Gas Volumes in the Mole Map

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. I have two balloons on my table, exactly the same size, at the same temperature and the same pressure. One is filled with methane, and the other with carbon dioxide. Now, one mole of methane weighs sixteen grams, and one mole of carbon dioxide weighs forty four grams. Almost three times heavier, right? So tell me: which balloon has more molecules? Methane, because its molecules are lighter? Hmm, think carefully. Today we will find the answer, and it may surprise you. And look, in JEE, gases come up again and again in mole concept questions. So, let's begin.

0:43Avogadro's law: one mole, one volume

So, look at the mole map on screen. Small n, the number of moles, sits in the middle, like a hub. From mass, you divide by the molar mass. From particles, you divide by N A, the Avogadro constant. Today we take the third road, the one going down: gas volume. See the arrows marked twenty two point four litres? Soon you will know exactly what they mean. First, a question. Carbon dioxide molecules are heavier than methane molecules. So does one mole of carbon dioxide need more room than one mole of methane? No, and this is the surprise. At the same temperature, T, and the same pressure, P, one mole of any ideal gas occupies the same volume. This is Avogadro's law. This volume of one mole has a special name: the molar volume, written V m, a capital V with a small m. Same temperature and pressure, same V m for every gas. Clear? Now, if one mole fills V m, how many moles fill a volume V? Simple: divide! n equals V divided by V m. On the map, going down from moles to volume, you multiply by V m; coming back up, you divide by it. Really simple. And now look what this means for our balloons. Same volume, same temperature, same pressure, so the same V m, and so the same n. And the same number of moles means the same number of molecules! It's a tie! And in example two, we will actually count how many molecules sit in just one millilitre of gas.

2:22Which molar volume? Read the question

Alright, so what is V m in numbers? At zero degree Celsius and one atmosphere, which we call STP, V m is twenty two point four litres per mole. On screen, L mol to the power minus one simply means litres per mole, okay? Now, a JEE trap. Is one mole of gas always twenty two point four litres? Most students say, yes, always! No, no, careful. See, it depends on the conditions. At zero degree Celsius and one bar, V m is twenty two point seven litres per mole. The two answers differ only in the third figure, but JEE options are often that close. So if a question gives you twenty two point seven, use twenty two point seven, not your old habit, okay? And if the question says nothing? Use twenty two point four litres per mole. That's it. But always read the conditions first. Got it? And for other temperatures and pressures, you will meet P V equals n R T in chapter six. For now, these two values are all you need.

3:31Density and molar mass of a gas

Now, back to our two balloons. Same volume, same number of molecules. So, do they weigh the same as well? No, no! Each carbon dioxide molecule is heavier, so the carbon dioxide balloon is heavier. Same volume, different mass. That means a different density. So let's find the density of a gas. Take exactly one mole. Its mass is the molar mass, capital M, and its volume is V m. Density is mass divided by volume. So rho equals capital M divided by V m. This curly letter is rho, the Greek letter we use for density, okay? Since V m is the same for every gas, the bigger the molar mass, the denser the gas. That is why our carbon dioxide balloon is heavier. Now turn it around. Suppose you weigh a gas, and small m grams of it fill a volume V, okay? The mole map gives you two roads to n: small m by capital M, and V by V m. Both roads reach the same n, right? So they must be equal. Rearrange, and capital M equals small m by V, times V m. Here is the topper's trick: small m by V is just the density. So the molar mass is simply density times V m. Weigh a known volume of a gas, and you know its molar mass. Very powerful, isn't it?

4:50Example 1: molar mass of a gas from 5.6 L at STP

Let's do example one. Five point six litres of a gas at STP weighs seven point zero grams. Take V m as twenty two point four litres per mole. Find its molar mass. Pause the video, and try it first. Done? Okay, let's check. Many students do this: molar mass is mass divided by volume, so seven divided by five point six, right? No, no, careful! That gives grams per litre, which is a density. Molar mass is grams per mole. Always check the unit. So, convert to moles first. n equals five point six divided by twenty two point four. Look, five point six is exactly one fourth of twenty two point four, so n is zero point two five moles. Now, capital M equals mass divided by moles: seven point zero divided by zero point two five. Dividing by one fourth is the same as multiplying by four. Twenty eight grams per mole. And a quick check with our formula: seven point zero by five point six, times twenty two point four. Again, twenty eight. Absolutely correct. So the gas could be nitrogen, N two, or carbon monoxide, C O, okay? Both have molar mass twenty eight.

6:11Example 2: molecules in 1 mL of gas, Loschmidt number

Now a slightly harder one, JEE level. Remember our balloons? Now we actually count molecules. How many molecules are there in one millilitre of any gas at STP? Take V m as twenty two point four litres per mole. Pause and try. Hint: watch the units. Done? Okay, let's check. Moles equal volume divided by V m, so one divided by twenty two point four, right? No, no, careful! This is the JEE trap. The volume is in millilitres, but V m is in litres. Mix them, and your answer is a thousand times too big. So first, make the units match. One litre is a thousand millilitres, so twenty two point four litres is twenty two thousand four hundred millilitres. Now, n equals one divided by twenty two thousand four hundred moles. Just a tiny fraction of a mole, isn't it? Next, the molecules. Capital N equals n times N A: one over twenty two thousand four hundred, times six point zero two two times ten to the power twenty three. Work it out, and you get two point six nine times ten to the power nineteen molecules. In just one millilitre! This number has a name: the Loschmidt number. And our puzzle? The question said any gas, right? Methane or carbon dioxide, one millilitre at STP holds the same count. Same volume, same temperature, same pressure, same number of molecules. So our two balloons hold exactly the same number of molecules. Yes, it was a tie all along!

7:50Quick revision: molar volume, gas density, molar mass

So, quick revision. At the same temperature and pressure, one mole of any ideal gas fills the same volume, V m. Moles equal V by V m, density equals capital M by V m, and molar mass equals small m by V, times V m. And use twenty two point four litres per mole at STP, unless the question says otherwise, with litres matched to litres. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!