Question

During an adiabatic process, the pressure of a gas is found to proportional to the cube of its absolute temperature. Identify the correct statement(s) about the gas used in the process.

• A. P$^{\gamma-1}$ T$^{\gamma}$ = constant
• B. $\gamma = \frac{3}{2}$
• C. P$^{1-\gamma}$ T$^{\gamma}$ = constant
• D. $\gamma = \frac{3}{4}$

Answer 1

Answer: (B), (C)

Options B and C

Answer 2

Here's a breakdown of the solution:

1. Adiabatic Process and Ideal Gas Law:

   • For an adiabatic process: $TV^{\gamma-1} = \text{constant}$.
   • Using the ideal gas law ($PV = nRT$), we can express volume as $V = \frac{nRT}{P}$.

2. Deriving the Relationship:

   • Substituting $V$ in the adiabatic equation: $T \left( \frac{nRT}{P} \right)^{\gamma-1} = \text{constant}$.
   • Simplifying: $\frac{T^\gamma}{P^{\gamma-1}} = \text{constant}$, which can be rearranged to $P^{\gamma-1} T^{-\gamma} = \text{constant}$.

3. Applying the Given Condition:

   • We are given that $P \propto T^3$, which means $P = kT^3$ for some constant $k$.
   • Substituting this into the derived equation: $(kT^3)^{\gamma-1} T^{-\gamma} = \text{constant}$.
   • This simplifies to $k^{\gamma-1} T^{3(\gamma-1) - \gamma} = \text{constant}$.
   • For this to be independent of $T$, the exponent of $T$ must be zero: $3(\gamma-1) - \gamma = 0$.

4. Solving for γ:

   • Solving the equation $3\gamma - 3 - \gamma = 0$ gives $2\gamma = 3$, so $\gamma = \frac{3}{2}$.

5. Checking the Options:

   • Option A: $P^{\gamma-1} T^{\gamma} = \text{constant}$. This is incorrect because our derived relation is $P^{\gamma-1} T^{-\gamma} = \text{constant}$.
   • Option B: $\gamma = \frac{3}{2}$. This is correct.
   • Option C: $P^{1-\gamma} T^{\gamma} = \text{constant}$. Since $P^{1-\gamma} = \frac{1}{P^{\gamma-1}}$, this option is equivalent to $\frac{T^{\gamma}}{P^{\gamma-1}} = \text{constant}$, which is correct.
   • Option D: $\gamma = \frac{3}{4}$. This is incorrect.

Therefore, the correct options are B and C.