Question

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of \(\frac{C_p}{C_v}\) for the gas is :

• A. \(\frac{5}{3}\)
• B. \(\frac{3}{2}\)
• C. \(\frac{7}{5}\)
• D. \(\frac{9}{7}\)

Answer 1

The Correct Option is B

To resolve this problem, we must establish the relationship between pressure \(P\), temperature \(T\), and the specific heat ratio \(\gamma = \frac{C_p}{C_v}\) under adiabatic conditions. The problem states that the gas pressure is directly proportional to the cube of its absolute temperature, which can be represented as:

\(P \propto T^3\)

For an adiabatic process, the pressure-temperature relationship is defined as:

\(PT^{-\frac{\gamma}{\gamma - 1}} = \text{constant}\)

Given \(P \propto T^3\), we equate the exponents of \(T\):

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

\(3 - \frac{\gamma}{\gamma - 1} = 0\)

Solving the equation yields:

\(3 = \frac{\gamma}{\gamma - 1}\)

\(3(\gamma - 1) = \gamma\)

\(3\gamma - 3 = \gamma\)

\(2\gamma = 3\)

\(\gamma = \frac{3}{2}\)

Thus, the specific heat ratio \(\frac{C_p}{C_v}\) is \(\gamma = \frac{3}{2}\).

The correct answer is:

\(\frac{3}{2}\)

Analysis of Options:

• The option \(\frac{5}{3}\) is characteristic of diatomic gases with additional energy degrees, which is not supported by the given condition.
• The option \(\frac{3}{2}\) was derived directly from the problem's stated condition and is the correct value.
• The option \(\frac{7}{5}\) is also typically associated with diatomic gases and does not align with the problem's stated condition.
• The option \(\frac{9}{7}\) does not correspond to the derived temperature-proportional relationship.