**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 log_{e}N = -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/N_{0})= e^{- λt}

Or N= N_{0}e^{-}^{ λt}

**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.