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. 10.8 MeV
B. 10.9 MeV
C. 11 MeV
D. 11.1 MeV

Answer

To find the Q value of the nuclear reaction where a nucleus \( ^{220}X \) emits an alpha particle and the energy of the daughter nucleus is given as 0.2 MeV, we can follow these steps:

### Step 1: Understanding the Reaction
The decay process can be represented as:
\[ ^{220}X \rightarrow ^{216}Y + \alpha \]
where \( ^{216}Y \) is the daughter nucleus and \( \alpha \) is the emitted alpha particle.

### Step 2: Define the Q Value
The Q value of a nuclear reaction is defined as the difference in the total energy of the reactants and the total energy of the products. Mathematically, it can be expressed as:
\[ Q = E_{\text{products}} - E_{\text{reactants}} \]

### Step 3: Initial Energy of the Reactants
Since the nucleus \( ^{220}X \) is at rest, its initial energy is purely its rest mass energy. However, for the purpose of calculating Q value, we consider the initial kinetic energy to be zero:
\[ E_{\text{reactants}} = 0 \quad (\text{since it is at rest}) \]

### Step 4: Energy of the Products
The energy of the products includes the kinetic energy of the daughter nucleus and the kinetic energy of the emitted alpha particle. The energy of the daughter nucleus is given as 0.2 MeV. Let’s denote the kinetic energy of the alpha particle as \( E_{\alpha} \).

### Step 5: Conservation of Momentum
Since the initial momentum is zero (the nucleus is at rest), the momentum of the daughter nucleus and the alpha particle must be equal and opposite:
\[ p_{\text{daughter}} + p_{\alpha} = 0 \]
This implies:
\[ p_{\text{daughter}} = -p_{\alpha} \]

### Step 6: Calculate the Kinetic Energy of the Alpha Particle
Using the relationship between momentum and kinetic energy, we can express the momentum of the daughter nucleus and the alpha particle:
\[ p_{\text{daughter}} = \sqrt{2 m_{\text{daughter}} E_{\text{daughter}}} \]
\[ p_{\alpha} = \sqrt{2 m_{\alpha} E_{\alpha}} \]
Setting these equal gives:
\[ \sqrt{2 \cdot 196 \cdot 0.2} = \sqrt{2 \cdot 4 \cdot E_{\alpha}} \]
Squaring both sides:
\[ 2 \cdot 196 \cdot 0.2 = 2 \cdot 4 \cdot E_{\alpha} \]
Simplifying:
\[ 39.2 = 8 E_{\alpha} \]
\[ E_{\alpha} = \frac{39.2}{8} = 4.9 \text{ MeV} \]

### Step 7: Total Energy of the Products
Now, we can find the total energy of the products:
\[ E_{\text{products}} = E_{\text{daughter}} + E_{\alpha} = 0.2 \text{ MeV} + 4.9 \text{ MeV} = 5.1 \text{ MeV} \]

### Step 8: Calculate the Q Value
Now substituting back into the Q value equation:
\[ Q = E_{\text{products}} - E_{\text{reactants}} = 5.1 \text{ MeV} - 0 = 5.1 \text{ MeV} \]

### Final Answer
Thus, the Q value of the reaction is:
\[ \boxed{5.1 \text{ MeV}} \]