### Video Transcript

A 12-liter balloon contains 0.52 moles of helium gas. A second balloon at the same temperature and pressure has a volume of 18 liters. How many moles of helium gas does the second balloon contain? Give your answer to two decimal places.

So, the first balloon has a volume of 12 liters. We can represent volume by a capital ๐. The balloon contains 0.52 moles of helium gas, where number of moles can be represented by a lowercase ๐. A second balloon at the same temperature and pressure has a volume of 18 liters. We want to find out how many moles of helium gas this balloon contains. The volume and number of moles of a gas are related by the equation ๐ equals ๐๐ m, where ๐ is the volume in liters, ๐ is the number of moles, and ๐ subscript m is the molar volume in liters per mole.

The molar volume is the volume occupied by one mole of a gas at a particular temperature and pressure. Since both balloons contain the same gas, helium, and theyโre at the same temperature and pressure as stated in the question, then the molar volume for the gases in both balloons will be the same. Letโs calculate the molar volume for the first balloon.

First, we need to rearrange the equation a bit. We want to make the molar volume the subject. If we divide both sides of the equation by ๐, the number of moles, then the ๐ terms on the right-hand side of the equation will cancel, leaving us with the equation ๐ over ๐ equals ๐ m. We have been given the values for ๐, the volume, and ๐, the number of moles, in the question. So, to find out the molar volume for the first balloon, we need to do 12 liters divided by 0.52 moles. This gives a value of 23.1 liters per mole to three significant figures. I will be quoting to three significant figures for simplicity, but for this type of question, you should use the full unrounded value for all parts of the calculation.

The molar volume for the second balloon will also be 23.1 liters per mole. And weโve been given the volume of the balloon in the question. We want to work out the number of moles of helium gas, so we need to rearrange the equation to make ๐ the subject. If we divide both sides of the equation by ๐ subscript m, which is the molar volume, then the molar volume terms on the right-hand side of the equation will cancel, leaving us with ๐ divided by ๐ m equals ๐.

The question tells us that ๐ is 18 liters. And we know that the molar volume is 23.1 liters per mole. If we divide 18 liters by the molar volume but ensuring to use the full unrounded number, then we calculate the number of moles of helium gas to be 0.78 moles. The question asks for our answer to two decimal places, which we already have. So, the answer to the question, โhow many moles of helium gas does the second balloon contain?โ is 0.78 moles.

We can check our answer by thinking about Avogadroโs law. Avogadroโs law states that at a constant temperature and pressure, the volume and number of moles of a gas are directly proportional. The question tells us that the balloons are at the same temperature and pressure. So, Avogadroโs law applies here. If we divide the volume of the second balloon by the volume of the first, then we find out that the second balloon is 1.5 times larger than the first balloon.

Since the volume and number of moles of a gas are directly proportional, then the second balloon should contain 1.5 times as many moles as the first balloon. If we divide 0.78 by 0.52, we find that it equals 1.5. So, the volume and number of moles of the gas are directly proportional. Therefore, we can be confident that the answer to the question, โhow many moles of helium gas does the second balloon contain?โ is 0.78 moles.