Problem

Find the mass defect and binding energy for a helium nucleus.

Data

• Number of protons ($Z$): $2$
• Mass of hydrogen ($M_H$): $1.007825 \, \mathrm{u}$
• Number of neutrons ($N$): $2$
• Mass of neutron ($M_n$): $1.008665 \, \mathrm{u}$
• Mass of helium nucleus ($M$): $4.002602 \, \mathrm{u}$
• Conversion factor ($c^2$): $931.5 \, \mathrm{MeV}/\mathrm{u}$

Prerequisite Concepts

1. Mass Defect: The difference between the total mass of the nucleons and the actual mass of the nucleus.

$$\Delta m = ZM_H + NM_n - M$$

2. Binding Energy ($E_B$): The energy equivalent of the mass defect, calculated using Einstein’s relation:

$$E_B = \Delta m \cdot c^2$$

Here, $c^2 = 931.5 \, \mathrm{MeV}/\mathrm{u}$.

Solution

Step 1: Calculate the Mass Defect ($\Delta m$)

Using the formula:

$$\Delta m = ZM_H + NM_n - M$$

Substitute the values:

$$\Delta m = 2(1.007825 \, \mathrm{u}) + 2(1.008665 \, \mathrm{u}) - 4.002602 \, \mathrm{u}$$ $$\Delta m = 2.01565 \, \mathrm{u} + 2.01733 \, \mathrm{u} - 4.002602 \, \mathrm{u}$$ $$\Delta m = 0.30378 \, \mathrm{u}$$

Step 2: Calculate the Binding Energy ($E_B$)

Using the formula:

$$E_B = \Delta m \cdot c^2$$

Substitute the values:

$$E_B = 0.30378 \cdot 931.5 \, \mathrm{MeV}$$ $$E_B = 28.297107 \, \mathrm{MeV}$$

Answer

1. Mass Defect ($\Delta m$): $0.30378 \, \mathrm{u}$
2. Binding Energy ($E_B$): $28.297107 \, \mathrm{MeV}$