In an `alpha` -decay, the kinetic energy of `alpha`-particles is `48 MeV` and `Q` value of the reaction is `50 MeV`. The mass number of the mother nucleus is (assume that daughter nucleus is in ground state)

Options

A. 96
B. 100
C. 104
D. none of these

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the alpha decay process
In an alpha decay, a mother nucleus (let's denote it as \( X \)) with mass number \( A \) and atomic number \( Z \) transforms into a daughter nucleus and an alpha particle. The daughter nucleus will have a mass number of \( A - 4 \) and an atomic number of \( Z - 2 \).

### Step 2: Write the equation for kinetic energy of the alpha particle
The kinetic energy \( KE \) of the emitted alpha particle can be expressed in terms of the Q value of the reaction and the mass numbers involved. The formula is:

\[ KE = \frac{(A - 4)}{(A - 4) + 4} \times Q \]

Where:
- \( KE \) is the kinetic energy of the alpha particle.
- \( Q \) is the Q value of the reaction.

### Step 3: Substitute the known values
From the problem, we know:
- \( KE = 48 \, \text{MeV} \)
- \( Q = 50 \, \text{MeV} \)

Substituting these values into the equation gives us:

\[ 48 = \frac{(A - 4)}{(A - 4) + 4} \times 50 \]

### Step 4: Simplify the equation
Simplifying the equation:

\[ 48 = \frac{(A - 4)}{A} \times 50 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:

\[ 48A = 50(A - 4) \]

### Step 6: Expand and rearrange the equation
Expanding the right side:

\[ 48A = 50A - 200 \]

Rearranging gives:

\[ 50A - 48A = 200 \]

### Step 7: Solve for \( A \)
This simplifies to:

\[ 2A = 200 \]

Dividing both sides by 2:

\[ A = 100 \]

### Conclusion
The mass number of the mother nucleus is \( A = 100 \).

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