Two nuclei P, Q have equal no.of atoms at t= 0. Their half-life are 3 hours, 9 hours. Compare their rates of disintegration after 18 hrs from the start.

Options

A. `3:16`
B. `16:3`
C. `1:3`
D. `3:1`

Answer

To solve the problem of comparing the rates of disintegration of two nuclei P and Q after 18 hours, we can follow these steps:

### Step 1: Determine the number of half-lives for each nucleus
For nucleus P:
- Half-life (T1) = 3 hours
- Total time (t) = 18 hours
- Number of half-lives (n1) = t / T1 = 18 hours / 3 hours = 6 half-lives

For nucleus Q:
- Half-life (T2) = 9 hours
- Total time (t) = 18 hours
- Number of half-lives (n2) = t / T2 = 18 hours / 9 hours = 2 half-lives

### Step 2: Calculate the remaining nuclei after the given time
Let the initial number of nuclei for both P and Q be N (at t=0).

For nucleus P:
- Remaining nuclei after 18 hours = N * (1/2)^(n1) = N * (1/2)^6 = N / 64

For nucleus Q:
- Remaining nuclei after 18 hours = N * (1/2)^(n2) = N * (1/2)^2 = N / 4

### Step 3: Calculate the rates of disintegration
The rate of disintegration (R) is proportional to the number of remaining nuclei. The formula for the rate of disintegration can be expressed as:
\[ R = \lambda N \]
Where λ (decay constant) can be expressed in terms of half-life:
\[ \lambda = \frac{\ln(2)}{T} \]

For nucleus P:
- Decay constant (λ1) = ln(2) / T1 = ln(2) / 3 hours
- Rate of disintegration (R1) = λ1 * (remaining nuclei of P) = (ln(2) / 3) * (N / 64)

For nucleus Q:
- Decay constant (λ2) = ln(2) / T2 = ln(2) / 9 hours
- Rate of disintegration (R2) = λ2 * (remaining nuclei of Q) = (ln(2) / 9) * (N / 4)

### Step 4: Calculate the ratio of rates of disintegration
Now, we can find the ratio of the rates of disintegration:
\[ \frac{R1}{R2} = \frac{(ln(2) / 3) * (N / 64)}{(ln(2) / 9) * (N / 4)} \]

The N and ln(2) cancel out:
\[ \frac{R1}{R2} = \frac{(1 / 3) * (1 / 64)}{(1 / 9) * (1 / 4)} \]

This simplifies to:
\[ \frac{R1}{R2} = \frac{1}{3} * \frac{4}{64} * 9 = \frac{9 * 4}{3 * 64} = \frac{36}{192} = \frac{3}{16} \]

### Conclusion
The ratio of the rates of disintegration of nuclei P and Q after 18 hours is 3:16.