Atoms & Nuclei: Comprehensive NEET Physics Formulae

1. Atoms

1.1 Rutherford's Nuclear Model

• Formula for Electrostatic Force:
  $F=\frac{1}{4\pi {ϵ}_0}\cdot \frac{Z{e}^2}{r^2}$
• • Explanation: The force of attraction between the nucleus (charge $Ze$) and the electron (charge $e$) is inversely proportional to the square of the distance $r$ between them.
  • Conditions: Assumes point charges and Coulombic interactions.
• Kinetic Energy of an Electron in Orbit:
  $K=\frac{1}{2}m{v}^2=\frac{Z{e}^2}{8\pi {ϵ}_{0}r}$
• • Explanation: The kinetic energy is derived from the centripetal force needed to keep the electron in its orbit.
• Potential Energy of Electron-Nucleus System:
  $U=-\frac{Z{e}^2}{4\pi {ϵ}_{0}r}$
• • Explanation: The potential energy is negative, indicating that work is required to separate the electron from the nucleus.
• Total Energy of an Electron:
  $E=K+U=-\frac{Z{e}^2}{8\pi {ϵ}_{0}r}$
• • Explanation: The total energy is the sum of kinetic and potential energy, and it is negative, indicating a bound system.

Common Mistake: Students often confuse the signs of potential energy and total energy. Remember that potential energy is negative, and total energy is less than zero in a bound system.

1.2 Bohr’s Model of Hydrogen Atom

• Quantization of Angular Momentum:
  $L=n\frac{h}{2\pi }$
• • Explanation: Angular momentum is quantized and is an integer multiple of $\frac{h}{2\pi }$.
• Radius of nth Orbit:
  $r_n=\frac{n^2{h}^2{ϵ}_0}{\pi m{e}^2}$
• • Explanation: The radius increases with the square of the principal quantum number $n$.
• Energy of Electron in nth Orbit:
  $E_n=-\frac{13.6\text{eV}}{n^2}$
• • Explanation: Energy levels are quantized and inversely proportional to the square of $n$.
• Frequency of Emitted Photon:
  $\nu =\frac{E_i-E_f}{h}$
• • Explanation: The frequency of the photon emitted during a transition between orbits is related to the energy difference.

NEET Tip: When calculating energy levels, always ensure that $n$ is correctly identified, as mistakes here can lead to incorrect energy values.

2. Nuclei

2.1 Mass-Energy Equivalence

• Einstein's Mass-Energy Relation:
  $E=m{c}^2$
• • Explanation: Energy and mass are interchangeable, with $c$ representing the speed of light in vacuum.

Common Mistake: Confusing the energy units—always ensure that mass is in kilograms and energy in joules for consistent results.

2.2 Nuclear Binding Energy

• Binding Energy per Nucleon:
  $BE/A=\frac{\Delta m{c}^2}{A}$
• • Explanation: Binding energy per nucleon indicates the stability of a nucleus.
• Binding Energy:
  $BE=\Delta m{c}^2$
• • Explanation: The energy required to disassemble a nucleus into its constituent protons and neutrons.

NEET Problem-Solving Strategy: Always calculate the mass defect before applying the mass-energy equivalence formula. Incorrect mass defect calculations will lead to wrong binding energy results.

2.3 Radioactive Decay

• Decay Law:
  $N(t)=N_0{e}^{-\lambda t}$
• • Explanation: The number of undecayed nuclei decreases exponentially over time.
• Half-Life:
  $T_{1/2}=\frac{\ln (2)}{\lambda }$
• • Explanation: The time required for half the nuclei in a sample to decay.
• Activity (Rate of Decay):
  $A=\lambda N$
• • Explanation: The activity of a radioactive substance is proportional to the number of undecayed nuclei.

Did You Know?: The concept of half-life is crucial in radiocarbon dating, which is used to determine the age of archaeological finds.

Quick Recap

• Electrostatic Force: $F=\frac{1}{4\pi {ϵ}_0}\cdot \frac{Z{e}^2}{r^2}$
• Energy of nth Orbit: $E_n=-\frac{13.6\text{eV}}{n^2}$
• Mass-Energy Equivalence: $E=m{c}^2$
• Radioactive Decay: $N(t)=N_0{e}^{-\lambda t}$

Practice Questions

1. Calculate the radius of the second orbit in a hydrogen atom using Bohr's model.
2. Determine the binding energy per nucleon for a nucleus with mass defect $\Delta m=0.03u$.
3. Given a half-life of 10 years, how much of a 100g sample remains after 30 years?

Solutions:

1. Using the formula rn​=πme2n2h