Degrees of Freedom and Equipartition

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Take two sealed bottles. The first holds one mole of helium, and the second holds one mole of oxygen, at the same temperature. We give each bottle exactly the same heat. Which gas gets hotter? Many of you will say both warm up equally, right? Hold that answer. One simple counting rule, called equipartition, will settle it. It comes up often in JEE, so let's begin.

0:31Equipartition: half kT per quadratic term

Here is the law of equipartition of energy. In thermal equilibrium, every independent quadratic term in a molecule's energy has the same average value, half k T, okay? Small k is Boltzmann's constant, and capital T is the temperature in kelvin. Quadratic means something squared, like a speed squared. Now, a molecule flying around in the box has translational kinetic energy half m v x squared, plus half m v y squared, plus half m v z squared. Here m is its mass, and v with a small x, y or z below is its velocity along that axis. Three quadratic terms, right? Each of these terms gets half k T on average. So tell me, what is the total average translational energy? Three times half k T, which is three halves k T, yes, exactly! So one molecule has an average translational energy of three halves k T, clear? Now, each independent term like this is called a degree of freedom. So per degree of freedom, one molecule gets half k T, and one mole gets half R T, where R is the gas constant, okay?

1:44Rigid molecules: f = translation + rotation

So, we call the number of degrees of freedom f. For a rigid molecule, one that does not vibrate, f is translation plus rotation, and translation along x, y and z always gives three. Monatomic gases are single atoms, like helium, neon and argon, with their symbols in the brackets. Rotation adds nothing here, so f equals three plus zero, which is three. With me? Next, diatomic molecules, like O two, oxygen; the small two below means two atoms. Such a molecule also rotates, which adds two terms, so f equals three plus two, which is five, okay? Now here is a tricky one. Carbon dioxide, C O two, has three atoms. So is its f five, or six? It is five, not six! Its three atoms lie on one straight line, and a linear molecule counts just like a diatomic one. Counting atoms and writing six is a classic trap. Got it? Finally, non-linear molecules, like water, H two O; ammonia, N H three; and methane, C H four. They rotate in three independent ways, so f equals three plus three, which is six. Right?

3:02What about vibration?

So far, our molecules were rigid. But at high temperature, the two atoms of a diatomic molecule can also vibrate, in and out, like two balls on a spring. So, one vibrational mode adds how much to f? Just one, isn't it? No, no, careful, this is exactly the trap! Now, a vibrating molecule stores kinetic energy of the motion and potential energy of the stretch. So each vibrational mode adds two to f, not one. Okay? That means one mode gives two times half k T, which is a full k T per molecule, not half k T, clear? For example, a diatomic molecule at high temperature has three for translation, two for rotation, and two for vibration. So f equals seven. Remember, rigid means no vibration, so add these twos only when vibration is active. With me?

3:55From f to U, Cv, Cp and gamma = 1 + 2/f

Now, the payoff. Each molecule gets f times half k T, so n moles get f over two, times n R T, where small n is the number of moles. That is the internal energy, capital U, of an ideal gas, okay? From this come the heat capacities per mole. C v, with a small v below, at constant volume, is f over two, times R, and C p, with a small p below, at constant pressure, is f over two plus one, in brackets, times R, right? And gamma, the Greek letter you see, is the ratio C p over C v. Divide them, and it comes out as one plus two over f. Clear? Now, here is a quick check: for a monatomic gas, f is three, so what is gamma? One plus two thirds, which is five thirds, very good! For a rigid diatomic gas, f is five, so gamma is one plus two fifths, which is seven fifths, or one point four. Both come just from counting f, got it? Here is a topper's trick: turn it around, and f equals two over, gamma minus one. So gamma of one point four gives f equals two over zero point four, which is five, okay?

5:08Back to the puzzle

Now, back to our puzzle. Same heat, same moles, and sealed bottles, so constant volume. Which one gets hotter? At constant volume, delta T, the rise in temperature, equals Q over n C v, where Q is the heat and the Greek letter delta means change. Helium has C v of three halves R, but oxygen, taken as rigid, has five halves R, with me? So yes, it is helium that gets hotter! Oxygen shares the heat among five degrees of freedom, helium among only three, so each share is bigger in helium. And that solves our puzzle, clear?

5:47Example 1: internal energy of 3 mol of oxygen at 300 K

Now, a problem from the course. Find the internal energy of three moles of oxygen, rigid molecules, at three hundred kelvin; m o l means moles, and capital K means kelvin. Pause the video and try it first. Done? Okay, let's check. Many students write three halves n R T straight away. Is that right? No, no, wait, three halves is only the translational part! Oxygen is diatomic and rigid, so it also has two rotations, and f equals five. Got it? So U equals five halves of n R T, which is five halves, times three, times eight point three one four, times three hundred, with R in joules per mole per kelvin. With me? That gives eighteen thousand seven hundred and six point five joules. The wavy equals sign means approximately, so U is about eighteen point seven kilojoules. Clear?

6:46Example 2: total and rotational energy of 1 g of hydrogen

Now a slightly harder one. Find the total internal energy of one gram of hydrogen at three hundred kelvin, and the rotational part of it. Here is a hint: find the moles first. Pause the video and try it. Done? Okay, let's check. Hydrogen gas is H two, so its molar mass, capital M, is two grams per mole, written g mol to the power minus one. So one gram is half a mole, and n equals zero point five. It is diatomic, with no vibration here, so f equals five, okay? So U equals zero point five, times five halves, times eight point three one four, times three hundred. That gives three thousand one hundred and seventeen point seven five joules, about three point one two kilojoules. Right? Now comes the interesting part: how much of this energy is rotational? Remember, every degree of freedom gets an equal share. Yes, exactly, two of the five degrees of freedom are rotational, so rotation holds two fifths of U, okay? Two fifths of three thousand one hundred and seventeen point seven five gives one thousand two hundred and forty seven point one joules, about one point two five kilojoules. Got it?

8:01Quick revision: f for each molecule, U, Cv, Cp and gamma

So, quick revision. Every degree of freedom gets half k T per molecule, or half R T per mole. For rigid molecules, f is three for monatomic, five for diatomic or linear, and six for non-linear, and each vibrational mode adds two. From f come U, C v, C p and gamma. Now try the practice problems on this page. All the best!