### Video Transcript

An ideal monatomic gas at a temperature of 542 kelvin expands adiabatically and reversibly to 4.00 times its volume. What is the gasβs final temperature?

Considering this problem statement, one of the key things weβre told is that when the gas expands, it does so adiabatically, that is, without exchanging heat with the system. In the case of an expansion like this, we can say that the temperature of the gas multiplied by the volume raised to the power of πΎ minus one β weβll explain what πΎ is in a second β is equal to a constant.

This means that if we have an initial state, that is, before the expansion, and a final state, after the expansion, we can write that π sub π times π sub π to the πΎ minus one is equal to π sub π times π sub π to the πΎ minus one.

Now about this factor πΎ, this factor πΎ is equal to a ratio: the specific heat of the gas at a constant pressure, π, to the specific heat of the gas at a constant volume, π. The fact that our ideal gas is monatomic in this example implies a particular value for πΎ. It implies that the ratio πΆ sub π to πΆ sub π is equal to five-thirds.

Going back to our expression for temperature then, we want to solve for the final temperature of the system, π sub π. Rearranging, thatβs equal to π sub π times the ratio π sub π over π sub π, all raised to the πΎ minus one power.

In the problem statement, weβre told that π sub π is 542 kelvin and that π sub π is four times greater than π sub π. In other words, π sub π over π sub π is one-quarter. Since we also know the value for πΎ, weβre ready to plug in and solve for π sub π. Five-thirds minus one is two-thirds. And when we plug this expression into our calculator, to three significant figures, we find that π sub π is 215 kelvin. Thatβs the final temperature of the gas.