Let's calculate the radioactive activity of the Cesium-137 sample. The activity decreases over time due to radioactive decay. We are given the half-life, the initial mass of the source, and the time elapsed since its preparation.

Cesium-137 Decay Parameters

Here are the given parameters for the Cesium-137 ($^{137}$Cs) decay:

• Half-life ($T_{1/2}$) = 30.08 years
• Initial mass (m) = 1 mg
• Preparation date: January 1, 1980
• Measurement date: January 1, 2021

Time Elapsed Calculation

First, we need to find the time elapsed ($t$) between the preparation date and the measurement date.

Time elapsed ($t$) = Measurement Year - Preparation Year

$t = 2021 - 1980 = 41$ years

Decay Constant Calculation

The decay constant ($\lambda$) is related to the half-life by the formula:

$\lambda = \frac{\ln(2)}{T_{1/2}}$

To calculate the activity in Becquerels (Bq), which is disintegrations per second, we need the decay constant in units of s^-1. First, let's find $\lambda$ in year^-1:

$\lambda_{year} = \frac{0.6931}{30.08 \text{ years}} \approx 0.02304 \text{ year}^{-1}$

Now, convert years to seconds (1 year $\approx$ 365.25 days $\times$ 24 hours/day $\times$ 3600 seconds/hour $\approx$ 31,557,600 seconds):

$\lambda_{sec} = \frac{0.6931}{30.08 \text{ years}} \times \frac{1 \text{ year}}{31,557,600 \text{ seconds}}$

$\lambda_{sec} = \frac{0.6931}{949,626,480} \text{ s}^{-1} \approx 7.30 \times 10^{-10} \text{ s}^{-1}$

Initial Number of Nuclei

We need to find the initial number of $^{137}$Cs nuclei ($N_0$) in 1 mg of the source. We can use the molar mass of $^{137}$Cs (approximately 137 g/mol) and Avogadro's number ($N_A = 6.022 \times 10^{23}$ mol^-1).

Initial mass ($m$) = 1 mg = $1 \times 10^{-3}$ g

Molar mass of $^{137}$Cs = 137 g/mol

Number of moles = $\frac{\text{Mass}}{\text{Molar mass}} = \frac{1 \times 10^{-3} \text{ g}}{137 \text{ g/mol}}$

$N_0 = \text{Number of moles} \times N_A = \left(\frac{1 \times 10^{-3}}{137}\right) \times (6.022 \times 10^{23})$

$N_0 = \frac{6.022 \times 10^{20}}{137} \approx 4.396 \times 10^{18}$ nuclei

Initial Activity Calculation

The initial activity ($A_0$) is given by $A_0 = \lambda_{sec} N_0$.

$A_0 = (7.30 \times 10^{-10} \text{ s}^{-1}) \times (4.396 \times 10^{18})$

$A_0 \approx 3.21 \times 10^9$ Bq

Activity at Time t

The activity ($A_t$) at a time $t$ after the initial time is given by the radioactive decay formula:

$A_t = A_0 e^{-\lambda_{year} t}$

We calculated $A_0 \approx 3.21 \times 10^9$ Bq, $\lambda_{year} \approx 0.02304$ year^-1, and $t = 41$ years.

$\lambda_{year} t \approx 0.02304 \times 41 \approx 0.94464$

$A_t \approx (3.21 \times 10^9) e^{-0.94464}$

Calculate $e^{-0.94464}$:

$e^{-0.94464} \approx 0.3888$

Now, calculate $A_t$:

$A_t \approx (3.21 \times 10^9) \times 0.3888$

$A_t \approx 1.248 \times 10^9$ Bq

The calculated activity on January 1, 2021, is approximately $1.248 \times 10^9$ Bq.

Comparing with Options

Let's compare our calculated value with the given options:

• Option 1: $1.79 \times 10^{16}$
• Option 2: $1.79 \times 10^9$
• Option 3: $1.24 \times 10^{16}$
• Option 4: $1.24 \times 10^9$

Our calculated value, $1.248 \times 10^9$, is closest to $1.24 \times 10^9$.

The activity of the Cesium-137 source on January 1, 2021, is closest to $1.24 \times 10^9$ disintegrations per second (Bq).