Question

An ideal gas is expanding such that \(PT^3 = \text{constant}\). The coefficient of volume expansion of the gas is :

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

Hint:

For any polytropic process of the form \(PV^x = \text{constant}\) or \(PT^y = \text{constant}\), the key is always to use the ideal gas law to establish a direct relationship between the two variables required for the derivative (in this case, V and T).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a process for an ideal gas described by the relation \(PT^3 = \text{constant}\). We need to find the coefficient of volume expansion (\(\gamma\)) for this gas under this specific process.
Step 2: Key Formula or Approach:
The coefficient of volume expansion is defined as \(\gamma = \frac{1}{V}\frac{dV}{dT}\).
We also need the ideal gas equation: \(PV = nRT\), where n and R are constants.
Our goal is to express volume \(V\) as a function of temperature \(T\) only, and then use the definition of \(\gamma\).
Step 3: Detailed Explanation:
First, we use the ideal gas equation to eliminate pressure \(P\) from the given process equation.
From \(PV = nRT\), we have \(P = \frac{nRT}{V}\).
Substitute this expression for \(P\) into the given relation \(PT^3 = k\) (where k is a constant).
\[ \left(\frac{nRT}{V}\right) T^3 = k \] \[ \frac{nRT^4}{V} = k \] Now, rearrange this equation to express \(V\) in terms of \(T\).
\[ V = \left(\frac{nR}{k}\right) T^4 \] Since \(n, R, k\) are all constants, we can say \(V \propto T^4\). Let \(C = \frac{nR}{k}\).
\[ V = CT^4 \] Next, we differentiate \(V\) with respect to \(T\) to find \(\frac{dV}{dT}\).
\[ \frac{dV}{dT} = \frac{d}{dT}(CT^4) = 4CT^3 \] Finally, we use the definition of the coefficient of volume expansion \(\gamma\).
\[ \gamma = \frac{1}{V} \frac{dV}{dT} \] Substitute the expressions for \(V\) and \(\frac{dV}{dT}\): \[ \gamma = \frac{1}{CT^4} (4CT^3) \] \[ \gamma = \frac{4T^3}{T^4} = \frac{4}{T} \] Step 4: Final Answer:
The coefficient of volume expansion of the gas for this process is \(\frac{4}{T}\).