# Activity

The rate at which nuclear transformations occur in radioactive material. Used as a measure of the amount of a radionuclide present. Unit becquerel, symbol Bq. 1 Bq = 1 transformation per second.

Source: IAEA

Details:

The number of decays per unit time interval, i.e. the activity $A$, is defined by

$A =$$dN/dt = kN$

It should be noted in this definition that it is assumed that $N$, is decreasing due to decay. In general the rate equation for a species $j$, contains a term for decay to the daughter and growth from the parent, i.e.

$dN_i/dt =$$k_iN_i + k_{i-1} N_{i-1}$

A situation could arise in which $\lambda_i N_i = \lambda_{i-1} N_{i-1}$ and thus $N_i$ is constant, i.e. $dNi/dt = 0$. Clearly the activity is not zero. In the definition of A above only decay is considered. In the general case where decay and growth occur, A is given by $A_i = â��k_iN_i$ . Hence $A_i$ is the number of disintegrations per second even though $N_i$ may be constant. The unit of activity is the Becquerel, i.e. 1 Bq = 1 disintegration per second.

A technical problem arises in the evaluation of quantities involving the activity in the case where the half-life is less than 1 s. The activity defined above gives the instantaneous disintegration rate. If the half-life is ≤ 1 s a significant amount of the material has decayed in the first second. The above definition of the activity will then overestimate the emitted radiation. The difficulty can easily be overcome by defining the activity per integral second, i.e.

$A_{1s}=\int_{0}^{1} kNdt\,=N(0)(1-e^k)$

where $k = ln 2/\tau(s)$ and $N(0)$ is the number of atoms at time $0$. For the calculation of the specific activity, denoted $spA$, $N(0)$ is the number of atoms in 1 g i.e. $N(0) = N_a/\mathcal{A}$ where $\mathcal{A}$ is the atomic mass. Hence

$spA = N_a (1- e^{â��k})/\mathcal{A}$

or

$spA(Bq/g) = 6.022 x 10^{23} Â·[1 - e^{-ln 2/\tau(s)} ]/\mathcal{A}$

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