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).

Key concepts

Mass Defect

When we talk about a mass defect, we are referring to the difference in mass between the components of a nucleus when they are separate and when they are bound together. This occurs because some of the mass is converted into binding energy that holds the nucleus together. In our helium nucleus example:

• The separate particles: 2 protons + 2 neutrons.
• The combined nucleus: helium.

The total mass of the individual protons and neutrons is greater than the mass of the helium nucleus. By calculating the mass defect, we find this difference.
The mass defect helps us understand the energetics of nuclear reactions and stability of nuclei.

Mass-Energy Equivalence

Mass-energy equivalence is a concept attributed to Albert Einstein through his famous equation, \(E = mc^2\). This principle states that mass can be converted into energy and vice versa.
In the context of nuclear physics, the mass defect we calculated can be translated into nuclear binding energy. The equation uses:

• \( m \): mass defect (converted into kg).
• \( c \): speed of light (approximately \(3 \times 10^8 \text{ m/s}\)).

By using this equation, we convert the small mass defect into a considerable amount of energy, which is the binding energy of the nucleus.
This energy is crucial for understanding nuclear stability and reactions.

Nuclear Physics

Nuclear physics is a branch of physics that deals with the constituents and interactions of atomic nuclei. This field examines phenomena such as radioactivity, nuclear fusion, and fission.
The study of these interactions is essential for applications in nuclear power and medical treatments like radiation therapy.
Understanding the principles of binding energy and mass defect helps physicists to design safer nuclear reactors and develop new medical technologies.
Nuclear physics also contributes to our understanding of fundamental forces in the universe, like the strong nuclear force that holds protons and neutrons in the nucleus.

Atomic Mass Unit

The atomic mass unit (amu) is a standard unit of mass used to express atomic and molecular weights. It is defined based on a specific isotope, carbon-12, where 1 amu is one-twelfth the mass of a carbon-12 atom.
This unit provides a convenient way to measure small particles, like protons and neutrons, which are crucial in the study of mass defect.

• Proton typically has a mass of approximately 1.0073 amu.
• Neutron typically has a mass of approximately 1.0087 amu.

Having a consistent unit makes it easier to compare atomic masses and perform calculations, such as finding the mass defect, without needing to constantly convert to larger or smaller units.