Definitions [4]
The atomic number of an element is equal to the number of protons in the nucleus.
Atomic number (Z) = Number of protons.
= Number of electrons.
The mass number of an element is the sum of the number of protons and neutrons in the nucleus of the atom of that element.
Mass number (A) = No. of protons (p)
+ No. of neutrons (n)
The atoms of the same element, having same atomic number Z, but different mass number A, are called isotopes.
OR
Atoms having the same atomic number (Z) but different mass numbers (A).
Define the term Electronic configuration.
Electronic configuration of an atom is defined as the distribution of its electrons in orbitals.
Formulae [7]
\[L=mvr=\frac{nh}{2\pi},\quad n=1,2,3\ldots\]
\[h\nu=E_2-E_1=\frac{hc}{\lambda}\]
\[r=\frac{n^2h^2}{4\pi^2mkZe^2}\]
Radius of the n-th Bohr Orbit (General):
\[r_n=\frac{\varepsilon_0n^2h^2}{\pi mZe^2}\]
\[\mathrm{i.e.,}r_n\propto n^2\mathrm{and}r_n\propto\frac{1}{Z}\]
Radius of n-th orbit for Hydrogen-like atom:
\[r_n=0.53\left(\frac{n^2}{Z}\right)\mathrm{\r{A}}\]
Total Energy of Electron in n-th Orbit (General):
- \[E_n=\frac{-Z^2me^4}{8\varepsilon_0^2n^2h^2}\]
Total Energy (Alternate form):
- \[E_n=-\frac{2\pi^2mk^2Z^2e^4}{n^2h^2}\]
Total Energy for Hydrogen-like Atom (Simplified):
- \[E_n=-13.6\frac{Z^2}{n^2}\mathrm{eV},\quad n=1,2,3\ldots\]
\[v=\frac{2\pi kZe^2}{nh}\]
For hydrogen atom (Z = 1):
\[v=\frac{2\pi ke^2}{nh}=\alpha\frac{c}{n}\]
where α is the fine structure constant and \[\alpha=\frac{1}{137}.\]
\[v_n=\frac{nh}{2\pi mr_n}\]
Theorems and Laws [1]
Bohr's First Postulate:
An atom consists of a small, massive central core called the nucleus, around which planetary electrons revolve. The centripetal force required for their rotation is provided by the electrostatic attraction between the electrons and the nucleus.
Bohr's Second Postulate (Quantum Condition):
The electrons are permitted to circulate only in those orbits in which the angular momentum of an electron is an integral multiple of \[\frac{h}{2\pi}\]; h being Planck's constant.
Bohr's Third Postulate:
While revolving in the permissible orbits, an electron does not radiate energy. These non-radiating orbits are called stationary orbits.
Bohr's Fourth Postulate:
An atom can emit or absorb radiation in the form of discrete energy photons only when an electron jumps from a higher to a lower orbit or from a lower to a higher orbit, respectively.
Key Points
Atoms are made up of three fundamental subatomic particles — electron, proton, and neutron. Their discovery was a milestone in understanding atomic structure.
Discovery Timeline:
| Particle | Year | Scientist | Experiment |
|---|---|---|---|
| Electron | 1897 | J.J. Thomson | Cathode ray tube experiment — cathode rays are streams of tiny, negatively charged particles |
| Proton | 1911 | Ernest Rutherford | Alpha-particle scattering on gold foil — hydrogen nucleus identified and renamed proton |
| Neutron | 1932 | James Chadwick | Nuclear reaction: bombardment of beryllium with alpha-particles produced neutral, massive particles |
Properties of Subatomic Particles
| Particle | Symbol | Absolute Charge (C) | Relative Charge | Mass (kg) | Mass (u) | Approx. Mass (u) |
|---|---|---|---|---|---|---|
| Electron | e⁻ | −1.6022 × 10⁻¹⁹ | −1 | 9.10938 × 10⁻³¹ | 0.00054 | 0 |
| Proton | p+ | +1.6022 × 10⁻¹⁹ | +1 | 1.6726 × 10⁻²⁷ | 1.00727 | 1 u |
| Neutron | no | 0 | 0 | 1.67493 × 10⁻²⁷ | 1.00867 | 1 u |
Isotopes are atoms of the same element that have the same atomic number but different mass numbers (different number of neutrons).
Same in isotopes:
- Atomic number (Z)
- Number of protons and electrons
- Electronic configuration
- Position in periodic table
- Chemical properties (nearly identical)
Different in isotopes:
- Mass number (A)
- Number of neutrons
- Physical properties
Examples: \[_1H^1and_1H^2\]
- Bohr modified Rutherford's model - electrons move in fixed orbital shells, each with fixed energy levels.
- The centripetal force for electron revolution is provided by electrostatic attraction between the electron and the nucleus.
- An electron does not radiate energy while revolving in a stationary orbit.
- Energy is emitted or absorbed only during electron transitions between orbits.
- Limitations of Bohr's Model:
- Fails to explain the Zeeman Effect (effect of high magnetic fields on atomic spectra).
- Contradicts the Heisenberg Uncertainty Principle.
- Unable to explain the spectra of larger/multi-electron atoms.
- The radius of Bohr's orbit is proportional to \[n^{2}\] and inversely proportional to Z.
- For hydrogen (Z = 1), the ground state (n = 1) radius is 0.53 Å, known as Bohr's radius.
- The velocity of an electron decreases as the orbital number (n) increases.
- For hydrogen, orbital speed of electron equals \[\alpha\frac{c}{n}\], where \[\alpha=\frac{1}{137}\].
- The total energy of an electron in any orbit is negative, indicating a bound state.
- For hydrogen-like atoms, the energy of an electron in the n-th orbit is \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .
Schrödinger Wave Equation:
Schrödinger developed the fundamental equation of quantum mechanics which incorporates the wave-particle duality of matter:
where H = Hamiltonian operator, Ψ (psi) = wave function, E = total energy of the system.
- Wave function (ψ): The solution of this equation has no physical significance by itself.
- ψ²: Probability density — gives the probability of finding an electron at a point within the atom.
- The region where the probability of finding an electron is maximum = atomic orbital.
