Problem 17 - A Complete Resource Book in Chemistry for JEE Main

Question

If the mass defect of \({ }_{4} \mathrm{X}^{9}\) is \(0.090 \mathrm{amu}\), then the binding energy per nucleon is \((1 \mathrm{amu}=921.5 \mathrm{MeV})\) (a) \(83.0 \mathrm{MeV}\) (b) \(9.315 \mathrm{MeV}\) (c) \(8.38 \mathrm{MeV}\) (d) \(931.5 \mathrm{MeV}\)

Step-by-Step Solution

Answer

The binding energy per nucleon is approximately \(9.315\, \text{MeV}\). Option (b) is correct.

Step 1: Understand the Mass Defect

In nuclear physics, the mass defect is the difference between the mass of the whole nucleus and the sum of the individual masses of the protons and neutrons. It represents the mass converted into energy to bind the nucleus.

Step 2: Calculate Total Binding Energy

The total binding energy is calculated as the mass defect multiplied by the energy equivalent per atomic mass unit (amu). In this case, the mass defect is given as \(0.090\, \text{amu}\), and the conversion factor is \(921.5\, \text{MeV/amu}\). Thus, the binding energy (\(E\)) is given by:\[E = 0.090 \times 921.5 = 82.935 \, \text{MeV}\]

Step 3: Determine Number of Nucleons

The isotope is \({ }_{4} \text{X}^{9}\). This notation indicates that there are \(9\) nucleons in total in the nucleus (the sum of protons and neutrons).

Step 4: Calculate Binding Energy per Nucleon

To find the binding energy per nucleon, divide the total binding energy by the number of nucleons:\[\frac{E}{ ext{nucleons}} = \frac{82.935}{9} = 9.215\, \text{MeV/nucleon}\]

Step 5: Match with Given Options

Compare the calculated binding energy per nucleon with the options provided. The closest match is option \((b)\ 9.315 \, \text{MeV}\).

Key Concepts

Mass Defect

In the realm of nuclear physics, the concept of mass defect is quite fundamental. It refers to the difference between the observed mass of a nucleus and the sum of the individual masses of its constituent protons and neutrons. This isn't merely an error in measurement but rather a reflection of a core principle: the mass defect represents the conversion of mass into energy, which is the binding energy that holds the nucleus together.
This concept is deeply rooted in Einstein's famous equation, \[E = mc^2\]which suggests that energy and mass are interchangeable. Here, \(E\) refers to energy, \(m\) is mass, and \(c\) is the speed of light. The mass defect results from this conversion, showcasing how even a tiny amount of mass can yield significant energy. This underscores a fundamental insight of nuclear physics, illustrating why nuclear processes release so much energy compared to chemical reactions.

Nucleons

Protons and neutrons, collectively termed as nucleons, are the primary building blocks of atomic nuclei. The distinction between these particles lies in their charge: protons are positively charged, while neutrons are neutral. The sum of these particles forms the nucleon number or mass number of an atom.
In nuclear notation, such as in the isotope representation \( {}_{4} ext{X}^{9}\),the number "9" represents the total nucleons. Understanding this nucleon count is vital when calculating various nuclear properties, including the binding energy per nucleon. It provides vital insight into the structure of the atom and its stability.
Nucleons are held together in the nucleus by nuclear forces, which are much stronger than the electromagnetic forces keeping electrons in orbit. This strong interaction is the primary force overcoming the repulsive electrostatic force between positively charged protons and thus preventing the atomic nucleus from flying apart.

Nuclear Physics

Nuclear physics is the study of the components and behavior of atomic nuclei. It explores phenomena such as nuclear decay, interactions in nuclear reactions, and the forces that operate within nuclei.
One of its core focus areas is the binding energy, which arises from the mass defect. Understanding binding energy helps explain why certain isotopes are more stable than others and is essential for applications including nuclear power generation and medical imaging technologies like PET scans.
Besides energy release, nuclear physics also delves into the implications of nuclear reactions in the cosmos. For example, fusion reactions in stars, where lighter nuclei combine to form a heavier nucleus, releasing energy that provides the immense light and heat of stars. This overreaching influence of nuclear physics spans both practical applications and fundamental cosmological processes, making it a rich field of study both in academia and industry.