A radioactive element decays by `beta-` emission. If mass of parent and daughter nuclide are `m_(1)` and `m_(2)` respectively, calculate energy liberated during the emission.

Options

A. `[m_1-m_2-2m_e]c^2`
B. `[m_2-m_1-2m_e]c^2`
C. `[m_e - m_2-2m_1]c^2`
D. `[m_2-m_e-2m_1]c^2`

Answer

To calculate the energy liberated during the beta- emission of a radioactive element, we can follow these steps:

### Step 1: Understand the Beta Decay Process
In beta- decay, a neutron in the nucleus of the parent nuclide (mass m1) converts into a proton, emitting a beta particle (an electron) and an antineutrino. The mass number remains the same, but the atomic number increases by one.

### Step 2: Write the Mass Relationship
The mass of the parent nuclide is denoted as \( m_1 \) and the mass of the daughter nuclide is denoted as \( m_2 \). During the decay, the emitted beta particle (electron) has a mass, which we can denote as \( m_e \).

### Step 3: Calculate the Mass Difference
The mass difference (Δm) between the parent and daughter nuclides can be expressed as:
\[ \Delta m = m_1 - (m_2 + m_e) \]
Here, \( m_e \) is the mass of the emitted electron.

### Step 4: Calculate the Energy Released
According to Einstein's mass-energy equivalence principle, the energy released (E) during the decay can be calculated using the formula:
\[ E = \Delta m \cdot c^2 \]
Substituting the expression for Δm, we get:
\[ E = (m_1 - (m_2 + m_e)) \cdot c^2 \]

### Step 5: Substitute the Mass of the Electron
The mass of the electron \( m_e \) is approximately \( 9.11 \times 10^{-31} \) kg. Therefore, we can rewrite the energy equation as:
\[ E = (m_1 - m_2 - m_e) \cdot c^2 \]

### Final Expression
Thus, the energy liberated during the beta- emission can be expressed as:
\[ E = (m_1 - m_2 - m_e) \cdot c^2 \]