### Video Transcript

The temperature of an ideal monatomic gas rises by 8.0 Kelvin, while the volume of the gas remains constant. What is the change in the internal energy per mole of the gas?

So in this question, weβve got an ideal monatomic gas. And its temperature rises by 8.0 Kelvin, while the volume of the gas remains constant. We need to find the change in the internal energy per mole of the gas.

Letβs start by recalling the first law of thermodynamics. The change in internal energy, Ξπ’, is equal to π, the heat supplied, plus π, the work done. And we can apply this to our gas. The change in internal energy of the gas is equal to the heat supplied to the gas plus the work done on the gas.

However, when heat is supplied at a constant volume, all of the heat supplied to a gas goes towards increasing the temperature. In other words, the work done is equal to zero. And the change in internal energy of the gas is equal to the heat supplied to the gas.

So basically, what this is telling us is that we donβt need to worry about any work being done on the gas. All we need to worry about is the change in internal energy due to the heat supplied to the gas. And that is the relevance of the constant volume bit.

Now we can recall that the internal energy of an ideal monatomic gas is given by πΈ sub int, the internal energy, is equal to three by two times ππ
π. The lowercase π is the number of moles of gas that we have. π
is simply the molar gas constant. And π is the temperature of the gas.

In this question, weβre trying to find the change in internal energy per mole of the gas. The internal energy per mole is given by the total internal energy of the gas, πΈ sub int, divided by π, the number of moles of gas that we have. And so thatβs just three by two ππ
π divided by π. The πs cancel out, leaving us with three by two π
π.

But weβre not just trying to find out the internal energy per mole of the gas. Weβre trying to find out the change in the internal energy per mole of the gas when the temperature rises by 8.0 Kelvin. This change in internal energy per mole is given by πΈ sub int over π sub final, the final internal energy after the temperature of the gas has risen by 8.0 Kelvin, minus πΈ sub int over π sub initial, the initial internal energy per mole of the gas before the temperature has risen.

And on the right-hand side, we can substitute in three by two π
π in each case. However, we know that three by two and π
are just constant. They do not change over time. So we can write the right-hand side of the expression like this. In other words, weβve pulled out three by two π
in each case because they donβt change. And at this point, we can actually factorize, which gives us that the right-hand side is equal to three by two π
multiplied by π sub final minus π sub initial.

In other words, this bracket is equal to the change in temperature. And we know that that change in temperature, Ξπ, is 8.0 kelvin because weβve been told this in the question. So our final mathematical expression for the change in internal energy per mole of the gas is three by two π
multiplied by Ξπ. At this point, we can substitute in the values, which gives us a value of 99.72 dot dot dot, so on and so forth, joules.

Now the value that weβve been given in the question and used in our calculation of 8.0 kelvin has been given to us to two significant figures. So we need to give our final answer to two significant figures as well. And so our final answer is that ΞπΈ sub int over π, the change in internal energy per mole of a gas, is 100 joules to two significant figures.