Question

The masses of neutron and Proton are \(1.0087\) amu and \(1.0073\) amu respectively. It the neuron and Protons combins to form Helium nucleus of mass \(4.0015\) amu then binding energy of the helium nucleus will be (A) \(14.2 \mathrm{MeV}\) (B) \(28.4 \mathrm{MeV}\) (C) \(27.3 \mathrm{MeV}\) (D) \(20.8 \mathrm{MeV}\)

Short answer

The binding energy of the Helium nucleus is \(28.4 \mathrm{MeV}\), which corresponds to option (B).

Step-by-step solution

Determine mass defect

First, we need to find the mass defect, which is the difference in mass between the initial particles (neutrons and protons) and the final Helium nucleus. The Helium nucleus has 2 protons and 2 neutrons. So we will find the total mass of the 2 protons and 2 neutrons, and then subtract the mass of the Helium nucleus.

Calculate masses of individual particles

The masses of a neutron and a proton are given as 1.0087 amu and 1.0073 amu, respectively. To find the total mass of the 2 protons and 2 neutrons, we need to multiply their respective masses by 2 and add them together: Total mass of 2 protons = 2 * 1.0073 amu = 2.0146 amu Total mass of 2 neutrons = 2 * 1.0087 amu = 2.0174 amu Combined mass of 2 protons and 2 neutrons = 2.0146 amu + 2.0174 amu = 4.032 amu

Find mass defect

Now that we have the combined mass of the protons and neutrons, we can find the mass defect by subtracting the mass of the Helium nucleus from the combined mass: Mass defect = 4.032 amu - 4.0015 amu = 0.0305 amu

Convert mass defect to energy

To convert the mass defect to energy, we need to use the mass-energy equivalence formula, E=mc^2, where E is energy, m is mass, and c is the speed of light (3.0 x 10^8 m/s). First, we need to convert the mass defect from amu to kg to be consistent with units: 1 amu = 1.66054 x 10^-27 kg Mass defect (in kg) = 0.0305 amu * (1.66054 x 10^-27 kg/amu) = 5.064 x 10^-29 kg Now, we can plug this mass defect into the mass-energy equivalence formula: E = (5.064 x 10^-29 kg) * (3 x 10^8 m/s)^2 = 4.557 x 10^-12 Joules

Convert energy from Joules to MeV

Finally, we need to convert the energy from Joules to MeV (Mega-electronvolts) for the final answer. Since 1 MeV = 1.60219 x 10^-13 Joules, the energy in MeV will be: Binding energy = (4.557 x 10^-12 Joules) / (1.60219 x 10^-13 Joules/MeV) = 28.4 MeV So the binding energy of the Helium nucleus is 28.4 MeV, which corresponds to option (B).