Animated Solution for Physics - Thermodynamics: When heat $Q$ is supplied to a diatomic gas of rigid molecules, at constant volume, its temperature increases by $\Delta T$. The heat required to produce the same change in temperature, at a constant pressure is

Imagine you have a certain amount of gas trapped in a cylinder. You want to heat it up so that its temperature rises by a specific amount, say $\Delta T$. But here is the catch: you can do this in two different ways.

In the first scenario, you lock the piston in place. The volume of the gas cannot change. When you supply heat $Q$, all of that energy goes directly into making the gas molecules jiggle faster, which increases the internal energy and thus the temperature. Mathematically, we write this as:

where $C_V$ is the molar heat capacity at constant volume.

Now, consider the second scenario. This time, the piston is free to move, maintaining a constant pressure. As you heat the gas, it wants to expand. To push the piston up against the external atmospheric pressure, the gas must do work.

This means the heat you supply, let's call it $Q^{\prime }$, has to do double duty: it must increase the internal energy by the exact same amount as before (to achieve the same $\Delta T$), AND it must provide the energy for the expansion work. Therefore, you will inevitably need more heat. The equation for this process is:

where $C_p$ is the molar heat capacity at constant pressure.

The problem asks for the new heat $Q^{\prime }$ in terms of the original heat $Q$. The most elegant way to find this is to take the ratio of the two equations. Watch how beautifully the common terms cancel out:

The number of moles $n$ and the temperature change $\Delta T$ vanish, leaving us with:

This ratio of specific heats is so important in thermodynamics that it has its own special symbol: $\gamma$ (gamma). So, $Q^{\prime }=\gamma Q$.

The final piece of the puzzle lies in the nature of the gas. The problem specifies a diatomic gas of rigid molecules (like $O_2$ or $N_2$ at room temperature). "Rigid" means the molecules can translate and rotate, but they don't vibrate. This gives them 5 degrees of freedom ($f=5$).

For any ideal gas, $\gamma$ is related to the degrees of freedom by the formula $\gamma =1+\frac{2}{f}$. Plugging in $f=5$, we get:

Substituting this back into our heat equation, we arrive at the final answer:

It takes 40% more heat to achieve the same temperature rise when the gas is allowed to expand at constant pressure!