Question

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 =$ constant. The coefficient of volume expansion of the gas is ____

• A. $\frac{2}{T}$
• B. $\frac{1}{T}$
• C. $\frac{4}{T}$
• D. $\frac{3}{T}$

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The coefficient of volume expansion is defined as $\gamma = \frac{1}{V} \frac{dV}{dT}$. We need to express $V$ as a function of $T$ using the given process equation and the ideal gas law ($PV = nRT$).
Step 2: Detailed Explanation:
Given $PT^3 = C$.
From the ideal gas law, $P = \frac{nRT}{V}$.
Substitute $P$ into the process equation:
\[ \left(\frac{nRT}{V}\right)T^3 = C \implies \frac{nRT^4}{V} = C \]
\[ V = \frac{nR}{C} T^4 \]
Differentiate $V$ with respect to $T$:
\[ \frac{dV}{dT} = \frac{nR}{C} (4T^3) \]
Now, calculate $\gamma$:
\[ \gamma = \frac{1}{V} \frac{dV}{dT} = \frac{1}{(nR/C)T^4} \cdot \frac{nR}{C} (4T^3) \]
\[ \gamma = \frac{4T^3}{T^4} = \frac{4}{T} \]
Step 3: Final Answer:
The coefficient of volume expansion is $\frac{4}{T}$.