The Kinetic Calculation of Pressure

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. At one atmosphere, the air pushes on every square centimetre of a wall with about ten newtons, roughly the weight of a one kilogram bag of sugar! Yet only air molecules hit that wall, and one nitrogen molecule has a mass of just four point six five times ten to the power minus twenty six kilograms. So how can such tiny things push so hard? Let's find out, using the motion of molecules. This comes up often in JEE, so let's begin.

0:36The model: what kinetic theory assumes

So, first we need a model of a gas. Kinetic theory makes four assumptions, and the first is this: a gas has a very large number of identical molecules, each of negligible size, moving at random. Okay? Second, the molecules collide elastically, with each other and with the walls. Now tell me, when a molecule bounces elastically off a wall, what happens to its speed? It stays the same, exactly! No kinetic energy is lost, so it comes back just as fast. And the last two together: there are no forces between the molecules except during collisions, and the time of a collision is negligible. Clear?

1:14One molecule, one wall

Now, let's build the formula with one molecule. Take a cube of side capital L, with one molecule of mass m inside. Its velocity component along the x axis, towards a wall, is v with a small subscript x, which we just call v x, okay? It hits that wall and bounces back. So how much momentum does the wall receive in one hit? Is it m times v x, right? No, that is the classic trap, so careful here! The bounce is elastic, so the molecule comes back with minus m v x. The arrow on screen means changes to: the momentum goes from m v x to minus m v x. So the wall gets two m times v x in every hit. Got it? Now the timing. After this bounce, how long before the same molecule hits the same wall again? It crosses to the opposite wall, a distance L, and comes back another L, along x. So it covers two L at speed v x, which takes two L over v x. Right? Force is the momentum delivered per unit time. So the average force is two m v x, divided by two L over v x. The twos cancel, and we get m times v x squared, over L, for one molecule on one wall. With me?

2:34Now add up all the molecules

Now, the gas in our box has capital N molecules, each with its own v x. So we add up m times v x squared, over L, for all of them. The bar on top means the average over all molecules, so the total force F is N times m over L, times the average of v x squared. Okay? But the velocity has three components, and v y and v z are the other two. So the speed squared, v squared, equals v x squared, plus v y squared, plus v z squared. Now, since the motion is random, how does the average of v x squared compare with the average of v squared? If you said they are equal, then careful, that is a trap! For random motion, no direction is preferred, so the three averages are equal. And together they add up to the average of v squared, so each one is one third of it. Clear?

3:29The pressure formula

Now, pressure is force per unit area, and the wall has area L squared. So small p, the pressure, is F over L squared, which is one third, times N m over L cubed, times the average of v squared. And L cubed is the volume, capital V, okay? Now look at N times m, over the volume. That is N molecules of mass m, in a volume V, so what is it? It is the density, yes, mass per unit volume, written with the Greek letter rho! And v with a small subscript r m s is the root mean square speed: the square root of the average of v squared, as the root sign shows. Got it? So here is the result: p equals one third, times rho, times v r m s squared. Write it down; the rest of kinetic theory builds on it. With me?

4:18Pressure and kinetic energy

Now one more useful form. The total translational kinetic energy is the energy of moving from place to place. We call it E with a small subscript trans, and it is one half, times N m, times the average of v squared. Okay? Now multiply our pressure formula by the volume. The volume cancels on the right, and p V equals one third, times N m, times the average of v squared. Right? So compare the two lines. One has one half, the other has one third. So p V is what fraction of E trans? It is two thirds, yes, exactly! One third divided by one half is two thirds, so p V equals two thirds of E trans. So the pressure of a gas is tied directly to the kinetic energy of its molecules: p is two thirds of the translational kinetic energy per unit volume. Clear?

5:14Example 1: gas pressure from density and v rms

Now, example one. Here it is: a gas of density one point two kilograms per cubic metre has a root mean square speed of five hundred metres per second. Find its pressure. Pause the video and try it first. Done? Okay, let's check. Straight into the formula: p equals one third, times rho, times v r m s squared. So, one third, times one point two, times five hundred squared. Okay? Five hundred squared is two point five times ten to the power five. And one third of one point two is zero point four. Right? So p equals zero point four times two point five times ten to the power five, which gives one point zero times ten to the power five pascals. And if you dropped the one third, you got three times ten to the power five, three times too big. Clear?

6:08Example 2: force of a nitrogen molecule beam on a wall

Now, example two, which answers our puzzle. Here it is: a beam of ten to the power twenty three nitrogen molecules per second, each of mass four point six five times ten to the power minus twenty six kilograms, strikes a wall normally, that is, head on, at five hundred metres per second, and rebounds elastically. Find the force on the wall. Pause the video and try it first. Done? Okay, let's check. Remember the trap from before. How much momentum does one molecule give to the wall? It is two m v, yes, very good! It arrives with m v and leaves with minus m v, because the bounce is elastic. Right? Now the numbers. Two m v equals two, times four point six five times ten to the power minus twenty six, times five hundred. Two times five hundred is one thousand, so we get four point six five times ten to the power minus twenty three kilogram metres per second. Okay? Force is momentum delivered per second, so multiply by the rate, the number of molecules hitting the wall each second. The powers of ten cancel, so the force is four point six five newtons. Each molecule is tiny, but ten to the power twenty three kicks every second add up to almost five newtons, and that solves our puzzle!

7:29Quick revision: p = 1/3 rho v rms squared, pV = 2/3 E trans

So, quick revision. One molecule gives the wall two m v x every two L over v x, a force of m times v x squared, over L, okay? Random motion gives the one third, so p equals one third, times rho, times v r m s squared. And p V equals two thirds of the translational kinetic energy. Now go and try the practice problems on this page. All the best!