JEE Chemistry Structure of the Atom Notes Set 04

Dual Nature of Matter and de Broglie's Relation

De Broglie suggested that a moving particle also behaves like a wave, with wavelength \(\lambda = \frac{h}{mv} = \frac{h}{p}\). A particle of mass m accelerated through a potential difference V has \(\lambda = \frac{h}{\sqrt{2meV}}\), and for kinetic energy K it has \(\lambda = \frac{h}{\sqrt{2mK}}\).

1. The wavelength is inversely proportional to momentum, so heavy everyday objects have wavelengths far too small to detect.
2. For an electron accelerated through V volts, \(\lambda \approx \frac{12.27}{\sqrt V}\) Å.
3. A Bohr orbit is stable when its circumference fits a whole number of waves: \(2\pi r = n\lambda\). This gives back Bohr's angular momentum condition.
4. The number of waves in the nth orbit is n.

Worked example: an electron moving at \(10^{6}\) m/s has \(\lambda = \frac{6.626\times10^{-34}}{9.11\times10^{-31}\times10^{6}} = 7.27\times10^{-10}\) m, a size comparable to atoms.

Heisenberg's Uncertainty Principle

It is impossible to know both the exact position and the exact momentum of a microscopic particle at the same moment. The uncertainties obey \(\Delta x \cdot \Delta p \ge \frac{h}{4\pi}\), or \(\Delta x \cdot m\Delta v \ge \frac{h}{4\pi}\).

1. The principle matters for electrons, protons and other tiny particles, and it is negligible for a cricket ball or a car.
2. It rules out a definite orbit for the electron, which is why we talk about the probability of finding the electron (an orbital) instead.
3. If \(\Delta x\) is made very small, \(\Delta p\) becomes very large and vice versa.

Worked example: if the position of an electron is known to within \(10^{-10}\) m, \(\Delta v \ge \frac{6.626\times10^{-34}}{4\pi\times9.11\times10^{-31}\times10^{-10}} = 5.8\times10^{5}\) m/s.

Quantum Mechanical Model, Orbitals and Quantum Numbers

The behaviour of the electron is described by the Schrodinger wave equation. Its solutions \(\psi\) are called orbitals, and \(\psi^2\) at a point gives the probability density of finding the electron there. An orbital is a region of space around the nucleus where the probability of finding the electron is high.

1. A shell n has \(n^2\) orbitals and holds at most \(2n^2\) electrons.
2. A subshell \(\ell\) has \(2\ell + 1\) orbitals and holds at most \(4\ell + 2\) electrons: s 2, p 6, d 10, f 14.
3. Orbital angular momentum is \(\sqrt{\ell(\ell+1)}\,\frac{h}{2\pi}\). It is zero for an s electron and \(\sqrt2\,\frac{h}{2\pi}\) for a p electron.
4. Radial (spherical) nodes = \(n - \ell - 1\); angular nodes = \(\ell\); total nodes = \(n - 1\).

Shapes: s orbitals are spherical; p orbitals are dumbbell shaped along the x, y or z axis, with one nodal plane through the nucleus; four of the d orbitals have four lobes (d\(_{xy}\), d\(_{yz}\), d\(_{zx}\) lie between the axes while d\(_{x^2-y^2}\) lies along the x and y axes) and d\(_{z^2}\) has two lobes along the z axis with a ring in the xy plane.

Rules for Filling Electrons

1. Aufbau principle: electrons occupy the lowest energy orbitals first. The energy order follows increasing (n + \(\ell\)); if two orbitals have the same (n + \(\ell\)), the one with lower n has lower energy.
2. Pauli exclusion principle: no two electrons in an atom can have the same four quantum numbers, so an orbital holds at most two electrons, with opposite spins.
3. Hund's rule of maximum multiplicity: in a set of degenerate orbitals, electrons occupy separate orbitals with parallel spins before any pairing takes place.

The filling order is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. In a one-electron species like hydrogen all subshells of one shell have the same energy, but in a multi-electron atom the order within a shell is s < p < d < f.

Electronic Configurations and Magnetic Properties

1. Half-filled and fully filled subshells are extra stable because of symmetry and greater exchange energy. This is why chromium is [Ar] 3d5 4s1 and copper is [Ar] 3d10 4s1.
2. While forming cations from transition metals, the 4s electrons are removed before the 3d electrons. For example \(\ce{Fe^{2+}}\) is [Ar] 3d6 and \(\ce{Fe^{3+}}\) is [Ar] 3d5.
3. The spin-only magnetic moment is \(\mu = \sqrt{n(n+2)}\) Bohr magneton, where n is the number of unpaired electrons. A species with unpaired electrons is paramagnetic, and one with none is diamagnetic.
4. Filled and half-filled p or d subshells give a spherically symmetric electron cloud.

Worked example: \(\ce{Co^{2+}}\) has Z = 27 and the configuration [Ar] 3d7, which has 3 unpaired electrons. Its spin-only magnetic moment is \(\sqrt{3\times5} = 3.87\) BM.