Animated Solution for Physics - Thermodynamics: Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V}\propto T^4$ and pressure $p=\frac{1}{3}\left(\frac{U}{V}\right)$. If the shell now undergoes an adiabatic expansion, the relation between $T$ and $R$ is

Imagine a spherical shell filled not with ordinary gas molecules, but with a gas of photons—this is what we call black body radiation. The physics of a photon gas is beautifully unique. The problem provides us with two critical pieces of information. First, the energy density $u$ (which is the total internal energy $U$ divided by the volume $V$) is proportional to the fourth power of the temperature, $T^4$. Second, the radiation pressure $p$ exerted by these photons is exactly one-third of the energy density.

Let's formalize this mathematically. We can write the energy density as $u=k{T}^4$, where $k$ is a proportionality constant. Consequently, the total internal energy of the photon gas is $U=uV=k{T}^{4}V$. The pressure is simply $p=\frac{u}{3}=\frac{k{T}^4}{3}$.

The shell undergoes an adiabatic expansion. The word "adiabatic" is our cue that no heat is exchanged with the surroundings, meaning $dQ=0$.

According to the First Law of Thermodynamics, the heat added to a system equals the change in its internal energy plus the work done by the system:

Since $dQ=0$, our governing equation becomes:

Now, we need to substitute our expressions for $U$ and $p$ into the First Law. This is where we must be extremely careful. The internal energy $U=k{T}^{4}V$ depends on both temperature and volume, and both are changing during the expansion. Therefore, to find $dU$, we must apply the product rule of differentiation:

Next, we substitute $dU$ and $p=\frac{k{T}^4}{3}$ back into our First Law equation:

Notice that the constant $k$ appears in every term, so we can divide it out. Now, let's group the $dV$ terms together:

To solve this differential equation, we separate the variables $T$ and $V$. Dividing the entire equation by $4T^{4}V$ yields a remarkably clean expression:

Integrating both sides, we get:

Using the properties of logarithms, we can rewrite this as:

We are almost at the finish line. The volume of a spherical shell is given by $V=\frac{4}{3}\pi R^3$.

If we take the cube root of the volume, we see that $V^{1/3}$ is directly proportional to the radius $R$ ($V^{1/3}\propto R$). Substituting this proportionality into our derived relation $T{V}^{1/3}=\text{constant}$, we obtain:

Therefore, the temperature $T$ is inversely proportional to the radius $R$:

A word of caution: Many students (and even some solution manuals!) attempt to solve this problem by blindly applying the ideal gas law, $pV=nRT$. They substitute $p=\frac{U}{3V}$ and manipulate it to get the right answer. This is physically incorrect.

A photon gas is fundamentally different from an ideal gas of molecules. In a photon gas, the number of particles ($n$) is not conserved; photons are continuously created and absorbed by the walls of the container as the temperature changes. Because $n$ is not a constant, the equation $pV=nRT$ is entirely invalid. The rigorous and correct method is to always rely on the First Law of Thermodynamics, just as we did here.