Degrees of Freedom and Equipartition

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. Take one mole of helium and one mole of oxygen, at the same temperature, each in a closed box that cannot expand. Give both the same amount of heat. Same moles, same heat, so the same rise in temperature, right? But no, the helium really does get hotter! So where does the oxygen hide the extra energy? Let's find out. Questions on this come up often in JEE, so let's begin.

0:32Degrees of freedom: ways a molecule holds energy

So, what is a degree of freedom? Molecules store energy by moving, spinning and vibrating. Each independent quadratic term in its energy, a squared term, is one degree of freedom. We call their number small f. Okay? Start with single atoms, like helium. In how many independent directions can one atom move? Yes, exactly, three directions: along x, y and z. That is translation. So a monatomic gas, like helium, written H e, has f equals three; the double arrow means so. Right? Now oxygen: two atoms joined, like a tiny dumbbell. It moves in three directions, sure. But what else can a dumbbell do? It can spin, exactly! About two axes at right angles to its length. Three translation plus two rotation: f equals five. That is oxygen, O two, the small two meaning two atoms, and any linear molecule, like carbon dioxide, C O two. Rigid means the bond does not vibrate. Clear? Now water, H two O, which is bent, not in a line. It rotates about all three axes, so three plus three: f equals six. With me? And if the bond vibrates, like a spring? Each vibrational mode adds two, not one, because a spring stores both kinetic and potential energy. Okay?

2:04Equipartition: half kT per degree of freedom

Now the big law: equipartition, which means equal sharing. In thermal equilibrium, each degree of freedom gets, on average, half k T per molecule. Small k is the Boltzmann constant; capital T, the temperature in kelvin. Per mole, it is half R T, with capital R the gas constant. Right? So, n moles of a gas with f degrees of freedom. What is the total internal energy? Yes, exactly: f times half R T, for each of the n moles. So U equals f over two, times n R T, and it depends only on the temperature, as it should for an ideal gas. Clear? Now C v, capital C with a small v below: the molar heat capacity at constant volume. From the last video, it is d U over n d T, the rise in U per mole per kelvin. So C v equals f over two, times R, okay? C p, with the small p, is at constant pressure. Mayer's relation, C p minus C v equals R, holds per mole of an ideal gas. So C p equals f over two plus one, times R, got it? And gamma, the Greek letter, is C p over C v. Divide, and gamma equals one plus two over f. Write this down: one line gives gamma for any gas. With me?

3:33The values to know, and the vibration trap

Now put in the numbers. Monatomic, f equals three: C v is three over two R, and gamma is one plus two thirds, which is five over three. Right? Rigid diatomic, f equals five: C v is five over two R, and gamma is seven over five. Now, our puzzle! Same heat at constant volume. Helium puts it all into moving; oxygen shares it with two rotations as well. Per mole per kelvin, helium needs only three over two R, and oxygen needs five over two. So the helium gets hotter: mystery solved! Now the trap. Suppose hot oxygen starts to vibrate: one vibrational mode. Five plus one, so f is six, right? No, no, careful here! One vibrational mode adds two, kinetic and potential. So f is five plus two, seven: C v is seven over two R, and gamma is one plus two sevenths, which is nine over seven. Okay?

4:37Gas mixtures: weight by the moles

Last idea: mixtures. Mixed gases simply add their internal energies, so C v of the mixture, with the small word mix below, is a weighted average. The capital Greek sigma means sum, and the small i counts the gases. So it is the sum of n i times C v i, over the sum of n i. Right? And gamma of the mixture is the total n C p, over the total n C v. Weight both by the moles, then divide. Clear? Or, an equivalent form: n one plus n two, over gamma mix minus one, equals n one over gamma one minus one, plus n two over gamma two minus one. Handy when a question gives you the gammas. Okay?

5:23Example 1: Cv, Cp and gamma of a non-linear triatomic gas

Alright, the first example. Find C v, C p and gamma for a rigid non-linear triatomic gas: three atoms, not in a line. Pause the video and try it first. Done? Okay, let's check. First, f: carbon dioxide also has three atoms, and its f is five. So f is five here too, right? No, no, that is the trap! Carbon dioxide is linear. Ours is non-linear, so it rotates about all three axes. Three translation plus three rotation: f equals six. Okay? So C v is six over two R, that is three R, and C p is C v plus R, that is four R, right? And gamma is four R over three R, which is four over three. Quick check with our formula: one plus two over six is four over three again, perfect!

6:25Example 2: Cv, Cp and gamma of a helium and oxygen mixture

Now a harder one. Two moles of helium are mixed with three moles of oxygen, and the oxygen is rigid. Find C v, C p and gamma of the mixture. Pause the video and try it first. Done? Okay, let's check. Helium is monatomic: C v is three over two R, or one point five R, and rigid oxygen has five over two R, or two point five R, with me? Weight by the moles: two times one point five R, plus three times two point five R, gives ten point five R, and over the total, five moles, that is two point one R, right? C p the same way, with five over two R and seven over two R: fifteen point five R over five, so three point one R, which is just two point one R plus R, Mayer's relation again. Clear? Now gamma. Helium's gamma is five over three, and oxygen's is seven over five. Can we just average these two, weighted by the moles? No, no, careful here! Gamma is a ratio, so take the ratio of the averages: three point one R over two point one R, which is thirty one over twenty one, about one point four eight; the wavy equals sign means approximately. Okay? Averaging five over three and seven over five by moles gives about one point five one: wrong. So average C v and C p first, then divide. Got it? Quick check with the other form: five over gamma mix minus one equals two over two thirds, plus three over two fifths, which is ten point five. So gamma mix is thirty one over twenty one again, perfect!

8:09Quick revision: f, U, Cv, Cp, gamma = 1 + 2/f, mixtures

So, quick revision. Each degree of freedom carries half k T per molecule, and U equals f over two, n R T, for n moles. C v is f over two R, C p is one R more, and gamma is one plus two over f, with f equal to three, five or six for rigid molecules. Each vibrational mode adds two, and for a mixture, average C v and C p by moles, then divide. Now go and try the practice problems on this page. All the best!