JEE Chemistry Structure of the Atom Notes Set 03

Subatomic Particles and Atomic Models

An atom is electrically neutral, so the number of electrons equals the number of protons. The atomic number Z is the number of protons, and the mass number A is the number of protons plus neutrons. The number of neutrons is therefore A - Z.

Thomson, Rutherford and the Nuclear Atom

1. Thomson pictured the atom as a sphere of positive charge with electrons embedded in it like seeds in a watermelon.
2. In Rutherford's experiment, a beam of alpha particles was fired at a very thin gold foil. Most particles passed straight through, a few were deflected through large angles, and a very small number bounced back.
3. From this Rutherford concluded that the positive charge and almost all the mass sit in a tiny, dense nucleus, and the electrons move in the large empty space around it.
4. The nuclear model could not explain why electrons do not spiral into the nucleus, nor why atoms give line spectra. Bohr's model was proposed to deal with these problems.

Isotopes, Isobars, Isotones and Related Terms

The average atomic mass of an element is the abundance-weighted mean of its isotopic masses: \(\bar M = \frac{\sum x_i M_i}{100}\) when abundances are in percent.

Worked example: chlorine has isotopes of mass 35 (75%) and 37 (25%). Average mass = \(0.75 \times 35 + 0.25 \times 37 = 35.5\) u.

Electromagnetic Radiation and Planck's Quantum Theory

1. Light is an electromagnetic wave. All such waves travel in vacuum at \(c = 3\times10^{8}\) m/s, and the speed, frequency and wavelength are linked by \(c = \nu\lambda\).
2. The wave number is \(\bar\nu = \frac{1}{\lambda}\), counted as waves per unit length.
3. Planck proposed that energy is emitted or absorbed in whole-number multiples of a small packet, the quantum. For radiation of frequency \(\nu\), one quantum (photon) has \(E = h\nu = \frac{hc}{\lambda}\), with \(h = 6.626\times10^{-34}\) J s.
4. A handy form: \(E\,(\text{eV}) = \frac{1240}{\lambda\,(\text{nm})}\).
5. Photoelectric effect: electrons are ejected from a metal only if \(\nu \ge \nu_0\), the threshold frequency. The maximum kinetic energy is \(K = h\nu - h\nu_0\).

Worked example: the energy of a photon of wavelength 600 nm is \(\frac{1240}{600} = 2.07\) eV, or \(\frac{6.626\times10^{-34}\times3\times10^{8}}{600\times10^{-9}} = 3.31\times10^{-19}\) J.

Bohr's Model of the Hydrogen Atom

1. An electron revolves round the nucleus only in certain allowed circular orbits, called stationary states, without radiating energy.
2. The allowed orbits are those in which the angular momentum is a whole multiple of \(\frac{h}{2\pi}\): \(mvr = \frac{nh}{2\pi}\), with n = 1, 2, 3 and so on.
3. Energy is emitted or absorbed only when the electron jumps between orbits, and the photon energy is the difference of the two orbit energies: \(\Delta E = E_{high} - E_{low} = h\nu\).

Going outward (larger n), the radius increases, the speed and kinetic energy decrease, and the total and potential energies increase (become less negative). The energy needed to remove the electron from level n is \(13.6\,\frac{Z^2}{n^2}\) eV, so the ionisation energy of hydrogen is 13.6 eV.

Worked example: the first excitation energy of hydrogen is \(E_2 - E_1 = -3.4 - (-13.6) = 10.2\) eV. For \(\ce{He+}\) (Z = 2) the same transition needs \(4 \times 10.2 = 40.8\) eV.

Limitations of Bohr's model: it works only for one-electron species, cannot explain the splitting of spectral lines in magnetic or electric fields, and violates the uncertainty principle by giving the electron a fixed orbit.

Hydrogen Spectrum

When an electron falls from level \(n_2\) to a lower level \(n_1\), the wave number of the emitted line is \(\bar\nu = R\,Z^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\), where \(R = 1.097\times10^{7}\) m\(^{-1}\).

1. The first line of a series comes from \(n_2 = n_1 + 1\), and it has the longest wavelength of that series.
2. The series limit comes from \(n_2 = \infty\), and it has the shortest wavelength of that series.
3. The maximum number of lines when an electron falls from level n to the ground state in all possible ways is \(\frac{n(n-1)}{2}\).
4. When an electron falls from level \(n_2\) to level \(n_1\), the number of lines is \(\frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}\).
5. For a hydrogen-like ion, the wave number of the same transition is \(Z^2\) times that of hydrogen.

Worked example: the longest wavelength of the Balmer series (3 to 2) is given by \(\frac{1}{\lambda} = 1.097\times10^{7}\left(\frac14 - \frac19\right) = 1.523\times10^{6}\) m\(^{-1}\), so \(\lambda = 656.6\) nm, which is red light.