`_86A^222rarr_84B^210`. In this reaction, how many `alpha` and `beta` particles are emitted?

Options

A. `6alpha , 3 beta`
B. `3 alpha , 4 beta`
C. `4 alpha , 3 beta`
D. `3 alpha, 6 beta`

Explanation

To solve the problem of how many alpha and beta particles are emitted in the reaction `_86A^222 → _84B^210`, we will follow these steps:

### Step 1: Identify the Initial and Final States
- The initial element A has:
- Mass number (A) = 222
- Atomic number (Z) = 86

- The final element B has:
- Mass number (A) = 210
- Atomic number (Z) = 84

### Step 2: Calculate the Change in Mass Number
- The change in mass number (ΔA) can be calculated as:
\[ \Delta A = A_{initial} - A_{final} = 222 - 210 = 12 \]

### Step 3: Determine the Number of Alpha Particles Emitted
- Each alpha particle emission decreases the mass number by 4. Therefore, the number of alpha particles (nα) emitted can be calculated as:
\[ n_{\alpha} = \frac{\Delta A}{4} = \frac{12}{4} = 3 \]

### Step 4: Calculate the Change in Atomic Number
- The change in atomic number (ΔZ) can be calculated as:
\[ \Delta Z = Z_{initial} - Z_{final} = 86 - 84 = 2 \]

### Step 5: Relate Alpha and Beta Particle Emissions
- Each alpha particle emission decreases the atomic number by 2. Therefore, the contribution of alpha particles to the atomic number change is:
\[ \text{Change due to alpha particles} = 2 \times n_{\alpha} = 2 \times 3 = 6 \]

- Since the atomic number decreases by 6 due to alpha emissions, we need to find how many beta particles (nβ) are emitted to adjust the atomic number back to 84. Each beta particle increases the atomic number by 1. Thus, we can set up the equation:
\[ Z_{final} = Z_{initial} - 2n_{\alpha} + n_{\beta} \]
Plugging in the values:
\[ 84 = 86 - 6 + n_{\beta} \]

### Step 6: Solve for the Number of Beta Particles
- Rearranging the equation gives:
\[ n_{\beta} = 84 - 80 = 4 \]

### Conclusion
- The total emissions are:
- Number of alpha particles emitted (nα) = 3
- Number of beta particles emitted (nβ) = 4

Thus, the final answer is:
- **3 alpha particles and 4 beta particles are emitted.**

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