A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is (A) 50 (B) 75 (C) 350 (D) 525

 To determine the number of alpha-decays of nucleus Q in the first one hour, we need to follow these steps:

1. Understanding the Problem

• Half-life of Q: 20 minutes.

• Decay Probabilities:

  • Alpha-decay: 60%

  • Beta-decay: 40%

• Initial number of Q nuclei: 1000.

• Time period: 1 hour (60 minutes).

2. Calculating the Number of Half-lives in One Hour

Since the half-life of Q is 20 minutes, in one hour (60 minutes), the number of half-lives is:

$$\text{Number of half-lives}=\frac{60\text{ minutes}}{20\text{ minutes/half-life}}=3\text{ half-lives}$$

3. Determining the Remaining Nuclei After Each Half-life

The number of nuclei remaining after each half-life can be calculated using the formula:

$$N=N_0\times {\left(\frac{1}{2}\right)}^n$$

where:

• $N$ is the remaining number of nuclei,

• $N_0$ is the initial number of nuclei,

• $n$ is the number of half-lives.

Let's calculate the remaining nuclei after each half-life:

• After 1st half-life (20 minutes):

  $$N_1=1000\times {\left(\frac{1}{2}\right)}^1=500$$

• After 2nd half-life (40 minutes):

  $$N_2=500\times {\left(\frac{1}{2}\right)}^1=250$$

• After 3rd half-life (60 minutes):

  $$N_3=250\times {\left(\frac{1}{2}\right)}^1=125$$

4. Calculating the Number of Decayed Nuclei in Each Interval

The number of nuclei that decayed in each interval is the difference between the number of nuclei at the beginning and end of the interval.

• Decay in the 1st interval (0-20 minutes):

  $$\mathrm{\Delta }{N}_1=1000-500=500$$

• Decay in the 2nd interval (20-40 minutes):

  $$\mathrm{\Delta }{N}_2=500-250=250$$

• Decay in the 3rd interval (40-60 minutes):

  $$\mathrm{\Delta }{N}_3=250-125=125$$

5. Determining the Number of Alpha-decays

Given that the probability of alpha-decay is 60%, we can calculate the number of alpha-decays in each interval by multiplying the total decays by 0.60.

• Alpha-decays in the 1st interval:

  $${\alpha }_1=500\times 0.60=300$$

• Alpha-decays in the 2nd interval:

  $${\alpha }_2=250\times 0.60=150$$

• Alpha-decays in the 3rd interval:

  $${\alpha }_3=125\times 0.60=75$$

6. Total Alpha-decays in One Hour

Summing up the alpha-decays from all intervals:

$$\text{Total alpha-decays}={\alpha }_1+{\alpha }_2+{\alpha }_3=300+150+75=525$$

7. Conclusion

The number of alpha-decays of Q in the first one hour is 525.

Final Answer: (D) 525.