Internal Energy Change in Ideal Gas Processes

1. The Nature of Internal Energy in an Ideal Gas

First, what is internal energy? It's the sum of all microscopic energies within a system: the kinetic energy of particles (translation, rotation, vibration) and the potential energy from intermolecular forces.

A key assumption of the ideal gas model is that there are no intermolecular forces. This means the potential energy term is zero. Therefore, for an ideal gas, the internal energy $U$ is simply the sum of the kinetic energies of all its molecules.

Since the average kinetic energy of gas particles is directly proportional to the absolute temperature, it follows that the internal energy of an ideal gas is a function of its temperature only.

2. Quantifying Internal Energy: Degrees of Freedom

To derive a concrete formula for $U$, we need to know how many ways a molecule can possess kinetic energy. These are called degrees of freedom (f). The Law of Equipartition of Energy states that, at thermal equilibrium, the total energy is distributed equally among all degrees of freedom.

This short, animated video provides a clear and concise explanation of degrees of freedom and how they lead to the formula for internal energy.

Let's summarize the key results. The energy associated with each degree of freedom per mole is $\frac{1}{2}RT$. For $n$ moles of a gas with $f$ degrees of freedom, the total internal energy is:

For JEE problems, you should know these values of $f$ (ignoring vibrational modes, which are only active at very high temperatures):

• Monatomic gas (e.g., He, Ne, Ar): 3 translational d.o.f. $⟹f=3$
• Diatomic gas (e.g., O₂, N₂, H₂): 3 translational + 2 rotational d.o.f. $⟹f=5$
• Polyatomic non-linear gas (e.g., H₂O, CH₄): 3 translational + 3 rotational d.o.f. $⟹f=6$
• Polyatomic linear gas (e.g., CO₂): Same as diatomic. $⟹f=5$

3. The Universal Formula: ΔU = nC_VΔT

Now we arrive at the central point of the lesson. The formula $\Delta U=n{C}_V\Delta T$ is often introduced in the context of an isochoric (constant volume) process. But its true power lies in its universality for ideal gases.

This often leads to confusion: how can a formula with $C_V$ (specific heat at constant volume) be used when the volume is changing? The answer lies in the fact that $U$ is a state function.

Let's synthesize the argument:

1. As we established, for an ideal gas, internal energy $U$ depends only on temperature T. It does not depend on pressure or volume independently.
2. A state function is a property whose value depends only on the current state of the system, not on the path taken to reach that state. Since $U$ depends only on $T$, it is a state function.
3. This means the change in internal energy, $\Delta U$, between an initial state (Tᵢ) and a final state (T₟) is always the same, regardless of the thermodynamic path (isobaric, adiabatic, anything) connecting them.

So, to calculate $\Delta U$ between two temperatures, we can choose the most convenient path for our calculation—an isochoric (constant volume) path.
For an isochoric process:

• From the First Law: $\Delta U=Q_V-W$. Since $V$ is constant, $W=0$, so $\Delta U=Q_V$.
• By definition of molar heat capacity at constant volume, $Q_V=n{C}_V\Delta T$.
• Therefore, $\Delta U=n{C}_V\Delta T$.

Since $\Delta U$ is the same for any path between the same initial and final temperatures, this formula is universally true for any process involving an ideal gas.

Connecting $C_V$ to Degrees of Freedom

We can link our two formulas for $\Delta U$:

Comparing them immediately gives a fundamental relationship:

This is an extremely useful result for problem-solving, as it allows you to calculate $C_V$ (and by extension $C_P=C_V+R$ and $\gamma =C_P/C_V$) just by knowing the atomicity of the gas.

4. Application in Problem Solving

Let's apply this theory to some JEE-style problems. The following video clips demonstrate how to calculate internal energy and its change in typical scenarios, including for a mixture of gases.

The first problem with the gas mixture highlights an important point: the total internal energy of a mixture is the sum of the internal energies of its components.
$U_{\text{mixture}}=U_1+U_2=\frac{f_1}{2}{n}_{1}RT+\frac{f_2}{2}{n}_{2}RT$

Conclusion

Today we've demystified the internal energy term in the First Law for an ideal gas. By going back to first principles, you can now handle a wide variety of problems with confidence.

Key Takeaways:

• For an ideal gas, internal energy $U$ depends only on temperature, making it a state function.
• The formula $\Delta U=n{C}_V\Delta T$ is universally applicable for any process involving an ideal gas, because the change in a state function is path-independent.
• The most fundamental expression for internal energy change is $\Delta U=\frac{f}{2}nR\Delta T$, which directly links macroscopic energy changes to the microscopic structure (degrees of freedom, $f$) of the gas molecules.
• From these, we derived the crucial relationship $C_V=\frac{f}{2}R$.

Preview of the Next Lesson:
We have now mastered the calculation of both work done ($W$) and internal energy change ($\Delta U$). In the next lesson, we will put everything together. You will learn to apply the full First Law equation, $\Delta U=Q-W$, to solve for any of the three unknown quantities ($Q$, $W$, or $\Delta U$) in a single thermodynamic process, solidifying your problem-solving toolkit.