Question

One mole of an ideal gas is at temperature \( T \) K. The \( \gamma \) value of this gas is \( \frac{5}{3} \). Now the gas does 12R Joules of work adiabatically (R is the universal gas constant). Then the final temperature of the gas will be:

• A. \( T - 8 \, \text{K} \)
• B. \( T + 4 \, \text{K} \)
• C. \( T - 4.4 \, \text{K} \)
• D. \( T - 6 \, \text{K} \)

Hint:

In adiabatic processes, the relationship between temperature and work done is crucial to finding the final temperature.

Answer 1

The Correct Option is A

Step 1: For \( \gamma = \frac{5}{3} \), find \( C_v \) using the Mayer relation combined with the definition of \( \gamma \): \( C_p - C_v = R \) and \( \gamma = \frac{C_p}{C_v} \), giving \( C_v(\gamma - 1) = R \), so \( C_v = \frac{R}{\gamma-1} = \frac{R}{2/3} = \frac{3}{2}R \).

Step 2: Since the process is adiabatic, \( Q = 0 \), so the first law gives \( W = -\Delta U \), meaning the gas does work only by cooling down: \( W = nC_v(T_i - T_f) \).

Step 3: Substitute \( n=1 \), \( W = 12R \), \( C_v = \frac{3}{2}R \): \( 12R = \frac{3}{2}R(T - T_f) \), which gives \( T - T_f = 8 \).

Step 4: Solve for the final temperature: \( T_f = T - 8 \, \text{K} \).

\[ \boxed{T_f = T - 8 \, \text{K}} \]

Answer 2

An alternative way to reach the final temperature is through the internal energy of an ideal gas expressed via its degrees of freedom, \( U = \frac{f}{2}nRT \), rather than through the heat capacity ratio directly.

For \( \gamma = \frac{5}{3} \), the gas behaves as a monatomic ideal gas, which has \( f = 3 \) translational degrees of freedom. The internal energy per mole is therefore:

\[ U = \frac{3}{2}RT \]

For an adiabatic process, \( Q = 0 \), so by the first law, the work done by the gas equals the loss in internal energy:

\[ W = -\Delta U = \frac{3}{2}R(T_i - T_f) \]

With \( W = 12R \) and \( n=1 \):

\[ 12R = \frac{3}{2}R(T - T_f) \implies T - T_f = 8 \implies T_f = T - 8 \, \text{K} \]

Checking the options against this degrees-of-freedom picture:

1. Option A, \( T - 8 \, \text{K} \): matches exactly what the internal-energy loss of \( 12R \) joules corresponds to for a monatomic gas.
2. Option B, \( T + 4 \, \text{K} \): would require the gas to gain internal energy while doing positive work with no heat supplied, violating energy conservation.
3. Option C, \( T - 4.4 \, \text{K} \): would require \( f \approx 5.45 \), which is not a valid degrees-of-freedom count for any real gas model matching \( \gamma = \frac{5}{3} \).
4. Option D, \( T - 6 \, \text{K} \): would require \( f = 4 \), which corresponds to \( \gamma = 1.5 \), not \( \frac{5}{3} \).

The degrees-of-freedom approach confirms the same result as the heat-capacity approach.

Therefore, the correct answer is \( T - 8 \, \text{K} \).