`._(90)Th^(232) to ._(82)Pb^(208)`. The number of `alpha and beta-"particles"` emitted during the above reaction is

Options

A. `4alpha,7beta`
B. `6alpha,4beta`
C. `8alpha,7beta`
D. `4alpha,4beta`

Answer

To solve the problem of determining the number of alpha and beta particles emitted during the decay of Thorium-232 (90Th) to Lead-208 (82Pb), we can follow these steps:

### Step 1: Identify the Mass and Atomic Numbers
- The initial isotope is Thorium-232 with atomic number 90 (90Th232).
- The final isotope is Lead-208 with atomic number 82 (82Pb208).

### 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 (A) is calculated as:
\[ \Delta A = A_{\text{initial}} - A_{\text{final}} = 232 - 208 = 24 \]

### Step 3: Determine the Number of Alpha Particles Emitted
- Each alpha particle (α) has a mass number of 4.
- To find the number of alpha particles emitted, we divide the change in mass number by the mass number of an alpha particle:
\[ n_{\alpha} = \frac{\Delta A}{4} = \frac{24}{4} = 6 \]
- Thus, 6 alpha particles are emitted.

### 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 (Z) is calculated as:
\[ \Delta Z = Z_{\text{initial}} - Z_{\text{final}} = 90 - 82 = 8 \]

### Step 5: Determine the Number of Beta Particles Emitted
- Each alpha particle reduces the atomic number by 2, and each beta particle (β) increases the atomic number by 1.
- Let \( n_{\beta} \) be the number of beta particles emitted. The relationship can be expressed as:
\[ n_{\alpha} \cdot 2 - n_{\beta} = \Delta Z \]
- Substituting the known values:
\[ 6 \cdot 2 - n_{\beta} = 8 \]
\[ 12 - n_{\beta} = 8 \]
\[ n_{\beta} = 12 - 8 = 4 \]
- Thus, 4 beta particles are emitted.

### Final Answer
- The total emissions during the decay of Thorium-232 to Lead-208 are:
- Alpha particles: 6
- Beta particles: 4

### Summary
- The number of alpha particles emitted is 6.
- The number of beta particles emitted is 4.