The decay constant for a radioactive nuclide is 1.5 10^-5 s^-1. Atomic weight of the substance is 60 g mole^-1 (NA = 6 10^23). The activity of 1.0 g of the substance is:

Given: decay constant = 1.5 10^-5 s^-1, mass of sample m = 1.0 10^-6 g, molar mass M = 60 g mole^-1, and NA = 6 10^23. Find: the activity of the sample. The activity is given by A = N First calculate the number of moles: n = mM = 1 10^-660 = 10^-76 Now calculate the number of atoms: N = nNA = 10^-76 6 10^23 = 10^16 Therefore, the activity is A = N = 1.5 10^-5 10^16 = 15 10^10 Bq Therefore, the required numerical value is 15. Given: use A = N. 1. Number of moles: No. of moles = Mass of sample Molar mass = 1 10^-660 = 10^-76 2. Number of atoms: No. of atoms = n NA = 10^-76 6 10^23 = 10^16 3. Initial activity: A0 = N0 = 10^16 1.5 10^-5 = 15 10^10 Bq Conclude: the activity is 15 10^10 Bq, so the numerical answer is 15.

The decay constant for a radioactive nuclide is 1.5 10^-5 s^-1. A… — Solution | JEE Physics

NVAEasyJEE 2023Radioactive Decay & Half-Life

The decay constant for a radioactive nuclide is $1.5 \times 10^{-5} \, \text{s}^{-1}$ . Atomic weight of the substance is $60 \, \text{g mole}^{-1}$ ( $N_A = 6 \times 10^{23}$ ). The activity of $1.0 \, \mu\text{g}$ of the substance is:

Answer

Correct answer:15

Step-by-step solution

Standard Method

Given: decay constant $\lambda = 1.5 \times 10^{-5} \, \text{s}^{-1}$ , mass of sample $m = 1.0 \times 10^{-6} \, \text{g}$ , molar mass $M = 60 \, \text{g mole}^{-1}$ , and $N_A = 6 \times 10^{23}$ .

Find: the activity of the sample.

The activity is given by

A = \lambda N

First calculate the number of moles:

n = \frac{m}{M} = \frac{1 \times 10^{-6}}{60} = \frac{10^{-7}}{6}

Now calculate the number of atoms:

N = nN_A = \frac{10^{-7}}{6} \cdot 6 \times 10^{23} = 10^{16}

Therefore, the activity is

A = \lambda N = 1.5 \times 10^{-5} \times 10^{16} = 15 \times 10^{10} \, \text{Bq}

Therefore, the required numerical value is 15.

Stepwise Calculation

Given: use $A = \lambda N$ .

Number of moles:

\text{No. of moles} = \frac{\text{Mass of sample}}{\text{Molar mass}} = \frac{1 \times 10^{-6}}{60} = \frac{10^{-7}}{6}

Number of atoms:

\text{No. of atoms} = n \cdot N_A = \frac{10^{-7}}{6} \cdot 6 \times 10^{23} = 10^{16}

Initial activity:

A_0 = N_0\lambda = 10^{16} \cdot 1.5 \times 10^{-5} = 15 \times 10^{10} \, \text{Bq}

Conclude: the activity is $15 \times 10^{10} \, \text{Bq}$ , so the numerical answer is 15.

Common mistakes

Using the mass $1.0 \, \mu\text{g}$ directly without converting it to grams is incorrect because the molar mass is given in $\text{g mole}^{-1}$ . First convert $1.0 \, \mu\text{g} = 1.0 \times 10^{-6} \, \text{g}$ .
Calculating activity from moles directly is wrong because activity depends on the number of radioactive atoms, not the number of moles. After finding moles, multiply by $N_A$ to get $N$ , then use $A = \lambda N$ .
Writing the final numerical answer as $1.5 \times 10^{11}$ in the answer field is not appropriate for an NVA question when the platform expects only the coefficient asked from $15 \times 10^{10}$ . Enter only 15 as the numerical value.

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