Radioactive nuclei P and Q disintegrate into R with half lives 1 month and 2 months respectively. At time t=0 , number of nuclei of each P and Q is x.
Time at which rate of disintegration of P and Q are equal , number of nuclei of R is

Options

A. x
B. 1.25 x
C. 1.5 x
D. 1.75 x

Answer

To solve the problem, we need to determine the time at which the rates of disintegration of nuclei P and Q are equal, and then calculate the number of nuclei of R formed at that time.

### Step-by-Step Solution:

1. **Identify the Half-Lives and Initial Conditions**:
- Half-life of nucleus P, \( T_P = 1 \) month.
- Half-life of nucleus Q, \( T_Q = 2 \) months.
- Initial number of nuclei of P, \( N_P(0) = x \).
- Initial number of nuclei of Q, \( N_Q(0) = x \).

2. **Calculate the Decay Constants**:
The decay constant \( \lambda \) is related to the half-life by the formula:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]
- For P:
\[ \lambda_P = \frac{\ln(2)}{1} = \ln(2) \]
- For Q:
\[ \lambda_Q = \frac{\ln(2)}{2} = \frac{\ln(2)}{2} \]

3. **Write the Rate of Disintegration Equations**:
The rate of disintegration is given by:
\[ \frac{dN_P}{dt} = -\lambda_P N_P \]
\[ \frac{dN_Q}{dt} = -\lambda_Q N_Q \]
Substituting the decay constants:
\[ \frac{dN_P}{dt} = -\ln(2) \cdot N_P \]
\[ \frac{dN_Q}{dt} = -\frac{\ln(2)}{2} \cdot N_Q \]

4. **Express \( N_P \) and \( N_Q \) as Functions of Time**:
The number of nuclei remaining at time \( t \):
\[ N_P(t) = N_P(0) e^{-\lambda_P t} = x e^{-\ln(2) t} = x \cdot 2^{-t} \]
\[ N_Q(t) = N_Q(0) e^{-\lambda_Q t} = x e^{-\frac{\ln(2)}{2} t} = x \cdot 2^{-\frac{t}{2}} \]

5. **Set the Rates Equal**:
We want to find \( t \) when the rates of disintegration are equal:
\[ -\ln(2) \cdot N_P(t) = -\frac{\ln(2)}{2} \cdot N_Q(t) \]
Simplifying gives:
\[ \ln(2) \cdot x \cdot 2^{-t} = \frac{\ln(2)}{2} \cdot x \cdot 2^{-\frac{t}{2}} \]
Cancel \( x \) and \( \ln(2) \):
\[ 2^{-t} = \frac{1}{2} \cdot 2^{-\frac{t}{2}} \]
Cross-multiplying gives:
\[ 2^{-t + \frac{t}{2}} = \frac{1}{2} \]
Which simplifies to:
\[ 2^{-\frac{t}{2}} = 2^{-1} \]
Hence, we find:
\[ -\frac{t}{2} = -1 \implies t = 2 \text{ months} \]

6. **Calculate the Number of Nuclei of R Formed**:
Now we need to find the number of nuclei of R formed after 2 months.
- For P:
\[ N_P(2) = x \cdot 2^{-2} = \frac{x}{4} \quad \text{(remaining)} \]
Thus, the number of P converted to R:
\[ R_P = x - N_P(2) = x - \frac{x}{4} = \frac{3x}{4} \]
- For Q:
\[ N_Q(2) = x \cdot 2^{-\frac{2}{2}} = x \cdot \frac{1}{2} = \frac{x}{2} \quad \text{(remaining)} \]
Thus, the number of Q converted to R:
\[ R_Q = x - N_Q(2) = x - \frac{x}{2} = \frac{x}{2} \]

7. **Total Nuclei of R**:
The total number of nuclei of R formed is:
\[ N_R = R_P + R_Q = \frac{3x}{4} + \frac{x}{2} = \frac{3x}{4} + \frac{2x}{4} = \frac{5x}{4} \]

### Final Answer:
The number of nuclei of R formed is \( \frac{5x}{4} \) or \( 1.25x \).