Animated Solution for Physics - Thermodynamics: Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

Thermodynamics is not just about plugging numbers into formulas; it is about understanding the story of a gas as it undergoes various transformations. In this matrix match question, we are presented with four distinct thermodynamic processes. Our goal is to determine how the temperature changes and whether the gas absorbs or releases heat in each scenario. Let's dive into the physics behind each process!

The Free Expansion Anomaly

Imagine an insulated container divided into two chambers by a valve. One chamber holds an ideal gas, while the other is a perfect vacuum. When the valve is opened, the gas rushes into the vacuum. This is known as free expansion.

Because the container is insulated, no heat can enter or leave the system, meaning $Q=0$. Furthermore, since the gas is expanding into a vacuum, it does not push against any external pressure. Therefore, the work done by the gas is also zero ($W=0$).

According to the First Law of Thermodynamics, $\Delta U=Q-W$. With both $Q$ and $W$ being zero, the change in internal energy $\Delta U$ is exactly zero. For an ideal gas, internal energy is solely a function of temperature ($\Delta U=n{C}_V\Delta T$). Thus, the temperature of the gas remains perfectly constant.

The Steep Polytropic Curve

Next, we encounter a gas expanding such that its pressure follows the relation $p\propto \frac{1}{V^2}$. This can be rewritten as $p{V}^2=\text{constant}$.

To find out what happens to the temperature, we use the ideal gas law, $p=\frac{nRT}{V}$. Substituting this into our process equation gives $\left(\frac{nRT}{V}\right)V^2=\text{constant}$, which simplifies to $T\propto \frac{1}{V}$. As the gas expands to twice its original volume, the volume $V$ increases, which dictates that the temperature $T$ must decrease.

But what about the heat exchange? This is a polytropic process of the form $p{V}^x=\text{constant}$, where $x=2$. The molar heat capacity for a polytropic process is given by the master equation:

For a monoatomic gas, $C_V=\frac{3}{2}R$. Plugging in $x=2$:

Notice that the molar heat capacity $C$ is positive. Since the temperature decreased ($\Delta T<0$), the heat exchanged $Q=nC\Delta T$ must be negative. A negative $Q$ signifies that the gas loses heat to its surroundings.

The Magic of Negative Heat Capacity

Process C introduces a fascinating scenario: $p\propto \frac{1}{V^{4/3}}$, or $p{V}^{4/3}=\text{constant}$.

Using the same substitution from the ideal gas law, we find that $T\propto \frac{1}{V^{1/3}}$. Once again, as the volume increases, the temperature must decrease.

The real magic happens when we calculate the heat capacity. Here, the polytropic exponent is $x=\frac{4}{3}$.

The heat capacity is negative! What does this mean physically? Since the temperature decreased ($\Delta T<0$), the heat exchanged is $Q=n(-1.5R)(-\text{value})>0$. The heat $Q$ is positive! This means the gas actually gains heat while its temperature drops. The gas is doing so much expansion work that the heat supplied isn't enough to maintain its temperature, forcing it to consume its own internal energy.

The Linear Expansion

Finally, we look at a linear $p-V$ graph with a positive slope. As the volume increases from $V_1$ to $2V_1$, the pressure $p$ also visibly increases.

From the ideal gas law, $T=\frac{pV}{nR}$. Since both pressure and volume are increasing simultaneously, their product $pV$ is definitely increasing. Therefore, the temperature of the gas must increase.

Because the temperature increases, the change in internal energy $\Delta U$ is positive. Additionally, since the gas is expanding, the work done $W$ (the area under the $p-V$ curve) is also positive. By the First Law of Thermodynamics ($Q=\Delta U+W$), the sum of two positive quantities is positive. Thus, $Q>0$, and the gas gains heat.