Find the amount of heat generated by 1 mg of `Po^210` preparation during the mean life period of these nuclei if the emitted alpha particles are known to possess kinetic energy 5.3 MeV and practically all daughter nuclei are formed directly in the ground state.

Explanation

To find the amount of heat generated by 1 mg of Polonium-210 during its mean life period, we will follow these steps:

### Step 1: Calculate the number of nuclei in 1 mg of Polonium-210
We start by using the formula for the number of nuclei:
\[ N = \frac{m}{M} \times N_A \]
where:
- \(N\) = number of nuclei
- \(m\) = mass of the sample (1 mg = \(10^{-3}\) g)
- \(M\) = molar mass of Polonium-210 (approximately 210 g/mol)
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23}\) nuclei/mol)

Substituting the values:
\[ N = \frac{10^{-3} \text{ g}}{210 \text{ g/mol}} \times 6.022 \times 10^{23} \text{ nuclei/mol} \]
Calculating this gives:
\[ N \approx 2.87 \times 10^{18} \text{ nuclei} \]

### Step 2: Calculate the number of decayed nuclei during the mean life period
During one mean life period, approximately 63.8% of the nuclei decay. Thus, we calculate the number of decayed nuclei:
\[ N_{\text{decayed}} = 0.638 \times N \]
Substituting the value of \(N\):
\[ N_{\text{decayed}} = 0.638 \times 2.87 \times 10^{18} \approx 1.83 \times 10^{18} \text{ nuclei} \]

### Step 3: Calculate the total energy released
The energy released per decayed nucleus is given as 5.3 MeV. We need to convert this energy into joules:
\[ E_{\text{per nucleus}} = 5.3 \text{ MeV} = 5.3 \times 10^{6} \text{ eV} = 5.3 \times 10^{6} \times 1.6 \times 10^{-19} \text{ J} \]
Calculating this gives:
\[ E_{\text{per nucleus}} \approx 8.48 \times 10^{-13} \text{ J} \]

Now, we can find the total energy released by all decayed nuclei:
\[ E_{\text{total}} = N_{\text{decayed}} \times E_{\text{per nucleus}} \]
Substituting the values:
\[ E_{\text{total}} = 1.83 \times 10^{18} \times 8.48 \times 10^{-13} \text{ J} \approx 1.55 \times 10^{6} \text{ J} \]

### Final Answer
The amount of heat generated by 1 mg of Polonium-210 during its mean life period is approximately:
\[ E \approx 1.55 \times 10^{6} \text{ J} \]