Moles of a Gas from P, V and T

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. In a hospital, there is a ten litre cylinder of oxygen. Its gauge shows twenty four point six atmosphere, and the temperature is three hundred kelvin. Now tell me, how many grams of oxygen are inside? Of course, you cannot open the cylinder and pour the gas onto a weighing balance, right? Still, those three readings, pressure, volume and temperature, are enough to find the mass. Sounds like magic? It is just one equation. Today we learn how. And look, in JEE, almost every gas question in stoichiometry starts by turning the gas into moles. So, let's begin.

0:45One equation for every gas

So, everything today comes from one equation, the ideal gas equation. P V equals n R T, okay? P is the pressure of the gas, V is its volume, T is its temperature, and n is the number of moles. And this R? That is the gas constant, a fixed number. We will meet its values in a minute. Now, in our cylinder, what do we want? The moles, n, right? So tell me, how do we get n alone on one side? Yes, exactly! Divide both sides by R T, simple. So n equals P V, divided by R T, okay? Pressure times volume on top, gas constant times temperature at the bottom. Look at what this line really means. Measure three things, pressure, volume and temperature, and the moles come out. No balance needed! And once you have the moles, the mass and the number of molecules follow from the mole concept. Clear?

1:43One constant, three values of R

Now, the gas constant. Its value depends on the units you use. With pressure in atmosphere and volume in litre, R is zero point zero eight two one litre atmosphere per kelvin per mole, okay? With pressure in bar and volume in litre, R is zero point zero eight three one litre bar per kelvin per mole. Almost the same number, but different units, right? And in S I units, pressure in pascal and volume in cubic metre, R is eight point three one four joule per kelvin per mole. So, one constant, written three ways. So which one do you use in a question? Yes, the one that matches the units of pressure and volume in that question. Pressure in atmosphere and volume in litre? Then zero point zero eight two one. Keep this in mind, the next part is where students lose marks.

2:36JEE trap: temperature and units

Alright, now a question says the gas is at twenty seven degrees Celsius. So T equals twenty seven, right? Put it in and done? No, no, careful! This is the JEE trap. In the gas equation, temperature is always in kelvin. Always! So add two hundred seventy three: twenty seven degrees Celsius becomes three hundred kelvin, okay? Put in twenty seven instead, and your answer is completely wrong. Second rule. The units of pressure and volume must match the value of R you are using. Atmosphere and litre go with zero point zero eight two one. Pascal and cubic metre go with eight point three one four, with me? And the most common slip? The volume is given in millilitres, and you use R in litre atmosphere. Then your answer is a factor of one thousand out! So change millilitres to litres first. Got it?

3:29Molar volume: litres filled by one mole

Now put n equals one in the gas equation. Then V equals R T by pressure: the volume of one mole, called the molar volume. So tell me, at two hundred seventy three kelvin and one atmosphere, how many litres does one mole fill? Twenty two point four litres, yes! The famous number. Twenty two point four litres, at two hundred seventy three kelvin and one atmosphere. Keep the same temperature, but make the pressure one bar. Now it is twenty two point seven litres, okay? Close, but not the same, so always read the pressure unit carefully. Now warm the gas to two hundred ninety eight kelvin, at one atmosphere. Will one mole fill more space, or less? More, exactly! V equals R T by P, so higher temperature means bigger volume. Twenty four point five litres. And at two hundred ninety eight kelvin and one bar, twenty four point eight litres. These conditions are called SATP, okay? Now a topper's trick. When a gas sits at exactly one of these conditions, skip the full formula: moles equal the volume divided by the molar volume. Super fast!

4:42Density and molar mass from the same equation

One more gift from the same equation. Moles are mass divided by molar mass: n equals small m by capital M, okay? Put that into the gas equation, and P V equals m by M, times R T, that's it. Now move the capital M to the left, and the V to the right. We get mass divided by volume on one side. And mass divided by volume, what is that? Density, yes! We write it as rho, the Greek letter you see on screen. So rho equals P M, divided by R T, that's all. The density of a gas, from its pressure, molar mass and temperature, clear? And turn it around: capital M equals rho R T, divided by the pressure. Measure the density of an unknown gas, and you get its molar mass. JEE loves this one.

5:33Example 1: mass of oxygen in a 10 L cylinder

Now, remember our oxygen cylinder from the start? Let's finish it. Here it is: a ten litre oxygen cylinder at three hundred kelvin shows a pressure of twenty four point six atmosphere. Find the mass of oxygen in it. Take R as zero point zero eight two. Pause the video, and try it first. Done? Okay, let's check. First, the units. Pressure in atmosphere, volume in litre, and the temperature is already in kelvin. So this R is the right one, okay? n equals P V by R T: twenty four point six times ten, over zero point zero eight two times three hundred. On top, two hundred forty six. At the bottom, twenty four point six. Divide, and you get exactly ten moles. Nice and clean! But the question asks for mass, not moles. One mole of oxygen gas, O two, is thirty two grams. So ten moles is three hundred twenty grams. There you go, puzzle solved: three hundred twenty grams of oxygen inside, and we never opened the cylinder!

6:37Example 2: density of CO2 at 27°C and 2 atm

Now a slightly harder one, JEE style. Find the density of carbon dioxide at twenty seven degrees Celsius and two atmosphere. Take R as zero point zero eight two one. Pause and try. Hint: which formula gives the density directly? Alright, for density we use rho equals P M by R T, the formula we just made. Pressure two, molar mass forty four, R zero point zero eight two one, and temperature twenty seven, right? Wait, no! Remember the trap? Temperature in kelvin, always. Twenty seven plus two hundred seventy three: three hundred kelvin. And the molar mass of carbon dioxide is forty four grams per mole. Now put in the numbers. Rho equals two times forty four, over zero point zero eight two one times three hundred. Pressure is in atmosphere, which matches this R, so no unit problem, okay? On top, eighty eight. At the bottom, twenty four point six three. Divide: three point five seven grams per litre. Per litre, because our R was in litre atmosphere, right? See how the units guided us?

7:51Quick revision: n = PV/RT, density = PM/RT and units

So, quick revision. Moles of a gas: n equals P V by R T, and density: rho equals P M by R T, okay? Temperature always in kelvin, and match the units of pressure and volume to the value of R you use. And one mole fills twenty two point four litres at two hundred seventy three kelvin and one atmosphere. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!