E.1 Structure of the atom: แบบฝึกหัด

Which of the following experimental observations provides direct evidence for the existence of discrete energy levels within an atom?

An electron in a hydrogen atom transitions from an energy level of \(-1.51\text{ eV}\) to an energy level of \(-3.40\text{ eV}\). What is the frequency of the emitted photon?
(Use Planck's constant \(h = 6.63 \times 10^{-34}\text{ J s}\) and \(1\text{ eV} = 1.60 \times 10^{-19}\text{ J}\))

An alpha particle (charge \(+2e\)) is fired directly at a stationary gold nucleus (charge \(+79e\)) with an initial kinetic energy \(E_k\). At the distance of closest approach \(d\), the alpha particle momentarily stops. If the initial kinetic energy is doubled to \(2E_k\), what will be the new distance of closest approach in terms of \(d\)?

Which of the following transitions in a hydrogen atom results in the emission of a photon with the longest wavelength?

An electron in a hydrogen atom is in the ground state (\(n=1\)) with an energy of \(-13.6\text{ eV}\). The first excited state (\(n=2\)) has an energy of \(-3.4\text{ eV}\). What is the minimum energy a photon must possess to excite the electron from the ground state to the first excited state?

The first ionization energy of a hydrogen atom is \(13.6\text{ eV}\). Calculate the maximum wavelength of electromagnetic radiation, in nanometers, capable of ionizing a hydrogen atom from its ground state. (Use \(h = 6.63 \times 10^{-34}\text{ J s}\), \(c = 3.00 \times 10^8\text{ m s}^{-1}\), and \(1\text{ eV} = 1.60 \times 10^{-19}\text{ J}\)).

The Rutherford scattering experiment provided evidence for the structure of the atom.
(a) Describe two observations from the alpha-particle scattering experiment and the conclusions drawn from them regarding atomic structure.
(b) An alpha particle with kinetic energy \(6.0\text{ MeV}\) is fired at a silver nucleus (\(Z=47\)). Estimate the distance of closest approach for a head-on collision.

In a hydrogen-like atom, the energy levels are given by the formula \(E_n = -\frac{13.6 Z^2}{n^2}\text{ eV}\). Consider a hydrogen atom (\(Z=1\)).
(a) Calculate the energy of a photon emitted when an electron transitions from the \(n=4\) level to the \(n=2\) level.
(b) Determine the wavelength of this emitted photon.
(c) If this photon is subsequently incident on a metal with a work function of \(1.80\text{ eV}\), calculate the maximum velocity of the ejected photoelectron. (Electron mass \(m_e = 9.11 \times 10^{-31}\text{ kg}\)).