Following a Reaction by Pressure, Volume, Titre or Rotation

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. In a closed flask, gas A breaks into two gases, B and C, okay? The gauge reads two hundred at the start, and three hundred after ten minutes. The pressure went up, while A is being used up! So how much of A is left? The gauge alone does not say, right? Today we follow reactions by pressure, gas volume, titre and rotation, which comes up often in JEE kinetics. So, let's begin.

0:33One formula for any measured property

So, here is the idea. We measure some property, capital P: a pressure, a volume, a titre or a rotation. P with a small zero is its value at the start, P with a small t at time t, and P with the sideways eight, infinity, at the end. It must change linearly with the extent of reaction, okay? Now think. Which difference measures all of the reactant we started with? Yes, exactly! P infinity minus P naught, the whole change. The sign like a little fish means proportional to. Right? And the reactant still left at time t is measured by P infinity minus P t, the change still to come! With me? Now, first order says the rate constant, k, equals one over t, times natural log of the starting amount over the amount left. So k equals one over t, times natural log of P infinity minus P naught, over P infinity minus P t; l n is the natural log. One formula for every case today. Got it?

1:40Gas phase: the gauge reads the total

First, a gas reaction. Gas A gives gas B plus gas C; the small g in brackets means gas, and the arrow means gives. Volume V and temperature T are kept fixed, so each partial pressure measures the amount of that gas. Okay? At the start only A is there, at p naught. If its pressure drops by x, then p with a small A is p naught minus x, and B and C are at x each, right? But the gauge cannot tell the gases apart. So what does it read at time t? The total pressure, p t! Add all three: p naught minus x, plus x, plus x, which is p naught plus x. One molecule goes, two come, so the total rises. Clear? Now, many students put the gauge reading, p t, straight in for the pressure of A, unchanged. Is that right? Oh no, careful here! The total goes up while A goes down. From the total, x is p t minus p naught, so the pressure of A, p naught minus x, is two p naught minus p t. Got it? Now our formula: k equals one over t, times natural log of p naught over two p naught minus p t. For other gas reactions, write the pressure of A from the total the same way. With me?

3:06Gas evolved, and titres

Next, a reaction that gives off a gas: nitrogen, N two, from a diazonium salt, or oxygen, O two, from hydrogen peroxide. We read the gas volume, capital V; nothing is collected at the start, so V naught is zero, and k equals one over t, times natural log of V infinity over V infinity minus V t. Okay? Now a titration. Hydrogen peroxide, H two O two, is titrated against potassium permanganate, K M n O four. The titre measures the peroxide still present. Clear? So at the end, when all the peroxide is gone, what is V infinity? It is zero, exactly! Put V infinity equal to zero, and k equals one over t, times natural log of V naught over V t. Right? Now the opposite case: in ester hydrolysis we titrate the acid formed, a product, against N a O H, sodium hydroxide. In acid hydrolysis there is acid from the start, so V naught is not zero; the slashed equals sign means not equal. So keep V infinity minus V naught on top, and V infinity minus V t below. Got that?

4:21Optical rotation: inversion of cane sugar

Last, the inversion of cane sugar. Sucrose, that is cane sugar, has a rotation of plus sixty six point five degrees; the small circle means degrees. It gives glucose, plus fifty two point five, and fructose, minus ninety two. Okay? Now look at the signs of glucose and fructose. So which way will the mixture turn, plus or minus? It turns to minus! Fructose wins, so the mixture is laevorotatory. The sign of the rotation gets inverted, which is why we call it invert sugar. Nice, right? With small r for the rotation: k equals one over t, times natural log of r naught minus r infinity, over r t minus r infinity. Same formula, with top and bottom both flipped in sign so they stay positive, since the rotation falls. Clear?

5:13Example 1: A to B plus C, half-life from total pressure

Now, our puzzle from the start! The reaction, A gives B plus C, is first order. Starting with A alone at two hundred millimetres of mercury, the m m H g on screen, the total is three hundred millimetres after ten minutes; m i n is minutes. Find the half-life, and the total after twenty minutes. Pause the video and try it first. Done? Okay, let's check. The gauge reads the total, so first get A: two p naught minus p t, that is two times two hundred, minus three hundred, one hundred millimetres. That is how much A is left. Puzzle solved, right? See: two hundred became one hundred in ten minutes. Half of A is gone! The double arrow means so: the half-life, t with a small one over two, is ten minutes. Very good! Now twenty minutes. The total rose by one hundred in the first ten minutes, so another hundred in the next ten, to four hundred, right? No, no, careful here! First order halves whatever is left; it does not use up equal amounts in equal times. Twenty minutes is two half-lives, so one quarter of two hundred is left: fifty millimetres. Okay? Back to the total: p t equals two p naught minus the pressure of A, four hundred minus fifty, that is three hundred and fifty millimetres. Perfect!

6:41Example 2: cane sugar rotation, find k and half-life

Now a slightly harder one. Here is a cane sugar solution in acid: rotation plus thirty degrees at the start, plus ten after sixty minutes, and minus ten at the end. Find k and the half-life. Pause the video and try it first. Done? Okay, let's check. With signs: r naught is plus thirty, r t is plus ten at sixty minutes, and r infinity is minus ten. Now, some students write r t minus r infinity as ten minus ten: zero at the bottom! Is that right? No, no, wait a moment here! This r infinity is minus ten, so we subtract minus ten. Top: thirty minus minus ten, forty. Bottom: ten minus minus ten, twenty. So k is one over sixty, times natural log of forty over twenty. Clear? Forty over twenty is two, so k equals natural log of two over sixty: zero point zero one one six per minute, since the minus one power means per. Right? Now the half-life. Look at that ratio of two again. Can you see the answer without any more work? Yes, you can! The ratio two means half the sugar was left at sixty minutes. And t half, natural log of two over k, is sixty minutes too. Very good!

8:09Quick revision: one formula, gauge total, sign of r infinity

So, quick revision. First order, any linear property: k equals one over t, times natural log of P infinity minus P naught, over P infinity minus P t. Gas phase: the gauge reads the total; for A gives B plus C, the pressure of A is two p naught minus p t. Rotation: keep the sign of r infinity, negative for cane sugar. Now try the practice problems on this page. All the best!