Atomic weight of Th is 232 and its atomic number is 90. The number of `alpha`- and `beta`-particles which be lost so that an isotope of lead (atomic weight 208 and atomic number 82) is produced is

Options

A. `4 alpha + 6 beta`
B. `6 alpha + 4 beta`
C. `8 alpha + 2 beta`
D. `10 alpha + 2 beta`

Explanation

To solve the problem, we need to determine how many alpha (α) and beta (β) particles are emitted from thorium (Th) to produce lead (Pb).

### Step 1: Write the nuclear reaction
We start with thorium-232 (Th-232) and want to transform it into lead-208 (Pb-208). The reaction can be represented as:

\[ \text{Th}_{90}^{232} \rightarrow \text{Pb}_{82}^{208} + n_1 \alpha + n_2 \beta \]

where \( n_1 \) is the number of alpha particles and \( n_2 \) is the number of beta particles.

### Step 2: Calculate the change in mass number
The mass number of thorium is 232 and that of lead is 208. The change in mass number is:

\[ \Delta A = A_{\text{Th}} - A_{\text{Pb}} = 232 - 208 = 24 \]

### Step 3: Determine the contribution of alpha particles
Each alpha particle has a mass number of 4. Therefore, the total change in mass number due to alpha particles can be expressed as:

\[ n_1 \times 4 = 24 \]

From this, we can solve for \( n_1 \):

\[ n_1 = \frac{24}{4} = 6 \]

### Step 4: Calculate the change in atomic number
The atomic number of thorium is 90 and that of lead is 82. The change in atomic number is:

\[ \Delta Z = Z_{\text{Th}} - Z_{\text{Pb}} = 90 - 82 = 8 \]

### Step 5: Determine the contribution of alpha particles to atomic number
Each alpha particle decreases the atomic number by 2. Therefore, the total change in atomic number due to alpha particles is:

\[ \Delta Z_{\alpha} = n_1 \times 2 = 6 \times 2 = 12 \]

### Step 6: Calculate the net change in atomic number
Now, we need to account for the total change in atomic number:

\[ \Delta Z_{\text{total}} = \Delta Z_{\alpha} + n_2 \]

We know that:

\[ 12 + n_2 = 8 \]

### Step 7: Solve for \( n_2 \)
Rearranging gives:

\[ n_2 = 8 - 12 = -4 \]

Since \( n_2 \) must be positive, we realize that we need to account for the fact that beta particles increase the atomic number by 1. Thus, we can write:

\[ n_2 = 12 - 8 = 4 \]

### Final Answer
Thus, the number of alpha particles \( n_1 = 6 \) and the number of beta particles \( n_2 = 4 \).

### Summary
The final answer is:
- Number of alpha particles (n1) = 6
- Number of beta particles (n2) = 4

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