Mixtures, Speed of Sound and Moving Containers

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Helium has gamma, the ratio of its two heat capacities, about one point six seven. Oxygen has gamma one point four. Now we mix one mole of each. What is gamma for the mixture? Many students take the average, about one point five three. Sounds fair, right? Hold that answer. We will also see why a box of gas that stops suddenly gets warmer. These ideas do appear in JEE problems, so let's begin.

0:32Recall: one gas with f degrees of freedom

So, a quick recall from the last lesson: f is the number of degrees of freedom of a molecule. C v, with a small v as the subscript, is the molar heat capacity at constant volume, f over two times R, with R the gas constant. C p, with a small p, at constant pressure, is C v plus R, and gamma, the Greek letter here, is C p over C v, which is one plus two over f. Okay? Now, helium is monatomic, made of single atoms, so f equals three. Tell me, what are its C v and its gamma? Yes, exactly, C v is three halves R, and gamma is one plus two thirds, which is five thirds. The brackets hold f equals three, and the wavy equals sign means approximately, so gamma is about one point six seven. Right? And oxygen is diatomic. Treated as rigid, which means no vibration, it has f equal to five. So C v is five halves R, and gamma is seven fifths, one point four. Those are the two gammas from our puzzle. Clear?

1:36Mixtures: average Cv by moles, not gamma

Now, for a mixture, the heat capacities add up, mole by mole. C v with the subscript mix is the C v of the mixture, and the capital Greek sigma means add up over all the gases. On top, n i times C v i, where n with the subscript i is the number of moles of gas number i. At the bottom, the total number of moles. So it is a mole-weighted average. Got it? Then gamma of the mixture, gamma with the subscript mix, is C v mix plus R, over C v mix. This is just C p over C v for the whole mixture, okay? Now, back to our puzzle. Can we skip all this and simply average the two gammas? No, we cannot, and this is exactly the trap. Gamma of a mixture is not the average of the gammas. We average the C v values, weighted by moles, and only then find gamma. Right?

2:28Speed of sound in a gas

Next, let us look at sound. v s, with the subscript s for sound, is the speed of sound in a gas: the square root of gamma R T over capital M, okay? In the brackets, T is in kelvin, the capital K, and capital M is in kilograms per mole, written k g mol to the power minus one, not grams per mole. Got it? Now look at its shape, the root of R T over M, and tell me: which speed of the gas molecules has this same shape? Yes, the root mean square speed, v r m s, the square root of three R T over M, from the last chapter. Divide one by the other, and R, T and M cancel. The double arrow means it follows: v r m s over v s equals the root of three over gamma. With me? So, for a monatomic gas, gamma is five thirds. Which is larger, the r m s speed or the speed of sound? Three over five thirds is nine fifths, and its square root is three over root five, about one point three four. So the r m s speed is larger, about one point three four times the speed of sound. Clear?

3:37A moving container stops suddenly

Now the third idea: picture a closed, insulated box of gas moving with speed v, so the whole gas has kinetic energy. Suddenly the box stops. Where does that kinetic energy go? It goes into the gas itself, exactly! We assume no heat leaks out and the box takes none of it. The ordered kinetic energy, with all the molecules moving together, turns into internal energy, the random motion of the molecules; that is what the arrow means. So the gas gets warmer. Okay? So, let us balance the energy. On the left, half m times v squared, with m the mass of the gas. On the right, the rise in internal energy, n times f over two times R times delta T, with n the number of moles; delta T, with the Greek capital delta, is the rise in temperature. Right? Now, the mass m is n times capital M, the molar mass. Put that in. What happens to n? It cancels from both sides, very good! So delta T equals capital M times v squared, over f times R, and the amount of gas does not matter at all. Got it?

4:49Example 1: gamma of 1 mol helium plus 1 mol oxygen

Now, remember our puzzle? Let's finish it. Find gamma for a mixture of one mole of helium and one mole of oxygen, where m o l is short for mole. Pause the video and try it first. Done? Okay, let's check. Helium has C v equal to three halves R, oxygen five halves R, one mole each. So C v of the mixture is three halves R plus five halves R, over two moles, which gives two times R, okay? Now gamma of the mixture is C v mix plus R, over C v mix: two R plus R, over two R, which is three over two, or one point five. Clear? Now compare the shortcut from our puzzle: half times the bracket, five thirds plus seven fifths, is about one point five three. It is close, but it is wrong; the true answer is one point five. So the puzzle is solved: average the C v values, never the gammas.

5:48Example 2: temperature rise when a box of oxygen stops

Now the harder one, our box from the hook. Here it is: a box of oxygen moving at one hundred metres per second, written m s to the power minus one, is suddenly stopped. Find the rise in temperature. Pause the video and try it first. Done? Okay, let's check. For oxygen we need two things: its degrees of freedom f, and its molar mass, capital M, in S I units. What are they? So f is five, because oxygen is diatomic and we treat it as rigid. And capital M is thirty two grams per mole, written g mol to the power minus one, that is zero point zero three two kilograms per mole. Oh no, careful: thirty two straight into the formula gives an answer a thousand times too big. Okay? So delta T equals capital M times v squared, over five times R, with f equal to five. Here v squared is one hundred squared, ten to the power four, and R is eight point three one four. So we get zero point zero three two times ten to the power four, over five times eight point three one four. With me? On top, zero point zero three two times ten thousand gives three hundred and twenty. At the bottom, five times eight point three one four gives forty one point five seven. Then three hundred and twenty over forty one point five seven gives about seven point seven kelvin. That's it!

7:16Quick revision: gamma of mixture, sound speed, stopped box

So, quick revision. For a mixture, average C v by moles, then gamma mix is C v mix plus R, over C v mix; never average the gammas. For sound, v s is the root of gamma R T over M, and v r m s over v s is the root of three over gamma. And when a container stops suddenly, delta T equals capital M times v squared, over f times R, okay? Now go and try the practice problems on this page. All the best!