# 18.5.1 Activity and Decay Constant

Some nuclides are stable while others are not. Unstable nuclides are radioactive. A sample of radioactive nuclei will decay over time, emitting radiation in the process. The number of decays per unit time is called the activity, A.It has the SI unit of Becquerel (Bq). 1 Bq = 1 decay per second.

Unlike nuclear fusion and fission, radioactive decay is spontaneous and random. By spontaneous, we mean that the decay is not triggered by anything. It just decays when it does. This means that the rate of decay of a radioactive sample is not affected by external environmental factors such as temperature, pressure, etc. By random, we mean that each decay is a probabilistic event. So there is absolutely no way one can predict when an individual nucleus is going to decay. However, if every nucleus in the sample has the same probability of decay, the rate of decay for the population as a whole can be predicted accurately.

To put it in simple terms, since each decay is a random event, the activity A of a radioactive sample must be proportional to the number of undecayed nuclei N in the sample. This is the basis for the formula

$\displaystyle A=\lambda N$

The constant of proportionality l is called the decay constant. It is often defined as “the probability per unit time that a nucleus decays”. When l has a small numerical value (this depends on the chosen time unit), it is indeed approximately equal to the probability of decay during the chosen time unit. However, when l has a large numerical value, the number does not have much meaning. For example, a decay constant of 0.01 day-1 does imply that the probability that a nucleus decays in one day is (approximately) 0.01. Note that 0.01 day-1 can also be expressed as 3.65 yr-1. But 3.65 does not have any physical meaning.

Example

The alpha decay of radon-222 into polonium-218 has a decay constant of $\displaystyle 0.181\text{ da}{{\text{y}}^{{-1}}}$. Calculate

a) the probability that a radon-222 nucleus decay in 1 min.

b) the activity of a sample containing 2 billion undecayed radon-222 nuclei.

Solution

a)

$\displaystyle 0.181\text{ da}{{\text{y}}^{{-1}}}=0.181\div 24\div 60=1.26\times {{10}^{{-4}}}\text{ mi}{{\text{n}}^{{-1}}}$

The probability of that a nucleus decays in 1 minute is $\displaystyle 1.26\times {{10}^{{-4}}}$.

b)

\displaystyle \begin{aligned}A&=\lambda N\\&=(\frac{{0.181}}{{24\times 60\times 60}})(2\times {{10}^{9}})\\&=4190\text{ Bq}\end{aligned}

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