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. `mN_Alambda`

Explanation

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

### Step 1: Understand the Concept of Activity
The activity \( A \) of a radioactive sample is defined as the number of decays per unit time. It is given by the formula:
\[ A = \lambda N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive nuclei present in the sample.

### Step 2: Calculate the Number of Nuclei
To find \( N \), we need to determine how many radioactive particles (nuclei) are present in the sample. The number of nuclei can be calculated using the mass of the sample and its molecular weight:
\[ N = \frac{m}{M} \times N_A \]
where:
- \( m \) is the mass of the radioactive material,
- \( M \) is the molecular weight of the material,
- \( N_A \) is Avogadro's constant (the number of particles in one mole).

### 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
Rearranging the above equation gives us:
\[ A = \frac{\lambda m N_A}{M} \]

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

### Final Answer
The initial activity of the sample is \( \frac{\lambda m N_A}{M} \).

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