© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Problem
We are given that during an adiabatic process, the pressure of a gas, $P$, is proportional to the cube of its temperature, $T$. Symbolically:
$P \propto T^3$.
Step 2: Use the Adiabatic Relation
For an adiabatic process of an ideal gas, the following relation holds:
$P \propto T^{\frac{\gamma}{\gamma - 1}}$,
where $\gamma$ is the ratio of the specific heats, $\frac{C_p}{C_v}$.
Step 3: Equate the Exponents
We know that $P \propto T^3$ from the problem statement, and from the adiabatic relation, $P \propto T^{\frac{\gamma}{\gamma - 1}}$. Therefore, we set the exponents equal:
$$
\frac{\gamma}{\gamma - 1} = 3.
$$
Step 4: Solve for $ \gamma $
From
$$
\frac{\gamma}{\gamma - 1} = 3,
$$
we can solve as follows:
$$
\gamma = 3(\gamma - 1).
$$
This simplifies to
$$
\gamma = 3\gamma - 3 \quad \Rightarrow \quad 2\gamma = 3 \quad \Rightarrow \quad \gamma = \frac{3}{2}.
$$
Step 5: Conclude the Ratio $\frac{C_p}{C_v}$
Since $\gamma = \frac{C_p}{C_v}$, we have
$$
\frac{C_p}{C_v} = \frac{3}{2}.
$$
The correct answer is therefore $\frac{3}{2}$.
