A radioactive nucleus decays to a nucleus with a decay constant

PHYSICS · Multi-part · A Level · Physics · Nuclear physics · Nuclear physics

Problem

A radioactive nucleus

X

decays to a nucleus

Y

with a decay constant

\lambda_x = 0.1 \text{ s}^{-1}

Y

further decays to a stable nucleus

Z

with a decay constant

\lambda_y = 1/30 \text{ s}^{-1}

. Initially, there are only

X

nuclei and their number is

N_0 = 10^{20}

. The population of

Y

nucleus as a function of time is given by

N_Y(t) = (N_0 \lambda_x / (\lambda_x - \lambda_y)) \{ \exp(-\lambda_y t) - \exp(-\lambda_x t) \}

. (1) Set up the rate equations for the populations of

X

Y

and

Z

. (2) Find the time at which

N_Y

is maximum (3) determine the populations

X

and

Z

at that instant.