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

1. Atomic Mass Unit

• Formula: $1u=\frac{1}{12}\times {\text{mass of one }}^{12}C\text{ atom}$
• 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.

2. Einstein's Mass-Energy Equivalence

• 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 (~$3\times 10^8$ 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 $c$.

3. Nuclear Binding Energy

• Formula: $E_b=\Delta M\cdot c^2$
• Explanation: The binding energy $E_b$ is the energy required to separate a nucleus into its individual protons and neutrons. $\Delta M$ 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 $\Delta M=(Z\cdot m_p+N\cdot m_n)-M$.
  2. Multiply $\Delta M$ 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.

4. Nuclear Radius

• 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 $R_0$.

5. Binding Energy Per Nucleon

• 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.

6. Radioactive Decay Law

• Formula: $N(t)=N_0\cdot e^{-\lambda t}$
• Explanation: Describes the number of undecayed nuclei as a function of time, where $\lambda$ 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 $N_0$.

Tips to Avoid Common Mistakes:

• 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.

Final Review:

• 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.