A sample of radioactive material has mass `m`, decay constant `lambda`, and molecular weight `M`. Avogadro constant `=N_(A)`. The initial activity of the sample is:

Options

A. `lambdam`
B. `(lambdam)/(M)`
C. `(lambdamN_(A))/(M)`
D. `(lambdam)/(MN_(A))`

Answer

To find the initial activity of a sample of radioactive material, we can follow these steps:

### Step 1: Understand the formula for activity
The activity \( A \) of a radioactive sample is given by the formula:
\[ A = \lambda N \]
where:
- \( A \) is the activity,
- \( \lambda \) is the decay constant,
- \( N \) is the number of radioactive nuclei in the sample.

### Step 2: Calculate the number of nuclei \( N \)
The number of nuclei \( N \) can be expressed in terms of the mass \( m \), molecular weight \( M \), and Avogadro's constant \( N_A \) as follows:
\[ N = \frac{m}{M} \times N_A \]
This equation arises because:
- The number of moles of the substance is given by \( \frac{m}{M} \),
- Each mole contains \( N_A \) nuclei.

### Step 3: Substitute \( N \) into the activity formula
Now, we can substitute the expression for \( N \) into the activity formula:
\[ A = \lambda N = \lambda \left(\frac{m}{M} \times N_A\right) \]

### Step 4: Simplify the expression
This leads us to the final expression for the initial activity:
\[ A = \frac{\lambda m N_A}{M} \]

### Conclusion
Thus, the initial activity of the sample is given by:
\[ A = \frac{\lambda m N_A}{M} \]