# Half-life of substances

You must have noticed that half-life is defined for various radioactive elements and other substances too.Why is it that always the half-life is defined?Why not a quarter-life or full life?

The half-life $t_{1/2}$ is defined so that the amount of stuff you have $N(t)$ at some time $t$ is $$N(t) = N_0 \cdot 2^{-t/t_{1/2}}$$ where $N_0$ is how much stuff you start with and you're familiar with the constant $2$.

This definition has several advantages. First, it's easy to explain, since scientists of all ages are familiar with the constant $2$. Second, it's unambiguous. If you asked about a "quarter-life," do you mean when a quarter of your original material remains $t_{1/4}$, which is after two half-lives, or when a quarter of your original material has decayed? The second option is not $\frac12 t_{1/2}$, because the decay is exponential and not linear; it's $t_{3/4} = t_{1/2}\log_2\frac43$, ugh.

"Full life" is not an option because the number of remaining decay-ers approaches zero only asymptotically.

When you start to do calculus, the same sort $\log_2$ ugliness comes up again. Another, probably nicer, way to write the decay equation is $$N(t) = N_0 e^{-t/\tau}$$ where $e = \frac1{0!} + \frac1{1!} + \frac1{2!} + \cdots \approx 2.7183$ is the base of the natural logarithms. The constant $\tau$ is referred to by several names, depending on your mood and the experience level of the person you're communicating with: sometimes $\lambda \equiv 1/\tau$ is called the "decay constant," and sometimes $\tau$ itself is referred to as the "$e$-folding time" or simply "the lifetime."

A nice feature of the exponential form is that, if you assume that every decay is detectable (or that a known fraction of the decays are detectable) then you can find the measured activity of your source: $$A(t) = -\frac{dN}{dt} = \frac{N_0}{\tau}e^{-t/\tau}$$ This means that if you can measure what your source's activity is, and measure long enough to watch it change, you have effectively weighed the radioactive part of the source by fixing $N_0$, even if $N_0$ may be only a few millions of atoms.

Given the definition of half life is "the time taken for the radioactivity of a specified isotope to fall to half its original value", "full life" as you asked would always be zero.

This also means that quarter life is just twice the half life, E.G iodine-131 has a half life of 8.1 days, the quarter life would be 16.2 days

A half is just a nicer number to work with, but it could have been quarter life

• Half-life is the time taken by the reactants to convert to half of its initial value. In this sense ,full life would not be zero because the reactants will take some finite amount of time to be consumed.And quarter-life is an alternative too. Commented Oct 21, 2016 at 15:51
• @user26857 " is the time required for a quantity to reduce to half its initial value" - wikipedia, therefore quarter life, if defined in the same way, would be twice the half-life, and full life would be zero, you can't argue with the definition Commented Oct 21, 2016 at 15:53
• Quarter life would not always be twice the half-life!Reactions don't occur with the same speed at all concentrations! Commented Oct 21, 2016 at 15:54
• @user26857 it would, because the time to get to a quarter is a half of a half, or two half lifes, hence TWICE the half life Commented Oct 21, 2016 at 15:57
• @user26857 - indeed, listen to Cursed1701 - nuclear decay is not a chemical reaction, where you get to a balance of forward and backward paths. It is just Poisson statistics on a random process that results in nuclear decay. There is not reverse process to consider. Commented Oct 21, 2016 at 16:05

1.half lie is the time taken for half of the initial amount of radioactive element decay.

2.quarter life is the time taken for half of the half of radioactive element to decay eg.if half life (T1/2) = 6 then quarter life (T1/4). = x

                          1/2  is the same as 0.5
1/4 is alas the same as 0.25

therefore,         0.5  =    6
0.25=    X


by calculations, X = 3 meaning quarter life is half of its half life

3.full life is twice of its half life eg. if 6 is half life then full life is 6+6 which is 12.