The value of decay constant of `Co^(60)` is `2.5xx 10^(-7)"min"^(-1)` . The activity of 2.0 g of the sample is nearly :

Options

A. `5 xx 10^(5)` dpm
B. `2.5 xx 10^(10)`dpm
C. `5xx10^(15)` dpm
D. `10^(10)`dpm

Answer

To find the activity of a 2.0 g sample of \( \text{Co}^{60} \) given its decay constant, we can follow these steps:

### Step 1: Calculate the number of moles of \( \text{Co}^{60} \)

First, we need to find the molar mass of \( \text{Co}^{60} \). The atomic mass of cobalt (Co) is approximately 60 g/mol.

\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{2.0 \, \text{g}}{60 \, \text{g/mol}} = \frac{1}{30} \, \text{mol} \approx 0.0333 \, \text{mol} \]

### Step 2: Calculate the number of atoms in the sample

Using Avogadro's number (\( 6.022 \times 10^{23} \, \text{atoms/mol} \)), we can find the total number of atoms in the sample.

\[ \text{Number of atoms} = \text{Number of moles} \times \text{Avogadro's number} = 0.0333 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 2.01 \times 10^{22} \, \text{atoms} \]

### Step 3: Calculate the activity using the decay constant

The activity \( A \) of a radioactive sample is given by the formula:

\[ A = \lambda N \]

where \( \lambda \) is the decay constant and \( N \) is the number of atoms.

Given:
- Decay constant \( \lambda = 2.5 \times 10^{-7} \, \text{min}^{-1} \)
- Number of atoms \( N \approx 2.01 \times 10^{22} \, \text{atoms} \)

Now we can calculate the activity:

\[ A = (2.5 \times 10^{-7} \, \text{min}^{-1}) \times (2.01 \times 10^{22} \, \text{atoms}) \approx 5.025 \times 10^{15} \, \text{decays/min} \]

### Final Answer

The activity of the 2.0 g sample of \( \text{Co}^{60} \) is approximately \( 5.025 \times 10^{15} \, \text{decays/min} \).

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