ideal gas undergoing adiabatic expansion from a volume \( V \) to \( 2V \), we use the adiabatic process equation:

\( T_iV_i^{\gamma-1} = T_fV_f^{\gamma-1} \)

where:

• \( T_i \) is the initial temperature.
• \( V_i \) is the initial volume.
• \( T_f \) is the final temperature.
• \( V_f \) is the final volume.
• \( \gamma \) is the ratio of specific heats, given as \( \frac{5}{3} \).

Given:

• \( T_i = T \)
• \( V_i = V \)
• \( V_f = 2V \)

Substituting these values, we get:

\( TV^{\frac{5}{3}-1} = T_f(2V)^{\frac{5}{3}-1} \)

Simplifying the exponents:

\( TV^{\frac{2}{3}} = T_f \times 2^{\frac{2}{3}} \times V^{\frac{2}{3}} \)

Cancel \( V^{\frac{2}{3}} \) from both sides:

\( T = T_f \times 2^{\frac{2}{3}} \)

Solving for \( T_f \):

\( T_f = \frac{T}{2^{\frac{2}{3}}} \)

Thus, the final temperature \( T_f \) of the gas after adiabatic expansion is \( \frac{T}{2^{2/3}} \).