Adiabatic Free Expansion and Temperature Change

1. Temperature Change in an Ideal Gas

Let's start with the simplest case: an ideal gas. From the kinetic theory of gases, one of the defining features of an ideal gas is that its internal energy $U$ is a function of temperature only. There are no potential energies associated with intermolecular forces. The relationship is:

$\Delta U=n{C}_v\Delta T$

where $n$ is the number of moles and $C_v$ is the molar specific heat at constant volume.

From our last lesson, we know that for an adiabatic free expansion, $\Delta U=0$. Combining these two facts gives us:

$n{C}_v\Delta T=0$

Since $n$ and $C_v$ are non-zero, it must be that:

$\Delta T=T_{final}-T_{initial}=0$

This is a key result: For an ideal gas undergoing an adiabatic free expansion, the initial and final equilibrium temperatures are identical.

Let's watch a short video that walks through this logic.

This concept is also concisely explained in the resource from Physics LibreTexts.

2. A Crucial Distinction: Not an Isothermal Process

We've established that $T_{initial}=T_{final}$ for an ideal gas. This might tempt you to call the process "isothermal." However, this is incorrect and a very important conceptual trap to avoid in JEE Advanced problems.

An isothermal process is one where the temperature of the system remains constant throughout the entire process. This implies the system is always in a state of thermal equilibrium.

Free expansion, by its nature, is a rapid, chaotic, and irreversible process. When the partition is removed, the gas rushes into the vacuum. During this expansion, the system is far from equilibrium.

• The molecules that have just crossed into the vacuum have higher forward kinetic energy.
• The molecules left behind might have slightly lower kinetic energy.
• Pressure and temperature are not uniform throughout the container; they are not well-defined for the system as a whole.

The ideal gas law, $PV=nRT$, only applies to systems in equilibrium. You cannot use it to describe the intermediate states of a free expansion. We can only speak of the temperature of the gas in the initial equilibrium state (before expansion) and the final equilibrium state (long after the expansion is complete and the gas has settled).

In summary:

• Adiabatic Free Expansion (Ideal Gas): Initial and final temperatures are equal ($T_i=T_f$), but $T$ is undefined during the process. It is not an isothermal process.
• Isothermal Expansion: Temperature is constant at all times ($T=\text{constant}$). This is a slow, quasi-static process where the system is always in equilibrium.

3. Contrasting Free Expansion with Reversible Adiabatic Expansion

Another common point of confusion is: "If the gas is expanding adiabatically, why doesn't it cool down?" You've likely learned that adiabatic expansions cause cooling.

The difference lies in work.

• Adiabatic Free Expansion ($W=0$): The gas expands against zero external pressure (a vacuum). It does no work on its surroundings. From the First Law ($\Delta U=Q-W$), with $Q=0$ and $W=0$, we get $\Delta U=0$. For an ideal gas, this means $\Delta T=0$. No energy is expended, so the temperature doesn't drop.

• Reversible (Quasi-static) Adiabatic Expansion ($W>0$): The gas expands slowly against an external pressure (like a piston). It does positive work on its surroundings. From the First Law, with $Q=0$, we have $\Delta U=-W$. Since $W>0$, the internal energy must decrease ($\Delta U<0$). For an ideal gas, this means the temperature drops ($\Delta T<0$). The gas cools because it is converting its own internal energy into work done on the surroundings.

This distinction is crucial. Let's see a video that beautifully illustrates this difference.

4. A Look at Real Gases

So far, we have focused on ideal gases. What happens during an adiabatic free expansion of a real gas?

For a real gas, internal energy is a function of both temperature and volume, $U=f(T,V)$. This is because intermolecular forces (like van der Waals forces) exist. The potential energy associated with these forces depends on the average distance between molecules, which changes with volume.

The First Law still tells us that $\Delta U=0$ for an adiabatic free expansion. However, the change in internal energy now has two components:
$\Delta U=(\text{change due to }\Delta T)+(\text{change due to }\Delta V)\approx n{C}_v\Delta T+{\left(\frac{\partial U}{\partial V}\right)}_T\Delta V=0$

When the gas expands ($\Delta V>0$), the molecules move further apart.

• For most gases under normal conditions, attractive forces dominate. Work must be done against these attractive forces to pull the molecules apart.
• This "internal work" is done at the expense of the molecules' kinetic energy.
• As the kinetic energy drops, the temperature of the gas decreases ($\Delta T<0$). This is known as the Joule effect.

So, for a real gas undergoing free expansion, we typically observe cooling.

The formula shown in the image is:
$\Delta T=-\frac{a{n}^2}{C_V}\left(\frac{1}{V_f}-\frac{1}{V_i}\right)$
(Note: The formula in the image might be simplified or presented differently, but the core idea is the same). Since $V_f>V_i$, the term in the parenthesis is negative, making the overall $\Delta T$ negative (cooling), consistent with our reasoning.

Conclusion

This lesson wraps up our exploration of the First Law by examining the subtleties of free expansion, a conceptually rich topic.

Key Takeaways:

• For an ideal gas undergoing adiabatic free expansion, the change in internal energy is zero ($\Delta U=0$), which implies the change in temperature is also zero ($\Delta T=0$).
• Despite $T_{initial}=T_{final}$, the process is not isothermal because it is rapid and irreversible, with temperature being undefined during the expansion.
• A reversible adiabatic expansion causes cooling ($\Delta T<0$) because the gas does work on its surroundings, expending its internal energy.
• For a real gas, adiabatic free expansion typically results in cooling ($\Delta T<0$) because the gas does "internal work" against intermolecular attractive forces.

Preview of the Next Module:
We have now covered the fundamentals of the First Law and its application to both well-defined cycles and irreversible processes. In the next module, "Thermodynamic Processes and Calculations," we will systematically analyze the four cornerstone quasi-static processes: isobaric, isochoric, isothermal, and adiabatic. You will learn to derive and apply formulas to calculate work, heat, and internal energy changes for each, building a robust toolkit for solving complex thermodynamic problems.