State the law of radioactive decay, Physics

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State the law of radioactive decay. Show that radioactive decay is exponential in nature. Does the activity of a radioactive element depend on external physical conditions?

Law of radioactive Decay: This law states that the number of radioactive nuclei decreases exponentially with time. The rate of disintegration dN/dt, i.e., the number of atoms disintegrated per second is directly proportional to the number of atoms present (N) a that moment. This is known as Decay Law.

Explanation

Let N0 be the number of atoms present in a radioactive sample initially i.e., at t = 0.

Let N be the no of atoms left at a time t and dN be the no of atoms disintegrating in a short interval of time t.

The rate of disintegration = dN/dt

According to decay law = dN/dt α N or dN/dt =  -λN

Where, λ = decay or disintegration constant. The negative sign indicates that as time

increases, the number of atoms decreases

dN/dt=  -λN OR dN/N = -λdt

Integrating the above equation

∫dN/N = -λ∫dt or logeN = -λt +c ...(1)

Where c = integration constant and it can be found in the initial condition.

i.e., If t = 0, N = N0

Then, logeN0 = c ............ (2)

From equation (1)

LogeN = - λt + logeN0

LogeN - logeN0 = -λt (or)

Loge(N/N0)= e- λt

Or N= N0e- λt

2149_law.png

a) The number of radioactive nuclei decreases exponentially with time and reduced to zero after infinite time

b) The above equation represent radioactive decay is exponential in nature. The activity of a radioactivity element does not depend on external physical conditions like temperature, pressure etc.


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