A nucleus `X^232` initially at rest undergoes `alpha` decay, the `alpha`-particle produced in above process is found to move in a circular track of radius 0.22 m in a uniform magnetic field of 1.5 T. Find Q value of reaction.

Answer

To find the Q value of the reaction for the alpha decay of the nucleus \( X^{232} \), we can follow these steps:

### Step 1: Understand the Problem
The nucleus \( X^{232} \) undergoes alpha decay, producing an alpha particle and a residual nucleus. The alpha particle moves in a circular path in a magnetic field, and we are given the radius of this path and the strength of the magnetic field.

### Step 2: Use the Formula for Radius in a Magnetic Field
The radius \( R \) of the circular path of a charged particle in a magnetic field is given by the formula:
\[ R = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

### Step 3: Rearranging the Formula
We can rearrange this formula to find the kinetic energy \( E \) of the alpha particle:
\[ E = \frac{1}{2}mv^2 \]
From \( R = \frac{mv}{qB} \), we can express \( v \) as:
\[ v = \frac{qBR}{m} \]
Substituting this into the kinetic energy formula gives:
\[ E = \frac{1}{2}m\left(\frac{qBR}{m}\right)^2 = \frac{q^2B^2R^2}{2m} \]

### Step 4: Calculate the Charge and Mass of the Alpha Particle
The charge \( q \) of an alpha particle (which consists of 2 protons) is:
\[ q = 2e = 2 \times 1.6 \times 10^{-19} \, \text{C} = 3.2 \times 10^{-19} \, \text{C} \]
The mass \( m \) of an alpha particle is approximately:
\[ m = 4 \, \text{u} = 4 \times 1.66 \times 10^{-27} \, \text{kg} \approx 6.64 \times 10^{-27} \, \text{kg} \]

### Step 5: Substitute Values into the Energy Equation
Substituting the known values into the energy equation:
- \( R = 0.22 \, \text{m} \)
- \( B = 1.5 \, \text{T} \)

We get:
\[ E = \frac{(3.2 \times 10^{-19})^2 \times (1.5)^2 \times (0.22)^2}{2 \times (6.64 \times 10^{-27})} \]

### Step 6: Calculate the Energy
Calculating the above expression step-by-step:
1. Calculate \( (3.2 \times 10^{-19})^2 = 1.024 \times 10^{-37} \)
2. Calculate \( (1.5)^2 = 2.25 \)
3. Calculate \( (0.22)^2 = 0.0484 \)
4. Combine these:
\[ E = \frac{1.024 \times 10^{-37} \times 2.25 \times 0.0484}{2 \times 6.64 \times 10^{-27}} \]

### Step 7: Find Q Value
The Q value of the reaction can be calculated from the energy of the alpha particle and the mass difference before and after the decay. The Q value is given by:
\[ Q = E + (mass \, of \, X^{232} - mass \, of \, residual \, nucleus - mass \, of \, alpha \, particle) \times c^2 \]

### Final Calculation
After calculating the energy \( E \) and considering the mass difference, we can find the Q value.