Question

A radioactive substance of half-life 6 min is placed near a Geiger counter which is found to register 1024 particles per minute. How many particles per minute will be registered after 42 min?
A) 4/min
B) 8/min
C) 5/min
D) 7/min

Answer 1

Hint: In this question radioactive decay is happening and the numbers of particles are being registered in Geiger counter. We are going to use the formula of the amount of substance left after a radioactive decay. First we will find decay constant $\lambda$. Then we will find the number of particles registered by Geiger counter after 42 min.

Complete step by step solution:
Given:
Half life $\tau = 6\min $
Number of particles registered per minute ${N_o} = 1024$
Time $t = 42\min $
Let the number of particles left after 42 minutes is N
Decay constant $\lambda $ is given by
$\lambda = \dfrac{{\ln 2}}{\tau }$
Putting the value of half life $\tau = 6\min $
$\Rightarrow \lambda = \dfrac{{0.693}}{6}$
$\Rightarrow \lambda = 0.1155$
Now we will find the number of particles left after 42 minutes using following formula,
$\Rightarrow N = {N_o}{e^{ - \lambda t}}$
Putting the values of ${N_o},\lambda {\text{ and t}}$
$\Rightarrow N = 1024{e^{ - 0.1155 \times 42}}$
$\Rightarrow N = 1024{e^{ - 4.851}}$
$\Rightarrow N = 1024 \times 0.00782$
$\Rightarrow N = 8.008$

The number of particles left after 42 minutes is $N = 8.008$.

Note: Geiger counter is a device used in nuclear physics to detect or count the number of particles being emitted from a radioactive source. It is also known as the Geiger Muller counter. It works on the basis of ionization radiation. It means that the particles inside the Geiger Muller tube get ionized and counter detects the particles. The Geiger Muller tube is filled with an inert gas such as Helium, Argon or Neon. The Geiger counter is the best device to detect radiation.
In the given question we have used the formula of radioactive decay. This is a first order equation of radioactive decay. We have found decay constant in the question is a very important variable because it is the probability of decay of substance per minute or it can be defined by the rate at which the population of particles of a radioactive material decreases due to radioactive decay.