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 find the number of nuclei of R at the time when the rate of disintegration of nuclei P and Q are equal. Let's break this down step by step.

### Step 1: Understand the half-lives and decay constants
- The half-life of nucleus P (T_half_P) is 1 month.
- The half-life of nucleus Q (T_half_Q) is 2 months.
- The decay constant (λ) is related to the half-life by the formula:
\[ \lambda = \frac{\ln(2)}{T_{half}} \]
- Calculate the decay constants for P and Q:
\[ \lambda_P = \frac{\ln(2)}{1 \text{ month}} = \ln(2) \text{ month}^{-1} \]
\[ \lambda_Q = \frac{\ln(2)}{2 \text{ months}} = \frac{\ln(2)}{2} \text{ month}^{-1} \]

### Step 2: Write the expressions for the number of nuclei
- At time \( t \), the number of remaining nuclei of P (N_P) and Q (N_Q) can be expressed as:
\[ N_P(t) = x e^{-\lambda_P t} = x e^{-\ln(2) t} \]
\[ N_Q(t) = x e^{-\lambda_Q t} = x e^{-\frac{\ln(2)}{2} t} \]

### Step 3: Calculate the rates of disintegration
- The rate of disintegration for P (R_P) and Q (R_Q) is given by:
\[ R_P = \lambda_P N_P(t) = \ln(2) \cdot x e^{-\ln(2) t} \]
\[ R_Q = \lambda_Q N_Q(t) = \frac{\ln(2)}{2} \cdot x e^{-\frac{\ln(2)}{2} t} \]

### Step 4: Set the rates equal to each other
- To find the time \( t \) when the rates are equal:
\[ R_P = R_Q \]
\[ \ln(2) \cdot x e^{-\ln(2) t} = \frac{\ln(2)}{2} \cdot x e^{-\frac{\ln(2)}{2} t} \]
- Cancel \( x \) (assuming \( x \neq 0 \)) and \( \ln(2) \):
\[ e^{-\ln(2) t} = \frac{1}{2} e^{-\frac{\ln(2)}{2} t} \]

### Step 5: Solve for \( t \)
- Rearranging gives:
\[ e^{-\ln(2) t + \frac{\ln(2)}{2} t} = \frac{1}{2} \]
- This simplifies to:
\[ e^{-\frac{\ln(2)}{2} t} = \frac{1}{2} \]
- Taking the natural logarithm of both sides:
\[ -\frac{\ln(2)}{2} t = \ln\left(\frac{1}{2}\right) = -\ln(2) \]
- Solving for \( t \):
\[ t = 2 \text{ months} \]

### Step 6: Calculate the number of nuclei of R
- The total number of nuclei of R formed from P and Q can be calculated using the initial number of nuclei:
\[ N_R = x - N_P(t) + x - N_Q(t) \]
- Substitute \( N_P(t) \) and \( N_Q(t) \):
\[ N_R = x - x e^{-\ln(2) \cdot 2} + x - x e^{-\frac{\ln(2)}{2} \cdot 2} \]
- Simplifying gives:
\[ N_R = x - x \cdot \frac{1}{4} + x - x \cdot \frac{1}{2} \]
\[ N_R = x (1 - \frac{1}{4} + 1 - \frac{1}{2}) = x (2 - \frac{3}{4}) = x \cdot \frac{5}{4} \]

### Final Result
Thus, the number of nuclei of R at the time when the rates of disintegration of P and Q are equal is:
\[ N_R = \frac{5}{4} x \]