Chemistry · Physical Chemistry

Atomic Structure formulas for JEE

Every Atomic Structure formula you need for JEE, grouped by concept.

FormulasRevision notes
49 formulas3 concepts
01Atomic Models02Quantum Mechanical Model of Atom03Electronic Configuration of Atoms
01

Atomic Models

21 formulas

Bohr radius constant

a0=4πε0ℏ2mee2a_0=\frac{4\pi\varepsilon_0\hbar^2}{m_e e^2}

Definition of Bohr radius constant.

applies whenSI form; constants as usual.
Bohrconstants

Bohr quantization

mevnrn=nℏm_ev_nr_n=n\hbar

Quantization of angular momentum in Bohr model.

applies whenBohr model; hydrogen-like species.
Bohrquantization

Bohr energy levels

En=−13.6Z2n2 eVE_n=-13.6\frac{Z^2}{n^2}\ \text{eV}

Energy of electron in nth orbit (Bohr).

applies whenHydrogen-like ions; negative indicates bound state.
Bohrenergy_levels

Bohr orbit radius (H-like)

rn=a0n2Zr_n=\frac{a_0 n^2}{Z}

Radius of nth Bohr orbit.

applies whenHydrogen-like ions; Z is atomic number.
Bohrradius

Rydberg formula

1λ=R(1n12−1n22)\frac{1}{\lambda}=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)

Hydrogen spectrum wavelengths for transitions.

applies whenFor hydrogen-like species; n2>n1 for emission.
hydrogenspectrumRydberg

Quantized angular momentum

mevr=nh2πm_e v r = \frac{nh}{2\pi}

Bohr's quantization condition for angular momentum of an electron in a stationary orbit.

applies whenn=1,2,3...n = 1, 2, 3...
atomic-structurebohr-modelangular-momentum

Bohr energy (Hydrogen)

En=−2.18×10−18(1n2) JE_n = -2.18 \times 10^{-18} \left(\frac{1}{n^2}\right) \text{ J}

Energy of an electron in the nth stationary state of a hydrogen atom.

applies whenHydrogen atom (Z=1Z=1).
atomic-structurebohr-modelenergy

Bohr energy (H-like ions)

En=−2.18×10−18(Z2n2) JE_n = -2.18 \times 10^{-18} \left(\frac{Z^2}{n^2}\right) \text{ J}

Energy of an electron in the nth stationary state of a hydrogen-like species.

applies whenSingle electron species.
atomic-structurebohr-modelenergy

Bohr energy relations

TE=−KE=PE2TE = -KE = \frac{PE}{2}

Relationship between Total Energy, Kinetic Energy, and Potential Energy in a Bohr orbit.

applies whenCircular orbits in hydrogen-like species.
atomic-structurebohr-modelenergyjee-advanced

Bohr's frequency rule

ν=ΔEh=E2−E1h\nu = \frac{\Delta E}{h} = \frac{E_2 - E_1}{h}

Frequency of radiation absorbed or emitted during electronic transition.

applies whenTransition between two stationary states.
atomic-structurebohr-modelspectra

Bohr radius (Hydrogen)

rn=52.9n2 pmr_n = 52.9 n^2 \text{ pm}

Radius of the nth stationary state for a hydrogen atom.

applies whenHydrogen atom (Z=1Z=1).
atomic-structurebohr-modelradius

Bohr radius (H-like ions)

rn=52.9n2Z pmr_n = 52.9 \frac{n^2}{Z} \text{ pm}

Radius of the nth stationary state for hydrogen-like species.

applies whenSingle electron species (e.g., He+He^+, Li2+Li^{2+}).
atomic-structurebohr-modelradius

Bohr orbit time period

Tn=2πrnvn∝n3Z2T_n = \frac{2\pi r_n}{v_n} \propto \frac{n^3}{Z^2}

Time period of revolution of an electron in a Bohr orbit.

applies whenSingle electron species.
atomic-structurebohr-modeltime-periodjee-advanced

Bohr orbit velocity

vn=2.18×106(Zn) m/sv_n = 2.18 \times 10^6 \left(\frac{Z}{n}\right) \text{ m/s}

Velocity of an electron in the nth Bohr orbit of a hydrogen-like species.

applies whenSingle electron species.
atomic-structurebohr-modelvelocityjee-advanced

Quantization of charge

q=neq = n e

The magnitude of electrical charge is an integral multiple of the fundamental charge.

applies whenn=1,2,3...n = 1, 2, 3...
atomic-structuremillikanoil-drop

Mass number

A=Z+nA = Z + n

Total number of nucleons (protons Z and neutrons n) in a nucleus.

atomic-structurenucleusnucleons

Rydberg formula (Hydrogen)

νˉ=109677(1n12−1n22) cm−1\bar{\nu} = 109677 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \text{ cm}^{-1}

Wavenumber of spectral lines for hydrogen atom transitions.

applies whenn2>n1n_2 > n_1
atomic-structurespectrarydberg

Rydberg formula (H-like ions)

νˉ=RHZ2(1n12−1n22)\bar{\nu} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

Generalized Rydberg formula for hydrogen-like species.

applies whenSingle electron species.
atomic-structurespectrarydbergjee-advanced

Rydberg reduced mass correction

RM=R∞MM+meR_M = R_\infty \frac{M}{M + m_e}

Correction to the Rydberg constant accounting for the finite mass of the nucleus M.

applies whenHighly accurate spectroscopic calculations.
atomic-structurespectrarydbergjee-advanced

Specific charge of electron

e/me=1.758820×1011 C kg−1e/m_e = 1.758820 \times 10^{11} \text{ C kg}^{-1}

The ratio of electrical charge to the mass of an electron determined by J.J. Thomson.

applies whenValid for electrons (cathode rays).
atomic-structuresubatomic-particlesthomson

Transition energy

ΔE=hν=hcλ\Delta E=h\nu=\frac{hc}{\lambda}

Photon energy equals level energy difference.

applies whenRadiative transition; emission/absorption.
spectrumtransition
02

Quantum Mechanical Model of Atom

24 formulas

De Broglie wavelength

λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}

Wavelength associated with a material particle of mass m moving with velocity v.

applies whenNon-relativistic speeds.
atomic-structurede-brogliedual-nature

Electron de Broglie (accelerated)

λ=h2meeV≈150V A˚\lambda = \frac{h}{\sqrt{2m_e e V}} \approx \sqrt{\frac{150}{V}} \text{ \AA}

De Broglie wavelength of an electron accelerated through a potential difference V.

applies whenNon-relativistic electron.
atomic-structurede-brogliepotentialjee-advanced

De Broglie wavelength (Kinetic Energy)

λ=h2mK\lambda = \frac{h}{\sqrt{2mK}}

De Broglie wavelength in terms of kinetic energy K.

applies whenNon-relativistic speeds.
atomic-structurede-brogliekinetic-energy

Photon energy (Planck)

E=hνE = h\nu

Energy of a quantum of radiation is proportional to its frequency.

atomic-structureplanckquantum

Photon energy from wavelength

E=hcλE = \frac{hc}{\lambda}

Energy of a photon expressed in terms of its wavelength.

atomic-structureplanckquantum

Photon momentum

p=hλ=Ecp = \frac{h}{\lambda} = \frac{E}{c}

Momentum of a photon.

atomic-structurephotonmomentumjee-advanced

Wavenumber

νˉ=1λ\bar{\nu} = \frac{1}{\lambda}

Number of wavelengths per unit length.

atomic-structureelectromagnetic-radiationwave

Wave relation for EMR

c=νλc = \nu \lambda

Relationship between speed of light, frequency, and wavelength in vacuum.

applies whenIn vacuum.
atomic-structureelectromagnetic-radiationwave

Angular nodes

ll

Number of angular nodes (nodal planes/cones) for a given orbital.

atomic-structureorbitalsnodes

Radial nodes

n−l−1n - l - 1

Number of spherical/radial nodes for a given orbital.

atomic-structureorbitalsnodes

Total nodes

n−1n - 1

Total number of nodes (regions of zero probability density) for an orbital.

atomic-structureorbitalsnodes

Einstein photoelectric equation

hν=hν0+12mev2h\nu = h\nu_0 + \frac{1}{2}m_ev^2

Conservation of energy principle for the photoelectric effect.

applies whenIncident frequency must be strictly greater than threshold frequency (ν>ν0\nu > \nu_0).
atomic-structurephotoelectric-effecteinstein

Stopping potential

eVs=Kmax=hν−W0eV_s = K_{max} = h\nu - W_0

Relationship between stopping potential and maximum kinetic energy of emitted photoelectrons.

applies whenν>ν0\nu > \nu_0
atomic-structurephotoelectric-effectstopping-potentialjee-advanced

Threshold frequency

ν0=W0h\nu_0 = \frac{W_0}{h}

Minimum frequency of light required to cause photoelectric emission.

atomic-structurephotoelectric-effectthresholdjee-advanced

Work function

W0=hν0W_0 = h\nu_0

Minimum energy required to eject an electron from a metal surface.

atomic-structurephotoelectric-effectwork-function

Spin-only magnetic moment

μ=n(n+2) B.M.\mu = \sqrt{n(n+2)} \text{ B.M.}

Magnetic moment of an atom/ion based on the number of unpaired electrons n.

applies whenn = number of unpaired electrons.
atomic-structuremagnetic-momentjee-advanced

Orbital angular momentum

L=l(l+1)h2πL = \sqrt{l(l+1)}\frac{h}{2\pi}

Magnitude of the orbital angular momentum of an electron in a subshell.

atomic-structurequantum-numbersangular-momentumjee-advanced

Orbitals in a shell

n2n^2

Total number of allowed orbitals in a shell with principal quantum number n.

atomic-structurequantum-numbersorbitals

Spin angular momentum

S=s(s+1)h2πS = \sqrt{s(s+1)}\frac{h}{2\pi}

Magnitude of the spin angular momentum of an electron.

applies whenwhere s=1/2s = 1/2
atomic-structurequantum-numbersangular-momentumjee-advanced

Number of orbitals in subshell

2l+12l+1

Total number of orbitals in a subshell characterized by azimuthal quantum number l.

atomic-structurequantum-numbersorbitals

Spin magnetic quantum number

ms=±12m_s = \pm \frac{1}{2}

Allowed values for the intrinsic spin orientation of an electron.

atomic-structurequantum-numbersspin

Heisenberg uncertainty (Momentum)

Δx⋅Δpx≥h4π\Delta x \cdot \Delta p_x \ge \frac{h}{4\pi}

Uncertainty principle relating position and momentum.

applies whenMicroscopic particles.
atomic-structureheisenberguncertainty

Heisenberg uncertainty (Energy-Time)

ΔE⋅Δt≥h4π\Delta E \cdot \Delta t \ge \frac{h}{4\pi}

Uncertainty principle relating energy and time.

atomic-structureheisenberguncertaintyjee-advanced

Heisenberg uncertainty (Velocity)

Δx⋅Δvx≥h4πm\Delta x \cdot \Delta v_x \ge \frac{h}{4\pi m}

Uncertainty principle relating position and velocity.

applies whenMicroscopic particles with constant mass.
atomic-structureheisenberguncertainty
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03

Electronic Configuration of Atoms

4 formulas

Max electrons in shell

2n22n^2

Maximum number of electrons in a shell with principal quantum number n.

atomic-structurequantum-numberselectrons

Max electrons in subshell

2(2l+1)2(2l+1)

Maximum number of electrons a subshell can hold.

atomic-structurequantum-numberselectrons

Number of orbitals in subshell

2l+12l+1

Total number of orbitals in a subshell characterized by azimuthal quantum number l.

atomic-structurequantum-numbersorbitals

Spin magnetic quantum number

ms=±12m_s = \pm \frac{1}{2}

Allowed values for the intrinsic spin orientation of an electron.

atomic-structurequantum-numbersspin
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