If `bX^(a)` species emit firstly a positron then two `alpha` and `beta` last one `alpha` is also emitted and finally convert in `dY^(c )` species so correct the relation is

Options

A. a=c+12,d=b-5
B. a=c-8,d=b-1
C. a=c-6,d=b-0
D. a=c-4,d=b-2

Explanation

To solve the problem, we need to analyze the decay process of the species \( bX^{(a)} \) and derive the relationships between the mass numbers and atomic numbers of the parent and daughter species. Here is a step-by-step solution:

### Step 1: Identify the Initial and Final Species
- The initial species is \( bX^{(a)} \) where:
- Mass number (A) = \( a \)
- Atomic number (Z) = \( b \)

- The final species after decay is \( dY^{(c)} \) where:
- Mass number (C) = \( c \)
- Atomic number (D) = \( d \)

### Step 2: List the Emissions
According to the problem, the emissions during the decay process are:
1. One positron (\( \beta^+ \))
2. Two alpha particles (\( \alpha \))
3. One beta particle (\( \beta^- \))
4. One alpha particle (\( \alpha \))

### Step 3: Write Down the Contributions to Mass and Atomic Numbers
- **Positron**:
- Mass number = 0
- Atomic number = +1

- **Alpha particles** (2 emissions):
- Each alpha particle:
- Mass number = 4
- Atomic number = +2
- Total for two alpha particles:
- Mass number = \( 2 \times 4 = 8 \)
- Atomic number = \( 2 \times 2 = 4 \)

- **Beta particles** (2 emissions):
- Each beta particle:
- Mass number = 0
- Atomic number = -1
- Total for two beta particles:
- Mass number = \( 0 \)
- Atomic number = \( 2 \times (-1) = -2 \)

### Step 4: Conservation of Mass Number
Using the conservation of mass number:
\[ a = c + 0 + 8 + 0 + 4 \]
This simplifies to:
\[ a = c + 12 \]
So, we have our first relationship:
\[ c = a - 12 \]

### Step 5: Conservation of Atomic Number
Using the conservation of atomic number:
\[ b = d + 1 + 4 - 2 + 2 \]
This simplifies to:
\[ b = d + 5 \]
Rearranging gives us:
\[ d = b - 5 \]

### Step 6: Summary of Relationships
From the above steps, we have derived two key relationships:
1. \( c = a - 12 \)
2. \( d = b - 5 \)

### Step 7: Conclusion
The correct relationships derived from the decay process are:
- \( a = c + 12 \)
- \( d = b - 5 \)

Thus, the correct option is:
**Option 1: \( a = c + 12 \) and \( d = b - 5 \)**.

---