Animated Solution for Physics - Thermodynamics: Comprehension Passage

Imagine a perfectly insulated cylinder divided equally by a movable partition. Both sides start in perfect harmony: same temperature, same volume, same pressure. But then, we turn on a heater in the left chamber. This is where the physics begins.

A Reversible Adiabatic Journey

As the left gas heats up, it expands, pushing the partition to the right. Notice that the partition is explicitly stated to be insulating and moves very slowly. What does this mean for the right gas? It's being compressed without any heat exchange. This is a classic reversible adiabatic process.

To analyze this adiabatic compression, we first need the adiabatic exponent, $\gamma$. We are given the specific heat at constant volume as $C_V=2R$. Using the fundamental relation $\gamma =1+\frac{R}{C_V}$, we find that $\gamma =1+\frac{R}{2R}=\frac{3}{2}$.

Now, let's connect the initial and final states of the right gas. For an adiabatic process, the product of temperature and volume to the power $\gamma -1$ is constant:

We substitute the initial volume $V_0$ and the final volume $V_R=\frac{V_0}{2}$:

Solving this, the $V_0$ terms cancel out beautifully, leaving us with $T_R=\sqrt{2}{T}_0$. So, the ratio of $T_R$ to $T_0$ is simply $\sqrt{2}$. That's our answer for the first part of the problem!

Mechanical Equilibrium

Let's shift our focus to the left gas. Because the partition moves slowly, the forces on it are always balanced. This means the pressure on the left side is always equal to the pressure on the right side. They reach the same final pressure, $P_f$.

To find this final pressure, we can again use the right gas. For an adiabatic process, pressure times volume to the power $\gamma$ is constant:

Substituting the volumes, we find the final pressure:

Now we have everything we need for the left gas. The total volume is fixed at $2V_0$, so the left gas must occupy a volume of $V_L=2V_0-\frac{V_0}{2}=\frac{3V_0}{2}$. Using the ideal gas equation, we can set up a ratio to find its final temperature:

Substituting the values of final pressure and final volume:

Notice how much hotter it is compared to the right side!

The First Law of Thermodynamics

We are finally ready to find the total heat supplied, $Q$. The smartest way is to apply the First Law of Thermodynamics to the entire system as a whole. Since the outer cylinder is rigid, the total work done by the combined gases is zero ($W_{total}=0$). So, all the heat goes into changing the total internal energy:

Let's calculate the change in internal energy for both sides. For an ideal gas, it's simply $n{C}_V\Delta T$. We plug in the temperatures we just found:

Adding them up, we combine the $\sqrt{2}$ terms and the constant terms:

Factoring out a $2$, we arrive at our final expression:

The ratio $\frac{Q}{R{T}_0}$ is exactly $4(2\sqrt{2}-1)$. We've elegantly solved a complex thermodynamic puzzle by breaking it down into manageable, logical steps!