Question

The binding energy Per nucleon of deuteron \(\left({ }^{2}{ }_{1} \mathrm{H}\right)\) and Lielium nucleus \({ }_{2}{ }^{4}{ }_{2} \mathrm{He}\) ) is \(1.1 \mathrm{MeV}\) and \(7.0 \mathrm{MeV}\). respectively. If two deuteron react to form a single helium nucleus, the energy released is (A) \(23.6 \mathrm{MeV}\) (B) \(26.9 \mathrm{MeV}\) (C) \(13.9 \mathrm{MeV}\) (D) \(19.2 \mathrm{MeV}\)

Short answer

The energy released in the reaction when two deuterons react to form a single helium nucleus is \(23.6\,\text{MeV}\).

Step-by-step solution

Calculate the total binding energy of two deuterons

First, we need to find the total binding energy of two deuterons: Binding energy per nucleon of a deuteron: \(1.1\,\text{MeV}\) Each deuteron contains 2 nucleons, so the binding energy for a single deuteron is: \[\text{Binding energy of a deuteron} = 1.1\,\text{MeV/nucleon} \times 2\,\text{nucleons} = 2.2\,\text{MeV}\] Since we have two deuterons, the total binding energy for two deuterons is: \[\text{Total binding energy of two deuterons} = 2.2\,\text{MeV/deuteron} \times 2\,\text{deuterons} = 4.4\,\text{MeV}\]

Calculate the binding energy of a helium nucleus

Now, we need to find the total binding energy of the helium nucleus: Binding energy per nucleon of a helium nucleus: \(7.0\,\text{MeV}\) The helium nucleus contains 4 nucleons, so the binding energy for a single helium nucleus is: \[\text{Binding energy of a helium nucleus} = 7.0\,\text{MeV/nucleon} \times 4\,\text{nucleons} = 28.0\,\text{MeV}\]

Calculate the energy released in the reaction

The energy released in the reaction is the difference between the total binding energy of the initial system (two deuterons) and the final system (one helium nucleus): \[\text{Energy released} = \text{Binding energy of the helium nucleus} - \text{Total binding energy of two deuterons}\] \[\text{Energy released} = 28.0\,\text{MeV} - 4.4\,\text{MeV} = 23.6\,\text{MeV}\] The energy released in the reaction is \(23.6\,\text{MeV}\). Therefore, the correct answer is (A) \(23.6\,\text{MeV}\).

Key concepts

Binding Energy per Nucleon

Understanding the concept of nuclear binding energy per nucleon is key to comprehending why some nuclear reactions release energy. Each nucleus is composed of protons and neutrons, together called nucleons. The nuclear binding energy per nucleon is the energy required to disassemble a nucleus into its individual nucleons.

It's a measure of the stability of a nucleus; the higher the binding energy per nucleon, the more stable the nucleus is. To be precise:

• Binding Energy per Nucleon = Total Binding Energy / Number of Nucleons

In the given example, we have a deuteron with a binding energy of 1.1 MeV per nucleon and helium with 7.0 MeV per nucleon. This higher value for helium indicates it is more tightly bound and thus more stable than the deuteron.

When two deuterons fuse to form helium, they transition from a less stable to a more stable state, releasing energy in the process.

Nuclear Fusion

Nuclear fusion is a powerful process in which two light atomic nuclei combine to form a heavier nucleus. This process is occurring continuously within stars, including our sun, as hydrogen atoms fuse to form helium.

Fusion releases a considerable amount of energy, primarily because the resulting nucleus has a higher binding energy per nucleon. For instance, in the exercise we considered the fusion of two deuterons (each a hydrogen isotope) to create a helium nucleus.

During the fusion:

• The total mass of the resulting nucleus is slightly less than the sum of the original deuterons.
• This mass difference, known as mass defect, is converted into energy according to Einstein’s equation: \[ E=mc^2 \] where \( E \) is energy, \( m \) is mass defect, and \( c \) is the speed of light.

This energy is what powers stars and has the potential for human use in fusion reactors, as it is several times greater than what can be released by fission.

Energy Released in Nuclear Reactions

In nuclear reactions, energy can be released or absorbed based on changes in nuclear binding energy. When two nuclei come together (fusion) or a single nucleus is split (fission), the energy released or absorbed is the result of differences in binding energy before and after the reaction.

In our example, the fusion of two deuterons into a helium nucleus releases energy. This energy is calculated by comparing the total initial binding energy of the deuterons with that of the resulting helium nucleus.

Here’s a step-by-step breakdown:

• Calculate the binding energy for both initial and final nuclei using binding energy per nucleon values.
• Subtract the total binding energy of the initial state (two deuterons) from that of the final state (helium nucleus).
• The difference gives the energy released in the reaction.

This released energy is crucial, as it illustrates how nuclear reactions can provide vast amounts of energy, potentially enabling advanced technologies and impacting energy generation on Earth.