Animated Solution for Physics - Thermodynamics: One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K, respectively. Choose the correct statement.

Unlocking Internal Energy in Cyclic Processes

Imagine you are navigating a complex maze. You take twists, turns, and loops, but eventually, you end up exactly where you started. In thermodynamics, this is what we call a cyclic process. The beauty of such a journey lies in a magical property called a state function.

In this problem, we are given a cyclic process $A\to B\to C\to A$ for one mole of a diatomic ideal gas. We know the temperatures at the three checkpoints: $T_A=400\text{ K}$, $T_B=800\text{ K}$, and $T_C=600\text{ K}$. The question asks us to identify the correct statement regarding the change in internal energy ($\Delta U$) for various parts of the cycle.

Before we dive into the calculations, let's equip ourselves with the ultimate tool for internal energy. For an ideal gas, the change in internal energy is strictly a function of temperature. It does not care whether the process is isobaric, isochoric, or adiabatic. The formula is always:

Since we are dealing with a diatomic gas, its molar heat capacity at constant volume is $C_V=\frac{5R}{2}$. We also know that $n=1$ mole.

Let's systematically evaluate each option by applying our master equation.

Option (a): The whole cyclic process For any complete cycle, the gas returns to its initial state. This means the initial and final temperatures are identical, so $\Delta T=0$. Consequently, $\Delta U_{\text{cycle}}=0$. Option (a) claims it is $250R$, which is completely incorrect.

Option (b): Process $C\to A$ Let's calculate the change in internal energy as the gas moves from state C to state A.

Substituting the values:

Option (b) says $700R$, so it is incorrect.

Option (c): Process $A\to B$ Now, let's check the journey from A to B.

Option (c) claims $-350R$, which is also wrong.

Option (d): Process $B\to C$ Finally, let's evaluate the process from B to C. The problem mentions that this process is adiabatic. While that's an interesting physical detail (meaning no heat $Q$ is exchanged), it is actually a distractor for calculating $\Delta U$! We just need the temperatures.

Option (d) states exactly $-500R$. We have found our winner!

This problem is a classic test of conceptual clarity. Examiners often throw in extra information—like specifying that a process is adiabatic—to see if you will abandon your fundamental principles. Always remember: for an ideal gas, $\Delta U$ is blindly loyal to temperature. If you know the initial and final temperatures, you hold the key to the internal energy.