Structure of Atom — JEE Mains Chemistry Formulas and Traps

Photons, Planck's Quantum and the Photoelectric Effect

Photon energy from wavelength, frequency or wavenumber

Photon energy

E=h\nu=\frac{hc}{\lambda}=hc\bar\nu

Photoelectric effect and work function

Einstein's photoelectric equation

h\nu=h\nu_0+\tfrac{1}{2}mv^2

Common traps

Wavenumber in cm⁻¹ needs c in cm s⁻¹

E=hc\bar\nu

only works if the units match. With

\bar\nu

in

\mathrm{cm^{-1}}

, take

c=3\times10^{10}\ \mathrm{cm\,s^{-1}}

. Using

3\times10^{8}

makes the energy 100 times too small.

Energy ratio is the inverse wavelength ratio

\dfrac{E_1}{E_2}=\dfrac{\lambda_2}{\lambda_1}

. A 900 nm photon has one third the energy of a 300 nm photon, not three times.

Intensity changes the current, not the energy

Brighter light gives more electrons per second. It does not give faster electrons. Only a higher frequency raises the kinetic energy.

Put eV and J on the same footing

Work functions are usually in eV and

h\nu

comes out in J. Convert with

1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm{J}

before you subtract.

Bohr Model: Radius, Energy and Velocity

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Orbit radius and its scaling

Bohr radius

r_n=52.9\,\frac{n^2}{Z}\ \mathrm{pm}

Orbit energy, kinetic and potential energy, and speed

Bohr energy

E_n=-13.6\,\frac{Z^2}{n^2}\ \mathrm{eV}

Thomson, Rutherford, Bohr and the quantum model

Model	Picture of the atom	What it explained	Where it failed
Thomson	Uniform sphere of positive charge with electrons embedded	The atom is neutral overall	Predicts only small deflections, so cannot explain large-angle scatteringQ If Thomson were right, α-particles would pass through gold foil with only small deflections.
Rutherford	Tiny dense positive nucleus with electrons around it	Large-angle scattering; a few α-particles bounce back	An orbiting electron should radiate and spiral in; no line spectrum
Bohr	Electrons in fixed circular orbits, mvr = nh/2π	Stability and line spectrum of H, He⁺, Li²⁺	Many-electron atoms (even Li⁺), Zeeman and Stark splitting; a definite path breaks the uncertainty principleQ Li⁺ has two electrons, so Bohr's theory does not apply to it. Li²⁺ has one.
Quantum mechanical	Electron as a wave; orbitals are regions of probability ψ²	All atoms, including many-electron ones	Keeps stationary states and ΔE = hν, but drops the definite orbit

The one Bohr postulate the quantum model rejects: the electron moves in a definite circular orbit.

Common traps

The radius goes as n², not n

The sixth orbit of H is

\dfrac{36}{16}=2.25

times the fourth, not

\dfrac{6}{4}

times. Square the orbit numbers before you divide.

First excited state is n = 2

"First excited" is one step above the ground state, so

n=2

. Using

n=1

or

n=3

is the common slip.

Every orbit energy is negative

A bound electron always has

E_n<0

. An option with a plus sign on an orbit energy is wrong before you do any arithmetic. Kinetic energy is the only positive one.

KE is minus E, PE is twice E

KE=-E_n

and

PE=2E_n

. So for

E=-3.4\ \mathrm{eV}

,

KE=+3.4\ \mathrm{eV}

and

PE=-6.8\ \mathrm{eV}

.

Li⁺ is not a one-electron ion

Count the electrons:

\mathrm{Li^{+}}

has two. Only

\mathrm{H}

,

\mathrm{He^+}

,

\mathrm{Li^{2+}}

,

\mathrm{Be^{3+}}

and so on are hydrogen-like.

Hydrogen Spectrum and Spectral Series

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Rydberg equation for hydrogen-like species

Rydberg equation

\bar\nu=RZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)

Spectral series, line counts and other spectra

Series	Lands on (n₁)	First line	Series limit	Region
Lyman	1	2 → 1, about 122 nm	∞ → 1, about 91 nm	UltravioletQ
Balmer	2	3 → 2, about 656 nm	∞ → 2, about 365 nm	Visible
Paschen	3	4 → 3, about 1875 nm	∞ → 3, about 821 nm	InfraredQ With R rounded to 10⁵ cm⁻¹, the Paschen limit ∞ → 3 is 9/R = 900 nm, the infrared used for heat therapy.
Brackett	4	5 → 4, about 4052 nm	∞ → 4, about 1459 nm	Infrared
Pfund	5	6 → 5, about 7460 nm	∞ → 5, about 2280 nm	Infrared

Wavelengths are for hydrogen, from

1/R=91.2\ \mathrm{nm}

. For a hydrogen-like ion divide by

Z^2

.

Common traps

Count lines from the landing level

The third Paschen line is

6\to3

, not

3\to\ldots

or

5\to3

. The k-th line of a series starts from

n_1+k

.

Longest wavelength is the first line

The first line of a series has the smallest energy gap, so the longest wavelength. The shortest wavelength is the series limit, from

n_2=\infty

.

One electron is not many atoms

\dfrac{n(n-1)}{2}

counts every drop a large sample can make. A single electron cascading from

n=5

to the ground state gives at most 4 lines.

Moseley used atomic number, not mass

\sqrt\nu

is linear in

Z

. A statement that

\sqrt\nu

against atomic mass, or

\nu

against

Z

, is a straight line is false.

de Broglie Waves and the Uncertainty Principle

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de Broglie wavelength

de Broglie relation

\lambda=\frac{h}{mv}=\frac{h}{\sqrt{2mK}}

de Broglie waves in a Bohr orbit

Waves in the n-th orbit

\lambda_n=\frac{2\pi r_n}{n}=\frac{2\pi n a_0}{Z}

Heisenberg's uncertainty principle

Uncertainty principle

\Delta x\cdot m\Delta v\ge\frac{h}{4\pi}

Common traps

Equal wavelength means equal momentum

Equal

\lambda

gives

m_1v_1=m_2v_2

, not equal speeds. The lighter particle must move faster by the mass ratio.

Work in kg and J

Masses in grams or amu, and energies in eV, must be converted first.

1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm{J}

,

1\ \mathrm{amu}=1.66\times10^{-27}\ \mathrm{kg}

.

The wavelength grows as n, not n²

The radius goes as

n^2

, but

n

waves share the circumference. So

\lambda\propto n

: the fourth orbit of H has

\lambda=8\pi a_0

, not

32\pi a_0

.

Mass in kg

h

is in J s, so the mass must be in kg. If a question asks for the mass in grams, find it in kg first, then multiply by 1000.

Δx = Δp is not Δx = Δv

Equal uncertainties in position and momentum give

\Delta v=\dfrac{1}{2m}\sqrt{\dfrac{h}{\pi}}

. Setting

\Delta x=\Delta v

instead gives a different, wrong answer.

Quantum Numbers and Electron Counting

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Allowed values and counting orbitals and electrons

Orbital angular momentum

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

Common traps

l stops at n − 1

l=n

is never allowed. So

n=3, l=3

and

n=2, l=2

are invalid sets, whatever

m_l

is.

Angular momentum uses l, not n

L=\sqrt{l(l+1)}\,h/2\pi

. A 2s and a 3s electron both have

L=0

; a 2p electron has

\sqrt2\,h/2\pi

.

\sqrt6

belongs to

l=2

.

Ions lose the highest-n electrons first

\mathrm{Ga^+}

is

[\mathrm{Ar}]3d^{10}4s^2

: the 4p electron left, so the valence electron has

l=0

, not

l=1

.

"By Aufbau" ignores the exceptions

Cr is really

3d^54s^1

, but a question that says "in accordance with Aufbau" wants the strict order, where the 19th electron goes into 4s.

Orbitals: Nodes, Shapes and Probability Plots

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Radial and angular nodes

Node counts

\text{radial}=n-l-1,\qquad \text{angular}=l

Probability plots, boundary surfaces and shapes

Orbital	ψ² at the nucleus	Radial nodes	Peaks in 4πr²ψ²	Shape
1s	Maximum	0	One, at a₀ for H	SphereQ The density ψ² peaks at the nucleus; only the radial probability peaks at a₀.
2s	Maximum	1, at 2a₀ for H	Two, the outer one larger	SphereQ
2p	Zero	0	One, at 4a₀ for H	Two lobes, one nodal plane
3s	Maximum	2	Three	Sphere
3p	Zero	1	Two	Two lobes, one nodal plane
3d	Zero	0	One, at 9a₀ for H	Four lobes (d z² has two lobes and a ring), two nodal surfaces

For hydrogen, an orbital with no radial node (l = n − 1) has its radial peak at

n^2a_0

.

Common traps

Radial nodes are n − l − 1

Not

n-l

and not

n-2

. A 3s orbital has

3-0-1=2

radial nodes; a 3p has 1.

A nodal plane is an angular node

Nodal planes count

l

, not radial nodes. A 2s orbital has one radial node and no nodal plane.

Density is not radial probability

For 1s,

\psi^2

is largest at the nucleus, while

4\pi r^2\psi^2

is largest at

a_0

. Read the axis label before you read the curve.

Signs on lobes are not charges

The + and − on a p or d orbital are the sign of the wave function. They matter for bonding, not for charge.

Orbital Energies and Electronic Configuration

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The (n + l) rule in many-electron atoms

Order of orbital energy

E\uparrow\ \text{with}\ (n+l);\quad \text{tie}\Rightarrow\text{lower } n \text{ first}

One-electron atoms and the effect of Z

One-electron energy

E_n=-13.6\,\frac{Z^2}{n^2}\ \mathrm{eV}\quad(\text{no } l \text{ dependence})

Configurations, exceptions and electron counts

Electrons in a subshell

N_{max}=2(2l+1)

Common traps

A tie goes to the lower n

3d and 4p both have

n+l=5

. 3d is lower because its

n

is smaller. Do not order a tie by

l

.

m does not enter

Orbitals given as

(n, l, m)

are ordered by

n

and

l

only. Two sets that differ only in

m_l

or

m_s

have the same energy.

Hydrogen does not follow Aufbau

In H, 4s is above 3d and 2s equals 2p. The

(n+l)

order applies only to atoms with more than one electron.

Higher Z lowers the orbital

"Energies of orbitals in the same subshell increase with atomic number" is false. They decrease, because the nucleus pulls harder.

Exchange energy needs degenerate orbitals

Extra stability comes from same-spin electrons exchanging among orbitals of equal energy. A statement placing them in non-degenerate orbitals is false.

Half-filled is not filled

When counting completely filled orbitals with

m_l=0

, a p orbital holding one electron does not count. In Ge (

4p^2

) the 4p electrons fill no orbital.

Structure of Atom formulas

Photons, Planck's Quantum and the Photoelectric Effect

Photon energy from wavelength, frequency or wavenumber

Photoelectric effect and work function

Common traps

Bohr Model: Radius, Energy and Velocity

Orbit radius and its scaling

Orbit energy, kinetic and potential energy, and speed

Thomson, Rutherford, Bohr and the quantum model

Common traps

Hydrogen Spectrum and Spectral Series

Rydberg equation for hydrogen-like species

Spectral series, line counts and other spectra

Common traps

de Broglie Waves and the Uncertainty Principle

de Broglie wavelength

de Broglie waves in a Bohr orbit

Heisenberg's uncertainty principle

Common traps

Quantum Numbers and Electron Counting

Allowed values and counting orbitals and electrons

Common traps

Orbitals: Nodes, Shapes and Probability Plots

Radial and angular nodes

Probability plots, boundary surfaces and shapes

Common traps

Orbital Energies and Electronic Configuration

The (n + l) rule in many-electron atoms

One-electron atoms and the effect of Z

Configurations, exceptions and electron counts

Common traps

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