Question

The mass of helium atom of mass number 4 is \(4.0026\) amu, while that of neutron and proton is \(1.0087\) and \(1.0078\) respectively in the same scale. Hence, the nuclear binding per nucleon in the helium atom is (a) \(7.18 \mathrm{MeV}\) (b) \(6.18 \mathrm{MeV}\) (c) \(8.18 \mathrm{MeV}\) (d) \(9.18 \mathrm{MeV}\)

Short answer

The nuclear binding energy per nucleon in helium is approximately 7.08 MeV, which doesn't exactly match any given options.

Step-by-step solution

Calculate Total Mass of Nucleons

A helium atom with a mass number of 4 consists of 2 protons and 2 neutrons. The mass of 2 protons is \(2 \times 1.0078\) amu and the mass of 2 neutrons is \(2 \times 1.0087\) amu. Calculate the total mass of the nucleons: \[ \text{Total mass of nucleons} = 2\times1.0078 + 2\times1.0087 = 4.033\text{ amu} \]

Find Mass Defect

The mass defect is the difference between the total mass of the nucleons and the actual mass of the helium atom. Calculate the mass defect: \[ \text{Mass defect} = 4.033\text{ amu} - 4.0026\text{ amu} = 0.0304\text{ amu} \]

Convert Mass Defect to Energy

To find the energy equivalent of the mass defect, use Einstein's equation \(E=mc^2\), where \(1 \text{ amu}=931.5 \text{ MeV}\). Calculate the energy: \[ \text{Energy} = 0.0304 \text{ amu} \times 931.5 \text{ MeV/amu} = 28.32 \text{ MeV} \]

Calculate Binding Energy Per Nucleon

The binding energy per nucleon is the total binding energy divided by the number of nucleons. For helium, with 4 nucleons, the calculation is: \[ \text{Binding energy per nucleon} = \frac{28.32 \text{ MeV}}{4} = 7.08 \text{ MeV} \]

Key concepts

Mass Defect

In simple terms, the mass defect is the difference between the sum of the individual masses of the protons and neutrons in a nucleus, and the actual mass of the atomic nucleus. When nucleons (protons and neutrons) come together to form a nucleus, some of the mass is lost. This lost mass is called the mass defect. It's fascinating to think that this lost mass doesn't just disappear. Rather, it gets converted into energy that holds the nucleus together, which we refer to as nuclear binding energy.
To illustrate, consider a helium atom: by calculating the mass of its nucleons (2 protons and 2 neutrons), we get a total mass of 4.033 amu. The heliums actual mass is slightly less at 4.0026 amu. The difference, or mass defect, is 0.0304 amu. This very subtle yet significant difference is what delights physics enthusiasts!

Einstein's Equation

One of the most celebrated equations in physics is Einstein’s equation: \(E=mc^2\). This simple yet powerful formula relates mass \(m\) to energy \(E\) with \(c\), the speed of light, acting as a conversion factor. It implies that mass can be converted to energy and vice versa.
In our example of the helium atom, the mass defect of 0.0304 amu is converted directly into energy using this equation. As energy is needed to keep the nucleus together, this allows us to compute the binding energy. Given that 1 amu approximately equals 931.5 MeV, this conversion demonstrates how even a small amount of mass can be transformed into a substantial amount of energy, epitomizing the concept of nuclear binding.

Binding Energy Per Nucleon

Binding energy per nucleon represents the average energy that is needed to disassemble a nucleus into its individual protons and neutrons. It's calculated by dividing the total binding energy by the number of nucleons within the nucleus. This value helps us understand how stable a nucleus is; the higher the binding energy per nucleon, the more stable the nucleus.
In helium’s case, the total binding energy is calculated to be 28.32 MeV. Since helium consists of 4 nucleons (2 protons and 2 neutrons), the binding energy per nucleon is approximately 7.08 MeV. This number underscores helium’s relative stability compared to other elements, which is why it’s so abundantly found and used in numerous applications.