Physics · Modern Physics

Atoms and Nuclei formulas for JEE

Every Atoms and Nuclei formula you need for JEE, grouped by concept.

FormulasRevision notes
27 formulas2 concepts
01Atomic Models and Spectra02Nuclear Structure and Size
01

Atomic Models and Spectra

15 formulas

Coulomb Force on Alpha Particle

F=14πϵ0(2e)(Ze)r2F = \frac{1}{4\pi\epsilon_0} \frac{(2e)(Ze)}{r^2}

Electrostatic force between an incoming alpha particle and a target nucleus.

applies whenAssuming the heavy target nucleus remains stationary.
atomic-modelrutherfordforce

Energy of Electron in Bohr Orbit

En=−me48ϵ02h2n2=−13.6 eVn2E_n = -\frac{me^4}{8\epsilon_0^2 h^2 n^2} = -\frac{13.6 \text{ eV}}{n^2}

Total energy of the electron in the nth stationary state.

applies whenHydrogen atom. For hydrogen-like ions, total energy scales as Z^2/n^2.
bohrenergy

Bohr Quantization of Angular Momentum

L=mvr=nh2πL = mvr = \frac{nh}{2\pi}

The angular momentum of an electron in a stable orbit is an integral multiple of h/2π.

applies whenCircular allowed orbits in the Bohr model.
bohrangular-momentumquantization

Bohr Orbit Radius

rn=n2h2ϵ0πme2=a0n2r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2} = a_0 n^2

Radius of the nth allowed stationary orbit in a hydrogen atom.

applies whenHydrogen atom. For hydrogen-like ions (Z > 1), the radius scales as 1/Z.
bohrradius

Distance of Closest Approach

d=14πϵ02Ze2Kd = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{K}

The minimum distance an alpha particle reaches from the nucleus before reversing direction.

applies whenHead-on collision (impact parameter b=0).
rutherfordclosest-approachscattering

De Broglie Standing Wave Condition

2πrn=nλ=n(hmv)2\pi r_n = n\lambda = n\left(\frac{h}{mv}\right)

The circumference of an allowed orbit equals an integral multiple of the electron's de Broglie wavelength.

applies whenResonant standing waves on a circular orbit.
de-brogliewave-particleorbit

Impact Parameter

b=14πϵ0Ze2cot⁡(θ/2)Kb = \frac{1}{4\pi\epsilon_0} \frac{Z e^2 \cot(\theta/2)}{K}

Perpendicular distance of the initial velocity vector of the alpha particle from the central axis of the nucleus.

applies whenPure Coulomb scattering without penetration of the nucleus.
rutherfordimpact-parameterjee-advanced

Kinetic and Potential Energy Relations

K=−E,U=2EK = -E, \quad U = 2E

Relationship between kinetic energy (K), potential energy (U), and total energy (E) for a bound electron.

applies whenInverse-square central force (Coulomb field).
bohrenergy-relations

Moseley's Law for Characteristic X-Rays

ν=a(Z−b)\sqrt{\nu} = a(Z - b)

Empirical law relating the frequency of characteristic X-rays to the atomic number.

applies whenApplies to characteristic X-ray emission spectra.
x-raymoseleyjee-advanced

Energy Levels with Reduced Mass

En=−μe48ϵ02h2n2E_n = -\frac{\mu e^4}{8\epsilon_0^2 h^2 n^2}

Modified Bohr energy levels taking into account the finite mass of the nucleus.

applies whenWhen the mass of the nucleus is comparable to the electron mass. μ=mMm+M\mu = \frac{mM}{m+M}.
bohrreduced-massjee-advanced

Rydberg Formula

1λ=RHZ2(1nf2−1ni2)\frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)

Calculates the wavelength of emitted light for transitions between orbital levels.

applies whenHydrogenic atoms transitioning from n_i to n_f.
spectrumrydbergjee-advanced

Maximum Number of Emission Lines

N=n(n−1)2N = \frac{n(n-1)}{2}

Maximum possible number of emission lines when a gas of atoms is excited to the nth state.

applies whenA large sample of atoms.
spectrumlinesjee-advanced

Energy of Emitted Photon

hν=Ei−Efh\nu = E_i - E_f

Energy of a photon emitted when an electron transitions to a lower energy state.

applies whenTransition from a higher initial state (E_i) to a lower final state (E_f).
bohrspectrumphoton

Fine Structure Velocity Relation

vn=c137Znv_n = \frac{c}{137} \frac{Z}{n}

Speed of the electron expressed as a fraction of the speed of light.

applies whenHydrogenic atoms.
bohrvelocityjee-advanced

Electron Velocity in Bohr Orbit

vn=e22ϵ0hnv_n = \frac{e^2}{2\epsilon_0 h n}

Speed of the revolving electron in the nth Bohr orbit.

applies whenHydrogen atom. For hydrogen-like ions, velocity scales as Z/n.
bohrvelocity
02

Nuclear Structure and Size

12 formulas

Radioactive Activity

R=∣dNdt∣=λN=R0e−λtR = \left| \frac{dN}{dt} \right| = \lambda N = R_0 e^{-\lambda t}

The instantaneous rate of disintegration of a radioactive sample.

radioactivityactivityjee-advanced

Kinetic Energy of Alpha Particle

Kα=A−4AQK_\alpha = \frac{A-4}{A} Q

Kinetic energy of the emitted alpha particle in radioactive decay, derived from momentum conservation.

applies whenParent nucleus initially at rest.
alpha-decaykinematicsjee-advanced

Binding Energy per Nucleon

Ebn=EbAE_{bn} = \frac{E_b}{A}

The average energy per nucleon needed to separate a nucleus. This determines true nuclear stability.

binding-energystability

Total Binding Energy

Eb=ΔMc2E_b = \Delta M c^2

Energy required to completely separate a nucleus into its constituent free nucleons.

binding-energy

Radioactive Decay Law

N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Formula representing the exponential decay of a radioactive sample over time.

applies whenSpontaneous decay process.
radioactivitydecayjee-advanced

Nuclear Density

ρ=3mavg4πR03\rho = \frac{3m_{avg}}{4\pi R_0^3}

Density of nuclear matter, roughly constant for all nuclei.

applies whenAssuming a uniform spherical drop approximation; independent of mass number A.
nucleusdensity

Half-Life

T1/2=ln⁡2λ≈0.693λT_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}

Time taken for exactly half the active nuclei in a given sample to decay.

radioactivityhalf-lifejee-advanced

Mass Defect

ΔM=[Zmp+(A−Z)mn]−Mnucleus\Delta M = [Z m_p + (A-Z)m_n] - M_{nucleus}

The difference between the sum of the masses of individual nucleons and the actual mass of the nucleus.

binding-energymass-defect

Mass-Energy Equivalence

E=mc2E = mc^2

Einstein's relation showing that mass is another form of energy.

relativitymass-energy

Mean Life

τ=1λ=T1/2ln⁡2\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2}

The average lifespan of a radioactive nucleus, at which the population reduces to 1/e of the initial value.

radioactivitymean-lifejee-advanced

Q-Value (Disintegration Energy)

Q=(∑mreactants−∑mproducts)c2Q = (\sum m_{reactants} - \sum m_{products}) c^2

The net energy released or absorbed in a nuclear reaction due to mass difference.

applies whenIf Q>0Q > 0, the reaction is exothermic. If Q<0Q < 0, it is endothermic.
nuclear-reactionq-value

Nuclear Radius

R=R0A1/3R = R_0 A^{1/3}

Empirical formula for the radius of a nucleus based on its mass number.

applies whenR0≈1.2 fmR_0 \approx 1.2 \text{ fm}.
nucleusradius
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