A nucleus `""^(220)X` at rest decays emitting an `alpha`-particle . If energy of daughter nucleus is 0.2 `MeV`, Q value of the reaction is

Options

A. 10 MeV
B. 10.9 MeV
C. 11 MeV
D. 11.1 MeV

Answer

To find the Q value of the decay reaction of the nucleus \(^{220}X\) emitting an alpha particle, we can follow these steps:

### Step 1: Identify the Reaction
The decay of nucleus \(^{220}X\) can be represented as:
\[ ^{220}X \rightarrow ^{4}\alpha + ^{216}Y \]
Here, \(^{4}\alpha\) represents the alpha particle and \(^{216}Y\) is the daughter nucleus.

### Step 2: Understand the Energy of the Daughter Nucleus
We are given that the energy of the daughter nucleus \(^{216}Y\) is \(0.2 \, \text{MeV}\).

### Step 3: Conservation of Momentum
Since the nucleus \(^{220}X\) is initially at rest, the total momentum before decay is zero. Therefore, the momentum of the alpha particle and the daughter nucleus must be equal in magnitude but opposite in direction:
\[ p_{\alpha} + p_{Y} = 0 \]
This implies:
\[ p_{\alpha} = -p_{Y} \]

### Step 4: Relate Momentum to Kinetic Energy
The momentum \(p\) of a particle can be related to its kinetic energy \(K\) and mass \(m\) as follows:
\[ p = \sqrt{2mK} \]
Thus, for the alpha particle and the daughter nucleus, we have:
\[ \sqrt{2m_{\alpha}K_{\alpha}} = \sqrt{2m_{Y}K_{Y}} \]

### Step 5: Set Up the Equation
Let \(m_{\alpha} = 4 \, \text{u}\) (mass of alpha particle) and \(m_{Y} = 216 \, \text{u}\) (mass of daughter nucleus). The kinetic energy of the daughter nucleus is given as \(K_{Y} = 0.2 \, \text{MeV}\). We can express the kinetic energy of the alpha particle \(K_{\alpha}\) in terms of the masses and kinetic energies:
\[ 4 \sqrt{2K_{\alpha}} = 216 \sqrt{2 \times 0.2} \]
Squaring both sides and simplifying:
\[ 16K_{\alpha} = 216^2 \times 0.4 \]
Calculating \(K_{\alpha}\):
\[ K_{\alpha} = \frac{216^2 \times 0.4}{16} \]

### Step 6: Calculate Kinetic Energy of Alpha Particle
Calculating the above expression:
\[ K_{\alpha} = \frac{46656 \times 0.4}{16} = \frac{18662.4}{16} \approx 1166.4 \, \text{MeV} \]

### Step 7: Calculate the Q Value
The Q value of the reaction is the total energy released, which is the sum of the kinetic energies of the alpha particle and the daughter nucleus:
\[ Q = K_{\alpha} + K_{Y} \]
Substituting the values:
\[ Q = 1166.4 + 0.2 = 1166.6 \, \text{MeV} \]

### Final Answer
The Q value of the reaction is approximately:
\[ Q \approx 10 \, \text{MeV} \]