Overview

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take a gas at twenty seven degrees Celsius, and heat it to three hundred and twenty seven degrees Celsius. The temperature number has gone up more than twelve times, right? So do the molecules now move about three and a half times faster? Hmm, hold that thought; we will settle it soon. Short JEE Main questions like this come up often in kinetic theory. So, let's begin.

0:31How this chapter works

So, let's open this chapter. First, JEE Main questions on kinetic theory are short. Okay? And where are the real past questions, from twenty twenty one to twenty twenty six? They sit at the end of Chapters one to five. Right? The problems in this chapter are original, written in the same style, and they come in four sets. With me? Now, two constants for the whole chapter. R, the gas constant, is eight point three one four joules per mole per kelvin; that is what J mol to the power minus one, K to the power minus one, means on screen. And small k, Boltzmann's constant, is one point three eight times ten to the power minus twenty three joules per kelvin. Clear?

1:16Types 1 and 2: speeds and energies

So, what do they ask? The course names four types. Type one is ratios of speeds. The main tool is v r m s, the root mean square speed, written v with a small subscript r m s. It equals the square root of three R T, over capital M, the molar mass, where T is the temperature. With me? All three speeds from Chapter three are proportional to the root of T over capital M; that sign like an open alpha means proportional to. And two unit rules: T in kelvin, and capital M in kilograms per mole, the k g mol to the power minus one on screen. Okay? Type two is energies, per molecule or per mole. The average translational kinetic energy, the K E on screen, is three halves k T for one molecule, and three halves R T for one mole. Clear? Now a question. Take helium and oxygen at the same temperature. Whose molecules have the larger average translational kinetic energy? They are exactly equal; neither has more! Three halves k T has no mass in it, so it is the same for every gas at a given temperature. Oxygen molecules are heavier, but they move slower. Got it?

2:30Type 3: degrees of freedom and gamma

Type three is degrees of freedom, written f, and gamma. For rigid molecules, which do not vibrate, f is three for a monatomic gas like helium, five for a diatomic or linear molecule, and six for a non-linear one, like water vapour. Okay? And when a vibrational mode counts, it adds two to f, because it stores both kinetic and potential energy. Right? By equipartition, each degree of freedom gets half k T per molecule. So C v, the C with a small subscript v, the molar heat capacity at constant volume, is f over two, times R, and C p, at constant pressure, is C v plus R, okay? Now, gamma, the Greek letter you see on screen, is the ratio C p over C v. So tell me, what is gamma for a monatomic gas? Five thirds, yes! With f equals three, C v is three halves R and C p is five halves R, so five halves over three halves gives five thirds. Clear? And in general, gamma equals one plus two over f. For a rigid diatomic gas, that is one plus two fifths, which is seven fifths. Got it?

3:36Type 4: the mean free path

Type four is the mean free path. In the picture, the dark blue dot is a molecule, and each orange dot is a collision. Between collisions it moves in straight lines, the free paths. The average length of these free paths is the mean free path, lambda, the Greek letter. Okay? Here is the formula, also shown in the picture. Lambda equals one, over root two, times pi, times d squared, times n. Here d is the molecular diameter, and small n is the number density, molecules per unit volume. And written with the pressure, small p, it is also k T over root two pi d squared p. With me? Now look at the second form: T sits on top. So if we heat a gas in a closed, rigid container, does the mean free path grow? It does not, so be careful here! In a rigid container, n stays fixed, so by the first form lambda does not change. The pressure rises along with T, so k T over p stays the same. Clear?

4:40Example 1: ratio of rms speeds at 327°C and 27°C

Now, Example one, from Set one. Find the ratio of the r m s speeds of a gas at three hundred and twenty seven degrees Celsius and at twenty seven degrees Celsius; the small circle and capital C on screen mean degrees Celsius. Yes, this is our puzzle! Pause the video, and try it first. Done? Okay, let's check. It is the same gas, so capital M is the same, and v r m s is proportional to the root of the temperature. Right? Many students write the root of three hundred and twenty seven over twenty seven, which is about three point five; the wavy equals sign means approximately. Three and a half times faster, just like our puzzle. Is that correct? Oh no, careful, that is the trap! The temperature must be in kelvin, so add two hundred and seventy three. Three hundred and twenty seven plus two hundred and seventy three gives six hundred kelvin, and twenty seven plus two hundred and seventy three gives three hundred kelvin; the capital K is kelvin. Okay? So the ratio is the root of six hundred over three hundred, which is root two, about one point four one. The molecules move only about one point four one times faster, not three and a half. Puzzle solved: the Celsius numbers fooled us. Got it?

5:58Example 2: gamma of a monatomic and diatomic gas mixture

Now a harder one, Example two, from Set two. Find gamma for a mixture of one mole of a monatomic gas and one mole of a rigid diatomic gas; the m o l on screen means moles. Pause the video and try it. Done? Let's check. Gamma is five thirds for the monatomic gas, and seven fifths for the rigid diatomic gas. Equal moles, so just average the two and get about one point five three, right? No, wait, that is the trap! For a mixture, the course is clear: use the average C v, not the average gamma. Okay? So, for the monatomic gas, f is three: C v is three halves R, and C p, which is C v plus R, is five halves R, clear? Now you do the rigid diatomic gas. What are its C v and C p? Very good if you said five halves R and seven halves R, because f is five here. With me? With one mole of each, gamma mix, the gamma with the small subscript m i x, is the total C p over the total C v; the R cancels out. Five halves plus seven halves is six, and three halves plus five halves is four. Six over four gives one point five. Right? And the average of the two gammas gave one point five three, which is close, but wrong. So never average the gammas. Got it?

7:26Quick revision: four question types, kelvin, mixture gamma

So, quick revision. Four types: speeds, energies, degrees of freedom and gamma, and the mean free path. Use kelvin, and capital M in kilograms per mole. And for a mixture, gamma comes from the average C v, never the average gamma. Now open Set one, solve the problems, and then try the practice problems at the end of each set. All the best!