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Atomic Structure — JEE Short Notes
Atomic Structure — JEE Short Notes
Fundamental particles
Atomic number: \(Z=\) number of protons \(=\) number of electrons in a neutral atom.
Mass number: \(A=\) protons \(+\) neutrons.
Neutrons: \(N=A-Z\).
Isotopes: same \(Z\), different \(A\). Isobars: same \(A\), different \(Z\). Isotones: same \(N\).
Electromagnetic radiation
\(c=3.0\times10^8\,\mathrm{m\,s^{-1}}\), \(h=6.626\times10^{-34}\,\mathrm{J\,s}\).
Wavenumber: \(\bar\nu=\frac{1}{\lambda}\).⁷
Energy per mole of photons: \(E{\text{molar}}=NAh\nu\).
Higher \(\nu\) means higher energy and lower \(\lambda\).
Photoelectric effect
\(\phi\): work function; \(V_0\): stopping potential.
Emission occurs only if \(\nu\geq\nu0\), where \(\phi=h\nu0\).
de Broglie matter waves
For an electron accelerated through potential \(V\):
Bohr model for hydrogen-like species
Applicable to one-electron systems: \(\mathrm{H}\), \(\mathrm{He^+}\), \(\mathrm{Li^{2+}}\), etc.
Bohr postulates
Electrons move only in certain stable circular orbits without radiating energy.
Angular momentum is quantized:
Radiation is absorbed/emitted only during transitions:
Key formulae
For nuclear charge \(Z\) and orbit \(n\):
Thus \(U=-2K\) and \(E=-K=U/2\).
Transition and spectrum
Use \(ni>nf\) for emission and \(nf>ni\) for absorption.
where \(R=1.097\times10^7\,\mathrm{m^{-1}}\).
Ionization energy from \(n\): \(\frac{13.6Z^2}{n^2}\,\mathrm{eV}\).
For hydrogen, ground-state ionization energy = \(13.6\,\mathrm{eV}\) per atom = \(1312\,\mathrm{kJ\,mol^{-1}}\).
Number of spectral lines when an electron descends from \(n\):
Maximum electrons in shell \(n\):
Limitations of Bohr model
Fails for multi-electron atoms.
Cannot explain fine spectrum, Zeeman effect, Stark effect, or wave nature of electrons.
Assumes definite electron paths, contradicting the uncertainty principle.
Heisenberg uncertainty principle
It is impossible to know the exact position and exact momentum of a microscopic particle simultaneously.
Consequence: an electron does not move in a fixed circular orbit; it exists in an orbital, a probability region.
Quantum-mechanical model and orbitals
Schrödinger wave equation gives wave function \(\psi\).
\(\psi^2\) gives probability density of finding the electron.
An orbital is the three-dimensional region around the nucleus where probability of finding an electron is high (about \(90\)–\(95\%\)).
One orbital can hold a maximum of \(2\) electrons with opposite spins.
Quantum numbers
(a) Principal quantum number \(n\)
Values: \(n=1,2,3,\ldots\)
Identifies shell, size, and approximate energy.
Shell names: \(K,L,M,N\) correspond to \(n=1,2,3,4\).
Number of orbitals in shell \(n\): \(n^2\).
Maximum electrons in shell \(n\): \(2n^2\).
(b) Azimuthal/angular momentum quantum number \(l\)
Values: \(l=0\) to \(n-1\).
Identifies subshell and shape.
Number of orbitals in a subshell: \(2l+1\).
Maximum electrons in a subshell: \(2(2l+1)\).
Magnitude of orbital angular momentum:
(c) Magnetic quantum number \(m_l\)
Values: \(m_l=-l\) to \(+l\), including zero.
Identifies orientation of an orbital in space.
(d) Spin quantum number \(m_s\)
Specifies electron spin orientation.
Nodes
Total nodes:
Angular nodes:
Radial nodes:
Examples: \(2p\): one angular node and zero radial nodes; \(3s\): zero angular nodes and two radial nodes.
Electronic configuration
Aufbau principle
Electrons occupy orbitals in increasing order of \(n+l\). If \(n+l\) is equal, the orbital with lower \(n\) fills first.
Order: \(1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s\).
Pauli exclusion principle
No two electrons in an atom can have the same set of all four quantum numbers. Maximum two electrons per orbital, with opposite spin.
Hund’s rule
For degenerate orbitals, electrons fill singly with parallel spins before pairing.
Important exceptions
\(\mathrm{Cr}\): \([\mathrm{Ar}]\,3d^5 4s^1\)
\(\mathrm{Cu}\): \([\mathrm{Ar}]\,3d^{10}4s^1\)
On forming cations, electrons are removed first from the shell with highest \(n\): for transition metals, remove \(4s\) before \(3d\).
High-yield JEE facts
For hydrogen-like atoms, energy depends only on \(n\); all orbitals of one shell are degenerate.
In multi-electron atoms, energy depends mainly on both \(n\) and \(l\).
Number of possible values of \(l\) in shell \(n\) is \(n\).
Number of possible values of \(m_l\) for a given \(l\) is \(2l+1\).
A \(p\) subshell begins at \(n=2\), \(d\) at \(n=3\), and \(f\) at \(n=4\).
In an orbital diagram, unpaired electrons determine paramagnetism; all electrons paired means diamagnetism.
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