Question

During an adiabatic process the cube of the pressure of a gas is proportional to the fifth power of its volume. The ratio of specific heat $C_p/C_v$ for that gas is

• A. 3/5
• B. 4/3
• C. 3/2
• D. 5/3

Answer 1

Answer: (D)

5/3

Answer 2

The problem states that during an adiabatic process, the cube of the pressure of a gas is proportional to the fifth power of its volume. We need to find the ratio of specific heats $C_p/C_v$ for this gas.

Let P be the pressure and V be the volume of the gas.
According to the problem statement, $P^3 \propto V^5$.
This can be written as $P^3 = k V^5$, where k is a constant.
Rearranging this, we get $P^3 V^{-5} = k$.

The general equation for an adiabatic process is $PV^\gamma = \text{constant}$, where $\gamma = C_p/C_v$.

If we compare $P^3 V^{-5} = k$ with the standard adiabatic equation, we first need to make the pressure term match. Let's raise the standard adiabatic equation to the power of 3:
$(PV^\gamma)^3 = (\text{constant})^3$
$P^3 V^{3\gamma} = \text{constant}'$

Now, comparing $P^3 V^{-5} = k$ with $P^3 V^{3\gamma} = \text{constant}'$, we would get $3\gamma = -5$, which implies $\gamma = -5/3$. This value of $\gamma$ is not physically possible for a gas, as $\gamma$ must be positive and greater than 1 (since $C_p > C_v$).

In such physics problems, when "proportional to" is used, and it leads to a non-physical result, it often implies an inverse proportionality. Given the options, it is highly probable that the intended meaning is that the cube of the pressure is proportional to the inverse fifth power of its volume, or more simply, the product of the cube of the pressure and the fifth power of the volume is constant.
So, we assume the relationship is $P^3 V^5 = \text{constant}$.

Now, let's convert this assumed relation to the standard adiabatic form $PV^\gamma = \text{constant}$.
We have $P^3 V^5 = \text{constant}$.
To get P to the power of 1, we take the cube root of both sides:
$(P^3 V^5)^{1/3} = (\text{constant})^{1/3}$
$P^{(3 \times 1/3)} V^{(5 \times 1/3)} = \text{new constant}$
$P^1 V^{5/3} = \text{new constant}$

Comparing this equation, $P V^{5/3} = \text{new constant}$, with the standard adiabatic equation $PV^\gamma = \text{constant}$, we can directly identify the value of $\gamma$:
$\gamma = 5/3$

This value is physically plausible, as $\gamma = 5/3$ corresponds to a monatomic gas.