Chapter Summary: Important Physics Formulae from Chapter 13 "Nuclei" (NCERT Physics)

Formula: $1 u = \frac{1}{12} \times mass of one^{12} C atom$
Explanation: Atomic mass unit (u) is used to express atomic and nuclear masses. It is defined as 1/12th of the mass of a carbon-12 atom.
Example Application: Calculating the mass of an atom in atomic mass units.
Common Mistakes: Confusing atomic mass unit (u) with kilograms.

Formula: $E = m c^{2}$
Explanation: This relation expresses the equivalence of mass and energy, where $c$ is the speed of light in a vacuum (~ $3 \times 1 0^{8}$ m/s).
Example Application: Calculating the energy equivalent of a given mass.
Common Mistakes: Misapplying the speed of light value or forgetting to square the value of $c$ .

Formula: $E_{b} = Δ M \cdot c^{2}$
Explanation: The binding energy $E_{b}$ is the energy required to separate a nucleus into its individual protons and neutrons. $Δ M$ is the mass defect, the difference between the mass of the nucleus and the sum of its individual nucleons.
Derivation Steps:
1. Calculate the mass defect $Δ M = (Z \cdot m_{p} + N \cdot m_{n}) - M$ .
2. Multiply $Δ M$ by $c^{2}$ to find the binding energy.
Example Application: Finding the binding energy for a given nucleus, e.g., $_{8}^{16} O$ .
Common Mistakes: Forgetting to use the correct mass units or neglecting electron mass in calculations.

Formula: $R = R_{0} \cdot A^{1/3}$
Explanation: The nuclear radius $R$ depends on the mass number $A$ , with $R_{0}$ being a constant (~1.2 fm).
Example Application: Estimating the size of a nucleus for an element.
Common Mistakes: Incorrectly calculating the mass number $A$ or using an incorrect value for $R_{0}$ .

Formula: $E_{bn} = \frac{E _{b}}{A}$
Explanation: It represents the average energy needed to remove a nucleon from the nucleus.
Example Application: Comparing the stability of different nuclei by their binding energy per nucleon.
Common Mistakes: Misinterpreting the stability implications of higher or lower binding energies per nucleon.

Formula: $N (t) = N_{0} \cdot e^{- λ t}$
Explanation: Describes the number of undecayed nuclei as a function of time, where $λ$ is the decay constant.
Example Application: Determining the remaining quantity of a radioactive substance after a given time period.
Common Mistakes: Misusing the decay constant or initial quantity $N_{0}$ .

Always double-check unit conversions, especially when switching between atomic mass units and kilograms.
Ensure consistency in using the speed of light ( $c$ ) value in energy calculations.
When calculating nuclear properties, ensure that all constants are accurately applied.

This summary covers key formulae from the chapter, focusing on their application and common pitfalls. It is intended to serve as a quick revision guide, enhancing understanding and problem-solving skills in preparation for NEET UG Physics.