Question

The counting rate observed from a radioactive source at t=0 second was 1600 counts per second and at t=8 second it was 100 counts per second. The counting rate observed as counts per second at t=6 seconds, will be
A. 200
B. 400
C. 300
D. None of these

Answer 1

Hint: The radioactivity of a radioactive material is directly proportional to the number of undecayed atoms left in the sample.

Complete step by step answer:
The rate of disintegration or count rate of a radioactive material is called the activity of the radioactive substance and it is denoted by ‘A’. The formula for Radioactivity is given by:
$A={{A}_{0}}{{(\dfrac{1}{2})}^{{t}/{T}\;}}$
Here ${{A}_{0}}$ is the radioactivity at time $t=0$
Given that,
${{A}_{0}}=1600$ at $t=0$
$A=800$ at $t=8$
Substituting these values in the above equation we can find the half-life (T) of the radioactive material.
$100=1600{{(\dfrac{1}{2})}^{\dfrac{8}{T}}}$
On solving we get
${{2}^{\dfrac{8}{T}}}=16$
$T=2\sec$
Now, at $t=6\sec$
$A=1600{{(\dfrac{1}{2})}^{\dfrac{6}{2}}}$
$A=200\text{counts/sec}$
Hence the correct answer is A. 200

Additional Information:
The radioactive decay law states that the no. of nuclei disintegrating per second at any instant Is directly proportional to number of undecayed nuclei present in the sample at that instant.
$d{{t}_{{}}}$
Here ${{N}_{0}}$ is the no. of nuclei present initially at t=0
And $dN$ is the number of radioactive nuclei disintegrating in time $dt$
The negative sign depicts that the number of undecayed nuclei are decreasing with time.

Note: The time interval in which half of the radioactive nuclei originally present in the radioactive sample disintegrates is called half-life.