If `-((dN)/(dt))_(o)` is the initial activity and `-((dN)/(dt))` is the activity at time t in a radioactive disintegration then :

Options

A. `-((dN)/(dt))_(o)=-((dN)/(dt))e^(-kt)`
B. `-((dN)/(dt))_(o)=((dN)/(dt))_(o)e^(-kt)`
C. `-((dN)/(dt))_(o)=-((dN)/(dt))_(o)e^(-kt)`
D. `-((dN)/(dt))_(o)=((dN)/(dt))_(o)e^(-kt)`

Explanation

To solve the problem regarding the relationship between the initial activity and the activity at time \( t \) in a radioactive disintegration, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Reaction Order**:
- Radioactive disintegration is a first-order reaction. This means that the rate of disintegration is proportional to the number of radioactive particles present.

2. **Define Initial Activity**:
- The initial activity, denoted as \( -\frac{dN}{dt}\bigg|_0 \), is the rate of disintegration at time \( t = 0 \). For a first-order reaction, this can be expressed as:
\[ -\frac{dN}{dt}\bigg|_0 = k \cdot N_0 \]
where \( k \) is the rate constant and \( N_0 \) is the initial number of radioactive particles.

3. **Define Activity at Time \( t \)**:
- The activity at time \( t \), denoted as \( -\frac{dN}{dt} \), can be expressed in terms of the remaining number of particles \( N \) at that time:
\[ -\frac{dN}{dt} = k \cdot N \]

4. **Relate Remaining Particles to Time**:
- For a first-order reaction, the number of particles remaining at time \( t \) is given by the equation:
\[ N = N_0 e^{-kt} \]

5. **Substitute \( N \) into Activity Equation**:
- Substitute the expression for \( N \) into the activity equation:
\[ -\frac{dN}{dt} = k \cdot (N_0 e^{-kt}) \]

6. **Express Activity in Terms of Initial Activity**:
- Now, we can relate this to the initial activity:
\[ -\frac{dN}{dt} = -\frac{dN}{dt}\bigg|_0 \cdot e^{-kt} \]
where \( -\frac{dN}{dt}\bigg|_0 = k \cdot N_0 \).

7. **Final Relationship**:
- Thus, we arrive at the relationship between the initial activity and the activity at time \( t \):
\[ -\frac{dN}{dt} = -\frac{dN}{dt}\bigg|_0 \cdot e^{-kt} \]