How many `beta`-particles are emitted during one hour by `1.0 mu g` of `Na^(24)` radionuclide whose half-life is `15` hours? [Take `e^((-0.693//15))=0.955=0.9555`, and avagadro number `=6xx10^(23)`]

Explanation

To solve the problem of how many beta particles are emitted during one hour by 1.0 µg of Na-24 radionuclide with a half-life of 15 hours, we can follow these steps:

### Step 1: Calculate the number of atoms in 1.0 µg of Na-24
1. **Convert the mass of Na-24 to grams**:
\[ 1.0 \, \mu g = 1.0 \times 10^{-6} \, g \]
2. **Calculate the number of moles of Na-24**:
The molar mass of Na-24 is approximately 24 g/mol.
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{1.0 \times 10^{-6} \, g}{24 \, g/mol} \approx 4.17 \times 10^{-8} \, mol \]
3. **Using Avogadro's number to find the number of atoms**:
\[ N_0 = \text{Number of moles} \times \text{Avogadro's number} = 4.17 \times 10^{-8} \, mol \times 6.022 \times 10^{23} \, atoms/mol \approx 2.51 \times 10^{16} \, atoms \]

### Step 2: Calculate the decay constant (λ)
The decay constant (λ) is given by the formula:
\[ \lambda = \frac{0.693}{T_{1/2}} \]
Where \( T_{1/2} \) is the half-life.
\[ \lambda = \frac{0.693}{15 \, hours} \approx 0.0462 \, hours^{-1} \]

### Step 3: Calculate the remaining number of atoms after 1 hour
Using the formula for radioactive decay:
\[ N = N_0 e^{-\lambda t} \]
Where \( t = 1 \, hour \).
\[ N = 2.51 \times 10^{16} \times e^{-0.0462 \times 1} \]
Given \( e^{-0.0462} \approx 0.955 \) (as provided in the question),
\[ N \approx 2.51 \times 10^{16} \times 0.955 \approx 2.398 \times 10^{16} \, atoms \]

### Step 4: Calculate the number of decayed atoms (beta particles emitted)
The number of decayed atoms (which corresponds to the number of beta particles emitted) is:
\[ \text{Decayed atoms} = N_0 - N \]
\[ \text{Decayed atoms} = 2.51 \times 10^{16} - 2.398 \times 10^{16} \approx 1.12 \times 10^{15} \, beta \, particles \]

### Final Answer
Approximately \( 1.12 \times 10^{15} \) beta particles are emitted during one hour by 1.0 µg of Na-24.

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