Atomic Structure

1. Introduction

The atom is the smallest particle of an element that takes part in a chemical reaction. Although Dalton treated it as indivisible, experiments in the late nineteenth and early twentieth centuries showed the atom is made of still smaller sub-atomic particles: the negatively charged electron ($\ce{e-}$), the positively charged proton ($\ce{p+}$), and the neutral neutron ($\ce{n}$). This chapter traces how our model of the atom developed from Rutherford, through Bohr, to the modern quantum-mechanical picture, and how electrons are arranged in orbitals.

Two defining numbers:

• Atomic number ($Z$) = number of protons = number of electrons in a neutral atom.
• Mass number ($A$) = number of protons + number of neutrons.

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2. Rutherford's Atomic Model (Nuclear Model, 1911)

Rutherford bombarded a thin gold foil with fast-moving alpha ($\alpha$) particles (the gold-foil / alpha-scattering experiment). Observations:

1. Most $\alpha$-particles passed straight through undeflected, so the atom is mostly empty space.
2. A few were deflected through large angles, and about $1$ in $20000$ bounced straight back, so the positive charge and nearly all the mass are concentrated in a tiny, dense central region.

Postulates of the model:

• The atom has a small, dense, positively charged centre called the nucleus, where almost all the mass resides.
• Electrons revolve around the nucleus in circular paths called orbits, like planets around the sun.
• The electrostatic attraction between the nucleus and the electrons provides the centripetal force for this motion.
• The atom is electrically neutral: the positive nuclear charge equals the total negative charge of the electrons.

2.1 Limitations of Rutherford's Model

1. Instability of the atom. According to classical (Maxwell's) electromagnetic theory, an accelerating (revolving) charged electron must continuously radiate energy. Losing energy, it should spiral into the nucleus in about $10^{-8}\ \text{s}$. But atoms are stable, which the model cannot explain.
2. No explanation of line spectra. A continuously spiralling electron would emit a continuous spectrum, whereas atoms actually give sharp, discrete line spectra.
3. It said nothing about how electrons are arranged (distributed) around the nucleus.

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3. Bohr's Atomic Model (1913)

To remove Rutherford's difficulties, Niels Bohr applied Planck's quantum theory to the hydrogen atom.

3.1 Postulates

1. Electrons revolve around the nucleus only in certain fixed circular orbits (called stationary states or energy levels) of definite energy. While in such an orbit the electron does not radiate energy.
2. Only orbits for which the angular momentum is an integral multiple of $\dfrac{h}{2\pi}$ are allowed (quantization of angular momentum):

$$mvr = n\frac{h}{2\pi}, \qquad n = 1, 2, 3, \dots$$

where $n$ is the principal quantum number, $h$ is Planck's constant, $m$ and $v$ the electron mass and speed, and $r$ the orbit radius.

3. Energy is absorbed or emitted only when an electron jumps from one orbit to another. The energy of the photon equals the difference between the two levels:

$$ \begin{aligned} \Delta E &= E_2 - E_1 \ &= h\nu \end{aligned} $$