Question

The count rate of a Geiger- Muller counter for the radiation of a radioactive material of half-life of 30 minutes decreases to $5s^{-1}$ after 2 hours. The initial count rate was
A. $25s^{-1}$
B. $20s^{-1}$
C. $80s^{-1}$
D. $625s^{-1}$

Answer 1

Hint: The Geiger-Muller counter is used for measuring the radiation emitted by a radioactive material. Half-life of the material is also given. After 2 hrs it rate decreases to $5s^{-1}$. We have to find the initial rate using the given data.

Formula used:
We can use the same formula connecting amount of the original sample, remaining amount of the sample, half-life and time taken for decay:
${\displaystyle \frac{N}{N_0}}={\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{t}{T}}}$
Where $\mathrm{N}$ is the amount of sample remaining after $t$ time or final rate.
$N_0$ is the original amount of sample or initial rate.
$\mathrm{T}$ is the half-life of the radioactive material.


Complete answer:
We have a radioactive material of half-life 30 minutes. And the count rate of Gieger-Muller counter

decreases to $5s^{-1}$ after 2 hrs. With these data we have to find the initial rate shown in
Geiger-Muller counter.
In order to find the initial rate, we have the equation connecting all the known factors in question as:
${\displaystyle \frac{N}{N_0}}={\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{t}{T}}}$
Here we have to find $N_0$ that is the initial count rate.
Final count rate, $N=5s^{-1}$
Half-life of the radioactive material, $T=30$ minutes
Time taken, $t=2hrs=120$ minutes
On substituting the values in the equation, we get:
${\displaystyle \frac{5}{N_0}}={\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{120}{30}}}$
Therefore, initial count rate is:
$N_0={\displaystyle \frac{5}{{\left({\displaystyle \frac{1}{2}}\right)}^4}}=5\times 2^4=16\times 5=80s^{-1}$

Thus, option (D) is correct.

Additional information: The Geiger-Muller counter is an instrument which measures and detects ionization produced by radiation. It can count particles at rates up to 10,000 per second. Radioactive particles produce radiation when it decays. So, it can also be used to measure decay of radioactive materials.



Note: Amount here is taken as the rate since the rate is given in the question. Radioactivity measurement means how much of radioactivity has decayed. So, we can replace the amount with the rate. Don’t forget to convert the unit of time before calculating.