Question

Find out value of $C_p/C_y$ for an ideal gas undergoing reversible adiabatic process for which $P ∝ T^3$ is given

• A. 4/3
• B. 4/3
• C. 5/4
• D. 5/3

Answer 1

The Correct Option is B

An ideal gas undergoing a reversible adiabatic process is described by the equation relating pressure \(P\) and temperature \(T\) such that \(P \propto T^3\). To find the value of \(\frac{C_p}{C_v}\), we need to derive it using adiabatic process relations and the given proportionality.

For an adiabatic process, the relationship between pressure \(P\) and temperature \(T\) is given by:

\(P \propto T^3\) can be written as \(P = k \cdot T^3\), where \(k\) is a constant. 

In a reversible adiabatic process, the adiabatic condition is:

\(P V^\gamma = \text{constant}\)

We also have the ideal gas law:

\(P V = nRT\)

From \(P \propto T^3\) and combining it with the ideal gas equation, we write:

\(R = \frac{C_p - C_v}{\mu}\)

Where \(\gamma = \frac{C_p}{C_v}\) and P = k \cdot T^3 and ideal gas relations:

Since \(PV^\gamma = \text{constant}\)

\((\frac{T^\gamma}{T^(3(\gamma-1))}) = \text{constant}\).

Equate powers of \(T\) to give \(3(\gamma-1)=\gamma\) which simplifies to \(\gamma = \frac{4}{3}\).

Thus, the value of \(\frac{C_p}{C_v}\) is 4/3, hence, the correct answer is: 4/3.