Atoms

This chapter traces the development of the modern atomic model, starting from early ideas and focusing on Rutherford's alpha-scattering experiment and the Bohr model for the hydrogen atom. Key topics include the concepts of impact parameter and distance of closest approach, which arose from Rutherford's work. The main part of the chapter covers Bohr's postulates, particularly the quantization of angular momentum, and their application to derive expressions for the radius, velocity, and energy of an electron in a stationary orbit.

By the end of this chapter, you should be able to apply Bohr's theory to hydrogen-like atoms (ions with a single electron). This involves calculating the energy levels and the wavelengths of photons emitted or absorbed during electronic transitions. You will learn to identify and work with the different spectral series, such as the Lyman, Balmer, and Paschen series, which correspond to transitions ending in different final states.

The chapter is mostly formula-based and is considered one of the more straightforward parts of modern physics. Difficulties can arise in keeping track of the signs for energy calculations (potential, kinetic, and total energy) and in correctly applying the Rydberg formula for the various spectral series. A solid understanding of basic electrostatics, especially Coulomb's law and electric potential energy, is necessary to understand the derivations. This chapter directly leads into the study of Nuclei.

Formulas for Atoms

Distance of closest approach: $r_0 = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{K}$ K is the initial kinetic energy of the alpha particle, Z is the atomic number of the nucleus.
Impact parameter and scattering angle: $b = \frac{1}{4\pi\epsilon_0} \frac{Ze^2 \cot(\theta/2)}{K}$ b is the impact parameter and θ is the scattering angle.
Bohr's quantization condition: $m_e v_n r_n = \frac{nh}{2\pi}$ The angular momentum of an electron in the nth orbit is an integral multiple of h/2π.
Radius of the nth orbit: $r_n = (0.529 \text{ Å}) \frac{n^2}{Z}$ For a hydrogen-like atom with atomic number Z and principal quantum number n.
Velocity of electron in the nth orbit: $v_n = (2.18 \times 10^6 \text{ m/s}) \frac{Z}{n}$ For a hydrogen-like atom with atomic number Z and principal quantum number n.
Total energy of electron in the nth orbit: $E_n = (-13.6 \text{ eV}) \frac{Z^2}{n^2}$ The negative sign indicates that the electron is bound to the nucleus.
Wavelength of emitted photon (Rydberg formula): $\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ R is the Rydberg constant; electron transitions from initial state n_i to final state n_f.