Animated Solution for Physics - Thermodynamics: One kg of a diatomic gas is at a pressure of $8\times 10^4{\text{ Nm}}^{-2}$. The density of the gas is $4{\text{ kgm}}^{-3}$. What is the energy of the gas due to its thermal motion?

Imagine a sealed container holding exactly $1\text{ kg}$ of a diatomic gas. The gas is exerting a pressure of $8\times 10^4{\text{ N/m}}^2$ on the walls of the container, and its density is given as $4{\text{ kg/m}}^3$.

Our goal is to find the energy of the gas due to its thermal motion. For an ideal gas, the energy associated with the random, chaotic thermal motion of its molecules is entirely represented by its internal energy ($U$). There are no intermolecular potential energies to worry about.

According to the kinetic theory of gases and the law of equipartition of energy, the internal energy of an ideal gas is given by:

where $f$ is the degree of freedom, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature.

However, looking at our given data, we don't have the temperature $T$ or the number of moles $n$. But we do have the pressure $p$. This is where the ideal gas equation comes to our rescue! We know that:

By substituting $pV$ for $nRT$ in our internal energy equation, we get a much more useful form for this specific problem:

To use our new equation, we need three things: the degree of freedom ($f$), the pressure ($p$), and the volume ($V$).

The problem states that we are dealing with a diatomic gas. A diatomic molecule (like $O_2$ or $N_2$) looks like a tiny dumbbell. At normal room temperatures, it can move in 3 independent directions (translational degrees of freedom) and rotate about 2 independent axes perpendicular to the bond connecting the atoms (rotational degrees of freedom). Therefore, the total degree of freedom is:

We are given the mass ($m=1\text{ kg}$) and the density ($\rho =4{\text{ kg/m}}^3$). Since density is mass per unit volume ($\rho =\frac{m}{V}$), we can easily find the volume:

Now we have everything we need! Let's substitute the values into our master equation:

Notice how beautifully the numbers align. The $2$ and the $4$ in the denominators multiply to $8$, which perfectly cancels out the $8$ in the numerator:

And there we have it! The thermal energy of the gas is $5\times 10^4\text{ J}$.