Question

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

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

Answer 1

Answer: (C)

$\frac{3}{2}$

Answer 2

Given that:

$P \propto T^3,$

where $P$ is the pressure and $T$ is the absolute temperature.

Step 1: Using the Ideal Gas Law
From the ideal gas law, we have:

$\frac{PV}{T} = nR = \text{constant}.$

Therefore:

$P \propto \frac{T}{V}.$

Step 2: Relating Pressure and Temperature
Given that:

$P \propto T^3,$

we can write:

$P = kT^3,$

where $k$ is a proportionality constant.

Step 3: Applying the Adiabatic Process Equation
For an adiabatic process, the relation is given by:

$PV^\gamma = \text{constant},$

where $\gamma = \frac{C_P}{C_V}$ is the adiabatic index.

Step 4: Comparing the Relations
From the given proportionality:

$P \propto T^3 \quad \text{and} \quad P \propto V^{-\gamma}.$

Equating the exponents:

$\gamma = 3.$

Thus, the ratio of $\frac{C_P}{C_V}$ is:

$\frac{C_P}{C_V} = \gamma = \frac{7}{5}.$

Therefore, the correct answer is $\frac{7}{5}$.