Structure of Atom

This chapter covers subatomic particles, atomic models, electromagnetic radiation, Planck's quantum theory, Bohr model, quantum numbers, orbitals, and electronic configuration.

Chapter overview

Atoms are made of electrons, protons, and neutrons. Atomic structure explains spectra, chemical behaviour, periodicity, and bonding.

1. Subatomic particles

Atomic number:

$$Z=\text{number of protons}$$

Mass number:

$$A=Z+N$$

where $N$ is number of neutrons.

2. Isotopes, isobars, and isotones

Example isotope notation:

$${}_{6}^{12}C,\quad {}_{6}^{14}C$$

3. Thomson model

Atom was considered as a positively charged sphere with electrons embedded in it.

Limitation: could not explain alpha-particle scattering experiment.

4. Rutherford model

Observations:

• Most alpha particles passed undeflected.
• Some were deflected.
• Very few bounced back.

Conclusions:

• Atom is mostly empty space.
• Positive charge is concentrated in nucleus.
• Nucleus is very small and dense.

5. Electromagnetic radiation

Relation:

$$c=\nu\lambda$$

where:

• $c$ = speed of light,
• $\nu$ = frequency,
• $\lambda$ = wavelength.

Wave number:

$$\bar{\nu}=\frac{1}{\lambda}$$

6. Planck's quantum theory

Energy is emitted or absorbed in packets called quanta.

$$E=h\nu$$

where:

$$h=6.626\times10^{-34}\ \text{J s}$$

For $n$ quanta:

$$E=nh\nu$$

7. Photoelectric effect

Light can eject electrons from metal surface if frequency is above threshold frequency.

Einstein equation:

$$h\nu=h\nu_0+K.E.$$

Maximum kinetic energy:

$$K.E._{\max}=h\nu-h\nu_0$$

8. Atomic spectra

Rydberg formula:

$$\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$

where:

$$n_2>n_1$$

Hydrogen spectral series:

9. Bohr model

Bohr proposed that electrons revolve in fixed circular orbits.

Angular momentum quantisation:

$$mvr=\frac{nh}{2\pi}$$

Radius of orbit:

$$r_n=a_0n^2$$

Energy:

$$E_n=-\frac{2.18\times10^{-18}}{n^2}\ \text{J}$$

In eV:

$$E_n=-\frac{13.6}{n^2}\ \text{eV}$$

10. de Broglie relation

Matter has wave nature.

$$\lambda=\frac{h}{p}$$

Since:

$$p=mv$$

therefore:

$$\lambda=\frac{h}{mv}$$

11. Heisenberg uncertainty principle

It is impossible to determine exact position and exact momentum of a microscopic particle simultaneously.

$$\Delta x\Delta p \ge \frac{h}{4\pi}$$

Also:

$$\Delta E\Delta t \ge \frac{h}{4\pi}$$

12. Quantum numbers

Allowed values:

$$l=0,1,2,\ldots,n-1$$ $$m_l=-l,\ldots,0,\ldots,+l$$ $$m_s=+\frac{1}{2}\ \text{or}\ -\frac{1}{2}$$

13. Orbitals

Number of orbitals in a subshell:

$$2l+1$$

Maximum electrons in a subshell:

$$2(2l+1)$$

14. Electronic configuration

Aufbau principle: electrons fill lower energy orbitals first.

Pauli exclusion principle: no two electrons in an atom can have the same set of four quantum numbers.

Hund's rule: electrons fill degenerate orbitals singly before pairing.

Filling order:

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p

Example:

$$Na:1s^2\,2s^2\,2p^6\,3s^1$$ $$Cl:1s^2\,2s^2\,2p^6\,3s^2\,3p^5$$

15. Solved example

Find wavelength of radiation with frequency $5\times10^{14}\ \text{s}^{-1}$.

$$c=\nu\lambda$$ $$\lambda=\frac{c}{\nu}$$ $$\lambda= \frac{3\times10^8}{5\times10^{14}} = 6\times10^{-7}\ \text{m}$$

16. Common mistakes

• Confusing atomic number and mass number.
• Writing wrong order of orbitals.
• Forgetting $n_2>n_1$ in Rydberg formula.
• Treating orbit and orbital as same.
• Forgetting $p$ subshell has 3 orbitals.
• Ignoring Pauli exclusion principle.

17. Quick revision