Animated Solution for Physics - Thermodynamics: For an ideal gas

Unraveling the Thermodynamics of an Ideal Gas

Imagine you are the ultimate accountant, but instead of money, you are tracking energy. The universe is your ledger, and the First Law of Thermodynamics is your golden rule. In this thrilling exploration, we are going to dissect the behavior of an ideal gas across various thermodynamic journeys. We aren't just going to memorize formulas; we are going to feel the physics behind every expansion, compression, and temperature shift.

Let's tackle the first statement. When we talk about the internal energy of an ideal gas, we are talking about a state function. What does that mean? It means the gas has no memory! It doesn't care if it was heated at constant pressure, compressed adiabatically, or taken on a wild, squiggly path on a $p-V$ diagram.

For an ideal gas, there are no intermolecular forces. The molecules are just tiny, perfectly elastic spheres zipping around. Therefore, there is no potential energy between them. The entire internal energy is purely kinetic, and kinetic energy is directly tied to one thing: absolute temperature.

This brings us to the master equation:

This equation is universally true for an ideal gas, regardless of the process. Even if the process is isobaric (constant pressure), the change in internal energy is still calculated using $C_V$. Why? Because $C_V$ is fundamentally linked to the degrees of freedom of the gas, which dictates how energy is stored. So, statement (a) is a resounding truth.

Now, let's look at the adiabatic process. The word "adiabatic" comes from the Greek "adiabatos," meaning "impassable." In this process, the system is wrapped in a perfect thermal blanket. No heat can enter, and no heat can escape.

By definition, this means:

This immediately proves statement (d) to be correct. But let's dig deeper. What happens to our energy ledger, the First Law of Thermodynamics?

Since $\Delta Q=0$, the equation simplifies beautifully to:

What does this tell us physically? If the gas expands and does positive work ($\Delta W>0$), it must pay for that work using its own internal energy ($\Delta U<0$), causing it to cool down. Conversely, if we compress the gas ($\Delta W<0$), that work goes directly into increasing the internal energy ($\Delta U>0$), heating it up.

If we take the magnitude of both sides, we get:

This confirms that statement (b) is absolutely correct. The change in internal energy and the work done are perfectly balanced in magnitude.

Finally, let's examine the isothermal process. "Iso" means same, and "thermal" refers to temperature. In an isothermal journey, the gas is in perfect thermal contact with a massive heat reservoir, ensuring its temperature never wavers.

If the temperature is constant, then:

And since we've already established that the internal energy of an ideal gas is solely a function of temperature, it must follow that:

The internal energy remains perfectly constant. Any heat added to the system goes entirely into doing work ($\Delta Q=\Delta W$). Thus, statement (c) is also correct.

Thermodynamics is not just a collection of random equations; it is a beautifully consistent logical framework. By understanding the core definitions of internal energy and the First Law, we can effortlessly navigate through any process. All four statements provided in the question are correct, making this a perfect conceptual checklist for mastering ideal gas behavior!