Physics · Modern Physics

Atoms and Nuclei formulas for JEE

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

27 formulas2 concepts

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QuestionState the formula — Coulomb Force on Alpha Particle

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Coulomb Force on Alpha Particle

F = \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

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All 27 Atoms and Nuclei formulas

Atomic Models and Spectra

15 formulas

Coulomb Force on Alpha Particle

F = \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

E_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 = \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

r_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 = \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

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 = \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, \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

\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

E_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.

\mu = \frac{mM}{m+M}

bohrreduced-massjee-advanced

Rydberg Formula

\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 = \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

h\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

v_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

v_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

Nuclear Structure and Size

12 formulas

Radioactive Activity

R = \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

K_\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

E_{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

E_b = \Delta M c^2

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

binding-energy

Radioactive Decay Law

N(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

\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

T_{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

\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 = mc^2

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

relativitymass-energy

Mean Life

\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 = (\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 > 0

, the reaction is exothermic. If

Q < 0

, it is endothermic.

nuclear-reactionq-value

Nuclear Radius

R = R_0 A^{1/3}

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

applies when

R_0 \approx 1.2 \text{ fm}

nucleusradius

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