Problem 68 - A Complete Resource Book in Chemistry for JEE Main

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) \(9.18 \mathrm{MeV}\)

Step-by-Step Solution

Answer

(a) 7.18 MeV

Step 1: Determine Total Mass of Free Nucleons

A helium atom consists of 2 protons and 2 neutrons. First, we calculate the total mass of these free nucleons. The mass of the protons is \(2 \times 1.0078\) and the mass of the neutrons is \(2 \times 1.0087\). Adding these, we get: \[ 2 \times 1.0078 + 2 \times 1.0087 = 2.0156 + 2.0174 = 4.0330 \text{ amu} \]

Step 2: Calculate Mass Defect

The mass defect is the difference between the mass of the free nucleons and the actual mass of the helium nucleus. Calculate the mass defect using the formula: \[ \Delta m = (\text{total mass of free nucleons}) - (\text{mass of helium nucleus}) \] Substituting the values: \[ \Delta m = 4.0330 - 4.0026 = 0.0304 \text{ amu} \]

Step 3: Convert Mass Defect to Energy

Using Einstein’s equation \(E=mc^2\), convert the mass defect into MeV. First, convert amu to energy using the conversion factor: \(1 \text{ amu} = 931.5 \text{ MeV/c}^2\). Thus, the energy equivalent of the mass defect is: \[ E = 0.0304 \times 931.5 = 28.3246 \text{ MeV} \]

Step 4: Calculate Binding Energy Per Nucleon

The binding energy per nucleon is found by dividing the total binding energy by the number of nucleons. A helium atom has 4 nucleons, so: \[ \text{Binding energy per nucleon} = \frac{28.3246}{4} = 7.08115 \text{ MeV} \] After rounding to two decimal places, this value is approximately \(7.08 \text{ MeV}\).

Step 5: Match with Given Options

The calculated binding energy per nucleon, \(7.08 \text{ MeV}\), is closest to option (a), which is \(7.18 \text{ MeV}\). Therefore, option (a) is the correct choice.

Key Concepts

Mass Defect

Mass defect is a fascinating concept in nuclear physics that reveals the difference between the mass of an atom’s nucleus and the sum of the masses of its individual nucleons, which include protons and neutrons. This discrepancy arises because the actual nucleus is lighter than the individual masses of its components when unbound.

Understanding mass defect is crucial in explaining why energy is released during nuclear reactions. In the case of our helium atom example, the calculation involved determining the mass of two protons and two neutrons when they exist freely as opposed to being bound together within the nucleus.

The mass defect for helium was calculated as \( \Delta m = 4.0330 \text{ amu} - 4.0026 \text{ amu} = 0.0304 \text{ amu} \). This represents the mass lost due to binding, and it's integral to determining the binding energy.

Binding Energy Per Nucleon

Binding energy per nucleon is a term used to describe the energy required to separate a nucleus into its individual nucleons, averaged over the number of nucleons. This measure is crucial because it reflects the stability of a nucleus; the higher the binding energy per nucleon, the more stable the nucleus is.

In our helium example, once we computed the total binding energy using the mass defect and Einstein's equation, we then divided this energy by the number of nucleons, which for helium is four.

• Total binding energy was calculated as approximately \( 28.3246 \text{ MeV} \).
• Dividing by 4 nucleons resulted in a binding energy per nucleon of about \( 7.08 \text{ MeV} \).

This demonstrates helium's nucleic robustness and why it is resistant to certain decay processes.

Einstein's Equation E=mc^2

Einstein's famous equation \( E=mc^2 \) establishes a profound relationship between mass and energy, providing the foundation for understanding binding energy. This equation proposes that mass can be converted into an equivalent amount of energy, and vice versa.

In the context of nuclear binding energy, the mass defect calculated (the mass 'missing' between the free and bound states of nucleons) can be transformed into energy using this relationship. The conversion involves multiplying the mass defect by a factor of 931.5 MeV/c², reflecting the energy equivalent per atomic mass unit.

• In the helium example, the mass defect was \( 0.0304 \text{ amu} \).
• Using \( E = mc^2 \) and the conversion factor, the energy equivalent became approximately \( 28.3246 \text{ MeV} \).

This energy is the nuclear binding energy, representing the work done to bring nucleons together or the energy released when they are separated.