The Half-Life Method

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. You run a reaction starting at zero point two molar, and half of it is gone in fifty minutes. Now you double the start, to zero point four molar. This time, half is gone in just twenty five minutes. More reactant, and yet a shorter half-life? Strange, right? Can these two numbers alone tell us the order of the reaction? Yes, they can. Today we learn the half-life method, a standard tool in JEE kinetics. So, let's begin.

0:35Half-life of an order-n reaction

First, remember one thing. The order of a reaction comes from experiment, not from the balanced equation, okay? It can even be zero, or a fraction. The half-life method is one such experiment. We measure the half-life at two starting concentrations, and the order simply falls out. Right? Here is the half-life for a reaction of order n, with a single reactant. The t with a small one half below it is t half, the half-life. The square brackets around capital A mean its concentration, and the little zero below marks the start, so we read it as A naught. And k is the rate constant. So, t half equals two to the power n minus one, minus one, over n minus one, times k, times A naught to the power n minus one. With me? Now, see the equals sign with a slash through it: n is not equal to one. Why does the course keep first order out? Put n equals one and tell me what goes wrong. Yes, exactly! The bottom becomes zero, and the top, two to the power zero, minus one, is also zero. Zero over zero tells us nothing. So first order is treated on its own, and we will see its rule in a moment. Okay? For a given reaction, k and n are fixed, so t half is proportional to A naught to the power one minus n. The curly sign in the middle means proportional to. This one line is the whole method. Clear? And for a gas, we usually measure pressure instead. So the initial pressure, p naught, can replace A naught in all of this. Got it?

2:11What the half-life tells you

Now let us read the three common orders from that one line. Zero order first, so n equals zero. What is the power of A naught now? One minus zero is one, correct! So t half is proportional to A naught itself. More reactant, longer half-life. Okay? First order: the power is one minus one, which is zero. So t half does not depend on A naught at all. Start with a lot or a little, the half-life is the same. Right? Second order: the power is one minus two, minus one. So t half is proportional to one over A naught. Now tell me, if you double A naught, what happens to the half-life? It is halved, very good! So here is a quick check worth remembering whenever the start is doubled. Half-life doubles: zero order. Stays the same: first order. Halves: second order. And our puzzle? Fifty minutes became twenty five. It halved, so we already suspect second order. With me?

3:15Two runs give the order

But what if the half-life changes by an odd amount? Then we need a formula. Run one starts at concentration a one, with a small one below, and its half-life is t one. Run two starts at a two, with half-life t two. Okay? Write the proportionality for both runs and divide. t one over t two equals a one over a two, to the power one minus n. Now flip the fraction inside the brackets, and the power changes sign: a two over a one, to the power n minus one. Clear? Take the log of both sides and solve for n. So n equals one, plus a fraction: on top, the log of t one over t two; at the bottom, the log of a two over a one. Okay? Now look very carefully at this fraction. On top, the times go one over two. So at the bottom, the concentrations also go one over two, right? No, no, careful here! The concentrations are flipped: a two over a one. That flip is exactly where the power changed from one minus n to n minus one. Flip it the wrong way, and you get a completely wrong order. We will see this in the first example. Got it?

4:29Example 1: order from half-lives 50 and 25 min, flip trap

Now, our puzzle from the start, as a proper question. The half-life of a reaction is fifty minutes when A naught is zero point two molar, and twenty five minutes at zero point four molar. Here, capital M means molar, moles per litre, and m i n means minutes. Find the order. Pause the video and try it first. Done? Okay, let's check. t one over t two is fifty over twenty five, which is two. And a two over a one is zero point four over zero point two, also two. With me? So n equals one, plus log two over log two. That is one plus one, which is two. So the reaction is second order, exactly what our quick check suggested. Very good! Now see the trap. If you flip the concentrations by mistake, the bottom becomes the log of one half, which is minus log two. Then n is one minus one, which is zero, and you would wrongly say zero order. But at zero order the half-life grows with A naught, and here it fell. So that answer is wrong. Clear? So the reaction is second order, and t half is proportional to one over A naught. That is why doubling the start halved the half-life, and that solves our puzzle. Okay?

5:44Example 2: fractional order three halves from half-lives

Now a harder one. The half-life falls from forty seconds to twenty eight point three seconds when the initial concentration is doubled. The small s stands for seconds. Find the order. Pause the video and try it first. Done? Alright. The half-life fell when we doubled the start. And our quick check says that when it falls, it is second order, right? Oh no, careful here! At second order the half-life must halve, from forty to twenty seconds. But it fell only to twenty eight point three. So it is not second order, and we need the formula. With me? So, t one over t two is forty over twenty eight point three, about one point four one. The wavy equals sign means approximately. And doubled means a two over a one is two. Okay? Now n equals one, plus a fraction: on top, the log of forty over twenty eight point three; at the bottom, log two. Using log to the base ten, the top is about zero point one five zero, and log two is zero point three zero one. Right? Zero point one five zero over zero point three zero one is very nearly one half. So n equals one plus one half, which is three over two. The order is three halves. Fractional orders are perfectly fine, because order comes from experiment. Got it? And a quick check. Two to the power one half is root two, written with the square root sign, about one point four one. That matches our time ratio, so n minus one is one half. Clear?

7:20Quick revision: half-life vs concentration, n from two runs

So, quick revision. t half goes as A naught to the power one minus n: double the start, and it doubles, stays, or halves, for zero, first and second order. For two runs, n equals one plus the log of t one over t two, over the log of a two over a one, with the concentrations flipped. And order comes from experiment, so it can be a fraction. Now go and try the practice problems on this page. All the best!