# Half life and Radioactive Decay

[Sorry about the misleading title, as the query isn't entirely about half-life, but I couldn't find any better way to condense my question to make a brief enough title anyways..... and]

Radioactive decay is [as far as I know] an example of a First Order reaction. If I'm not mistaken, half-life [represented by t1⁄2] is the time elapsed for a sample to reduce BY 50% of its initial concentration.

Extending this piece of logic further, a book I've read states that t3⁄4 would represent the time elapsed for a sample to reduce BY 3/4th or 75% of the initial concentration. They were also kind enough to provide the following relation between t1⁄2 and t3⁄4,

t3⁄4 = 1.5 x [t1⁄2]


But I'm a little uncomfortable with this relation. Because if you think about this intuitively, it would take one t1⁄2 to reduce a 100g sample [say] to 50g, subsequently it would require a second t1⁄2 to reduce the 50g sample [obtained previously] to 25g. So it took two t1⁄2 to reduce the 100g sample to 25g, in other words, it took two t1⁄2 to reduce the 100g sample BY 75%.

So shouldn't the relation be,

t3⁄4 = 2 x [t1⁄2]


So am I correct? If not, where have I erred?

• What is the value of defining $t_{3/4}$? I can't think of a situation when it would be useful. – garyp Jun 30 '16 at 16:34
• Oh no, t3⁄4 showed up in the numerical exercise section of the book. It's just to check how well someone's grasped the concept.... @garyp – user122395 Jun 30 '16 at 16:42
• ... or in this case, grasping the wrong concept. :) – garyp Jun 30 '16 at 18:45

The number of atoms in your radioactive sample falls exponentially with time, so we get something like:

$$N = N_0 e^{-t/\tau}$$

where $\tau$ is a characteristic constant decay time called the mean lifetime. The half life is then defined by:

$$\frac{1}{2} = e^{-t_{1/2}/\tau}$$

or:

$$t_{1/2} = \tau\ln 2$$

By this reasoning a $3/4$ life would be defined by:

$$\frac{1}{4} = e^{-t_{3/4}/\tau}$$

or:

$$t_{3/4} = \tau\ln 4 = 2\tau\ln 2 = 2t_{1/2}$$

So I agree with you not the book!

• @AaronAbraham: I suppose it's just possible the book has some other definition of $t_{3/4}$, though it would have to be a rather odd definition ... – John Rennie Jun 30 '16 at 16:51
• Nope, there's only one [teeny tiny] section that deals with it.....and pretty much all it says is that t3/4 is one and a half times more than a t1/2.......still, really grateful for your answer :D – user122395 Jul 1 '16 at 16:39

Half life is a common physics misconception. Half life does not actually mean the cencentration of a substance being reduced to half by radio active disintegration. But it actially means the activity of the substance being reduced to half. For example if a sample of wood is found have uranium content with 5000 disintegration per second. And the half life is 100 years. Then it would take 100 years for the sample of wood to reduce to disintegrate at a rate of 2500 disintegration per second.

• If the rate of disintegrations has been reduced to 1/2 the original value, doesn't that imply that the concentration of that species has reduced to 1/2 of its original value? In any event, half-life has many more applications other than radioactive decay. – garyp Jun 30 '16 at 16:32
• @user122406 I suppose that half life could be considered as a measure of activity, strictly speaking that is. But as garyp mentioned, I fail to see any serious mistake in considering half life in terms of concentration.... – user122395 Jun 30 '16 at 16:37
• See my answer to an earlier question for some discussion of this. "Half life" is (and should be!) used in both way depending on the context. – dmckee --- ex-moderator kitten Jun 30 '16 at 22:48