Half-Life and Rate of Decay: MDCAT Physics notes

Half-life

The half-life $T_{1/2}$ of a radioactive element is the time in which half of the nuclei present decay. It is a fixed property of each nuclide and is not affected by temperature, pressure, chemical combination or the amount of sample. It has nothing to do with any wavelength.

After each half-life the number left is halved:

$$N=N_0\left(\frac12\right)^n,\qquad n=\frac{t}{T_{1/2}}$$

Half-lives passed	Fraction left	Fraction decayed
1	1/2	1/2
2	1/4	3/4
3	1/8	7/8
4	1/16	15/16
5	1/32	31/32

Relation with decay constant

From $N=N_0e^{-\lambda t}$, putting $N=N_0/2$ gives

$$T_{1/2}=\frac{\ln2}{\lambda}=\frac{0.693}{\lambda}$$

So the half-life is inversely proportional to the decay constant. Keep units consistent: if $T_{1/2}$ is in hours, convert to seconds for $\lambda$ in s$^{-1}$.

The decay curve

A graph of $N$ against $t$ is an exponential curve that never reaches zero. Whatever point you start from, the time taken for $N$ to halve is the same: that constant halving time is what makes the half-life a useful measure. Activity $A=\lambda N$ follows the same curve, so a counter reading also halves every half-life. Because decay is random, the half-life is a statistical quantity: it predicts the behaviour of a large number of nuclei, not the moment any single nucleus will decay. A short half-life means a large decay constant and a highly active, quickly exhausted sample; a long half-life means a slowly decaying, long-lasting sample.

Mean life

The mean (average) life is $\tau=\dfrac1\lambda=\dfrac{T_{1/2}}{0.693}\approx1.44\,T_{1/2}$. It is longer than the half-life, so in a time equal to one mean life more than half of the nuclei decay (about 63%).

Worked examples

1. Counting half-lives. A 32 g sample reduces to 2 g in 60 days. $32\to16\to8\to4\to2$ is 4 half-lives, so $T_{1/2}=60/4=15$ days.
2. Mass remaining. Radium, $T_{1/2}=1600$ years: 100 g to 25 g is 2 half-lives, i.e. 3200 years.
3. Carbon-14 ($T_{1/2}\approx5730$ years): after 22 920 years ($=4$ half-lives) 1/16 remains; after 11 460 years 1/4 remains.
4. Decay constant from half-life. $T_{1/2}=8.70$ h $=31\,320$ s, so $\lambda=0.693/31\,320\approx2.2\times10^{-5}$ s$^{-1}$.
5. Half-life from $\lambda$. $\lambda=4.3\times10^{-4}$ s$^{-1}$ gives $T_{1/2}=0.693/4.3\times10^{-4}\approx1.6\times10^3$ s.
6. Atoms left. 400 atoms after 3 half-lives: $400/8=50$ (statistically; decay is random).

Some standard half-lives

• Iodine-131: about 8 days.
• Carbon-14: about 5730 years.
• Radium-226: about 1600 years.

Key formulas

• $N=N_0(1/2)^{t/T_{1/2}}$
• $T_{1/2}=0.693/\lambda$
• $\tau=1/\lambda=1.44\,T_{1/2}$
• $A=\lambda N$; activity also halves every half-life

Common MDCAT traps

• After $n$ half-lives the amount left is $N_0/2^n$, not $N_0/n$ or $2^nN_0$.
• Distinguish "fraction left" from "fraction decayed" (after 2 half-lives, 3/4 has decayed).
• $\lambda=0.693/T_{1/2}$, not $0.693\times T_{1/2}$.
• Mean life is longer than half-life.

Quick revision

• Half-life: time for half the nuclei to decay.
• 3 half-lives leave 1/8.
• Half-life is independent of the amount of sample.
• Unit of $\lambda$: s$^{-1}$.
• I-131 half-life about 8 days.