Atomic Structure formulas for JEE Main & Advanced

16 formula entries for Atomic Structure, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Atomic Number (Z) & Mass Number (A)

Z = protons

A = protons + neutrons

When to use. When identifying elements, calculating neutrons, or dealing with isotopes.

Trap. Confusing atomic number with mass number; Z defines element identity.

Bohr's Model of Hydrogen Atom

m v r = \frac{n h}{2 π}

E_{n} = - \frac{13.6}{n^{2}} eV

r_{n} = 0.529 \times n^{2} \overset{˚}{A}

When to use. For hydrogen atom energy levels, radii, and spectral lines.

Trap. Using $E_{n}$ formula for multi-electron atoms; it's only for H-like species.

Energy of Electron in H-like Atom

E_{n} = - 13.6 \frac{Z^{2}}{n^{2}} eV

E_{n} = - 2.18 \times 10^{- 18} \frac{Z^{2}}{n^{2}} J

When to use. For H, He+, Li2+ etc. to find energy of any orbit.

Trap. Forgetting to square Z; using Z instead of Z^2.

Radius of Bohr Orbit

r_{n} = 0.529 \frac{n^{2}}{Z} \overset{˚}{A}

r_{n} = 5.29 \times 10^{- 11} \frac{n^{2}}{Z} m

When to use. To find orbit radius for H-like atoms.

Trap. Using formula for hydrogen (Z=1) for other species without adjusting Z.

Velocity of Electron in Bohr Orbit

v_{n} = 2.18 \times 10^{6} \frac{Z}{n} m/s

When to use. To find speed of electron in any Bohr orbit.

Trap. Using formula for hydrogen only; Z must be included.

Wavelength of Emitted Photon (Rydberg Formula)

\frac{1}{λ} = R_{H} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})

R_{H} = 1.097 \times 10^{7} m^{- 1}

Δ E = 13.6 (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}) eV

When to use. For calculating wavelength or energy of spectral lines in hydrogen.

Trap. Using n1 > n2; n1 is lower energy level, n2 is higher.

Calculating Number of Spectral Lines

Number of lines = \frac{n (n - 1)}{2}

where n is the initial principal quantum number.

When to use. When electron de-excites from a high energy level to ground state.

Trap. Using formula for transitions to a specific level; it's for all possible downward transitions.

de Broglie Wavelength

λ = \frac{h}{p} = \frac{h}{m v}

When to use. For particles like electrons, protons, neutrons to find wavelength.

Trap. Forgetting that h is Planck's constant; using wrong units.

Heisenberg Uncertainty Principle

Δ x \cdot Δ p \geq \frac{h}{4 π}

Δ x \cdot Δ (m v) \geq \frac{h}{4 π}

When to use. When dealing with microscopic particles and their dual nature.

Trap. Using $\frac{h}{2 π}$ instead of $\frac{h}{4 π}$ .

Number of Nodes

Total nodes = n - 1

Angular nodes = l

Radial nodes = n - l - 1

When to use. To find number of nodes in any orbital.

Trap. Confusing angular and radial nodes; angular nodes depend on l.

Photoelectric Effect

K E_{max} = h ν - ϕ

ϕ = h ν_{0}

(threshold frequency)

When to use. When light interacts with metal and ejects electrons.

Trap. Using intensity instead of frequency; intensity affects number, not KE.

Energy of Photon

E = h ν = \frac{h c}{λ}

When to use. For any electromagnetic radiation.

Trap. Using wrong units; h in J s, c in m/s, λ in m.

Ionization Energy and Spectral Transitions

I E = E_{\infty} - E_{1} = 13.6 eV

When to use. Calculating energy to remove electron.

Trap. Confusing ionization energy with excitation energy.

Calculating Wavelength from Energy Difference

λ = \frac{h c}{Δ E}

Δ E = 13.6 (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}) eV

When to use. For any electronic transition in hydrogen.

Trap. Forgetting to convert ΔE from eV to J if using h in J s.

Rydberg Constant and Its Units

R_{H} = 1.097 \times 10^{7} m^{- 1}

R_{H} = 109677 {cm}^{- 1}

When to use. In Rydberg formula for spectral lines.

Trap. Using R_H in cm^-1 when formula expects m^-1.

Energy Conversion: eV to Joules

1 eV = 1.602 \times 10^{- 19} J

When to use. Whenever energy is given in eV and formula uses Joules.

Trap. Forgetting conversion factor; using 1.6 × 10^-19 is approximate.

Values to remember

$13.6 eV$ Bohr's Model of Hydrogen Atom
$0.529 \overset{˚}{A}$ Bohr's Model of Hydrogen Atom
$2.18 \times 10^{- 18} J$ Energy of Electron in H-like Atom
$5.29 \times 10^{- 11} m$ Radius of Bohr Orbit
$2.18 \times 10^{6} m/s$ Velocity of Electron in Bohr Orbit
$1.097 \times 10^{7} m^{- 1}$ Wavelength of Emitted Photon (Rydberg Formula)
$6.626 \times 10^{- 34} J s$ de Broglie Wavelength
$3 \times 10^{8} m/s$ Energy of Photon
$1.6 \times 10^{- 19} J/eV$ Calculating Wavelength from Energy Difference
$400 - 700 nm (visible)$ Electromagnetic Spectrum Regions
$1.602 \times 10^{- 19} J/eV$ Energy Conversion: eV to Joules

Read the full chapter: Atomic Structure for JEE, with what JEE asks and the common traps.