Question

Obtain the law of radioactivity.

Answer 1

Law of radioactive decay:

At any instant t, the number of decays per unit time, called rate of decay \((\frac{dN}{dt})\) to the number of nuclei at the same instant.

\(\frac{dN}{dt}\)∝ N

By introducing a proportionality constant, the relation can be written as

\(\frac{dN}{dt}\) = −λN ……………………. (1)

Here proportionality constant X is called decay constant which is different for different radioactive sample and the negative sign in the equation implies that the N is decreasing with time. By rewriting the equation (1), we get 

dN = -XNdt …………… (2)

Here dN represents the number of nuclei decaying in the time interval dt. Let us assume that at time t = 0 s, the number of nuclei present in the radioactive sample is N_o . By integrating the equation (2), we can calculate the number of undecayed nuclei N at any time t. 

From equation (2), we get

N = N_o e^-λt  …………… (4) 

[Note: e^Inx = e^y ⇒ x = e^y ] 

Equation (4) is called the law of radioactive decay. Here N₀ denotes the number of undecayed nuelei present at any time t and N denotes the number of nuclei at initial time t = 0. Note that the number of atoms is decreasing exponentially over the time. This implies that the time taken for all the radioactive nuclei to decay will be infinite. Equation (4) is plotted.

We can also define another useful quantity called activity (R) or decay rate which is the number of nuclei decayed per second and it is denoted as

R = \(|\frac{dN}{dt}|\)

Note : That activity R is a positive quantity. From equation (4), we get.

The equation (6) is also equivalent to radioactive law of decay. Here R₀ is the activity of the sample at t = 0 and R is the activity of the sample at any time t. From equation (6), activity also shows exponential decay behavior. The activity R also can be expressed in terms of number of undecayed atoms present at any time t. From equation (6), since N = N₀ e^-λt we write 

R = λN    ………………… (7) 

Equation (4) implies that the activity at any time t is equal to the product of decay constant and number of undecayed nuclei at the same time t. Since N decreases over time, R also decreases.