Mixtures of Gases

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Mix equal masses of hydrogen and helium in one vessel. Hydrogen gas has a molar mass of two grams per mole, and helium has four. So the mean molar mass of the mixture is three, exactly halfway, right? Most students say three without a second thought. Hold that answer; we will test it at the end. Gas mixtures come up often in JEE problems on pressure and density. So, let's begin.

0:31Dalton's law: each gas behaves as if alone

So, here is the set-up. We put a few gases in one closed vessel. They do not react with each other, and each behaves as an ideal gas, okay? Every gas spreads through the whole vessel, so they all share the same volume, capital V, and the same temperature, capital T, in kelvin. With me? Here is the key idea of Dalton's law: each gas behaves as if it were alone in the vessel. So what pressure does one gas exert, if it alone filled this whole volume at this temperature? Just the ideal-gas equation for that one gas, yes, exactly! We call it the partial pressure. On screen, p with a small subscript i, read as p sub i, is the partial pressure of gas number i, and n sub i is its number of moles. So p sub i equals n sub i times R T, over V, where R is the gas constant. Okay? Now, the total pressure is easy. Just add the partial pressures: p equals the sum of p sub i, over all the gases. The big Greek letter on screen is capital sigma, which means add them all up: p one, plus p two, and so on, which is what the three dots mean. This is Dalton's law of partial pressures. Clear? Now let us add them. Every term has the same R T over V, so take it out as common, and the moles add up: n one plus n two plus the rest. So the whole mixture obeys p V equals n R T, where small n is the total number of moles. We will use this in example one. Got it?

2:09Each gas's share of the pressure

Now, a quick question. Divide p sub i, the partial pressure of one gas, by p, the total pressure. Both have the same R T over V, so what cancels, and what is left? Yes, R T over V cancels, and p sub i over p equals n sub i over n, that gas's share of the total moles. This ratio is called the mole fraction, written x sub i on screen. Okay? So, multiply across, and p sub i equals x sub i times p, the total pressure. Each gas takes a share of the total pressure equal to its share of the moles. Right? And read this line carefully: pressure is shared by moles, not by mass. Splitting the pressure by grams is a common trap, and we will catch it in example one!

2:59Mean molar mass of a mixture

Next, molar mass. There is no single mixture molecule, so we define a mean molar mass: capital M with the subscript mix, for mixture. M mix equals the sum of n sub i times M sub i, over the sum of n sub i, where M sub i is the molar mass of that gas. Okay? Now look at the top. Moles times molar mass, n sub i times M sub i: what does that give, for one gas? It gives the mass of that gas, correct, because moles times grams per mole is just grams. So the top is the total mass of the mixture, and the bottom is the total number of moles. With me? So, M mix is the total mass over the total moles. Notice that it is an average weighted by moles, so the gas with more moles pulls the average towards its own molar mass. Our puzzle depends on this. Clear?

3:55Example 1: total and partial pressure of He and O2 in 10 L

Time for example one. Four grams of helium and thirty two grams of oxygen are in a ten litre vessel at three hundred kelvin; small g is grams, capital L is litres, and capital K is kelvin. Find the total pressure, and the partial pressure of oxygen. Pause the video, and try it first. Done? Okay, let's check. The first step is always to find the moles. Helium, written H e, has a molar mass of four grams per mole, so four grams is one mole. Oxygen gas, O two, has molar mass thirty two, so thirty two grams is also one mole. Right? So the total is one plus one, which gives two moles. And the volume must be in cubic metres. Since a litre is ten to the power minus three cubic metres, ten litres is ten to the power minus two, written m cubed. Okay? Now the mixture obeys p V equals n R T with the total moles. So p equals two, times eight point three one four, times three hundred, over ten to the power minus two, where eight point three one four is R in joules per mole per kelvin. With me? Work it out, and p is about four point nine nine times ten to the power five pascals, written capital P, small a, on screen. That is the total pressure. Very good! Now, oxygen's partial pressure. Oxygen is thirty two grams out of thirty six, about ninety percent of the mass. So it gives about ninety percent of the pressure, right? No, no, careful, that is the trap! Pressure is shared by moles, not by mass. Oxygen is one mole out of two, so its mole fraction, x sub O two, is one half. So p sub O two is half of p, which is two point four nine times ten to the power five pascals, and helium has the other half. Perfect!

5:59Example 2: mean molar mass of equal masses of H2 and He

Now example two, which settles our puzzle from the start. Equal masses of hydrogen and helium are mixed. Find the mean molar mass of the mixture. Pause the video and try it. Here is a hint: the masses are equal, but are the moles? Done? Okay, let's check. Hydrogen is two grams per mole and helium is four, so the answer is just three, halfway, right? No, no, wait, three is wrong here! Halfway works only when the moles are equal, and here the masses are equal. So call the mass of each gas small m, the same for both. Hydrogen gas, H two, has molar mass two, so its moles, n sub H two, are m over two, while helium's moles, n sub H e, are m over four. So hydrogen has twice as many moles. See? Now use our formula: total mass over total moles. The total mass is two m, counting both gases. The total moles are m over two, plus m over four, which gives three m over four. Okay? So M mix is two m, divided by three m over four. Small m cancels, and we get eight over three, about two point six seven grams per mole, written g mol to the power minus one. So the answer is not three; it is closer to hydrogen's two, because hydrogen has more moles and pulls the average down. That solves our puzzle. Very good!

7:31Quick revision: Dalton's law, mole fraction, mean molar mass

So, quick revision. Dalton's law: for non-reacting ideal gases, each gas behaves as if alone, with p sub i equal to n sub i times R T over V, and the total pressure is the sum of the partial pressures. Pressure is shared by moles, so p sub i equals x sub i times p, the total pressure. And the mean molar mass is the total mass over the total moles, an average weighted by moles. Now go and try the two practice problems on this page. All the best!