Exercises: Atomic Models and Energy Levels

1. a) Describe the Thomson "plum pudding" model of the atom.
2. b) Describe the Rutherford nuclear model.
3. c) Name the experiment that led to the Rutherford model.

1. a) State the two main postulates of Bohr's model of the hydrogen atom.
2. b) What is meant by the ground state?

An electron in an atom drops from energy level $E_2 = -3.4\ \text{eV}$ to $E_1 = -13.6\ \text{eV}$.

1. a) Calculate the energy of the photon emitted, in eV and in joules.
2. b) State the relationship used, $E_2-E_1=hf$.

Hydrogen energy levels are $E_n=-13.6/n^2\ \text{eV}$. Use $h=6.63\times10^{-34}\ \text{J s}$, $c=3.0\times10^{8}\ \text{m s}^{-1}$, $1\ \text{eV}=1.6\times10^{-19}\ \text{J}$.

1. a) Calculate the energies of levels $n=1,2,3$.
2. b) Calculate the frequency of the photon emitted in the transition $n=3\to n=2$.
3. c) Calculate its wavelength and state whether it is visible.

In the Rutherford scattering experiment, alpha particles were fired at thin gold foil. Most passed straight through, a few were deflected through large angles and very few bounced back.

1. a) Explain what each observation shows about the structure of the atom.
2. b) Why did this contradict the Thomson model?
3. c) State one limitation of the Rutherford model that Bohr's model addressed.

A hydrogen atom in the ground state absorbs a photon and is excited to $n=3$.

1. a) Calculate the minimum photon energy, in eV.
2. b) List all the possible photon energies emitted as the atom returns to the ground state.
3. c) Explain why the emission spectrum consists of discrete lines rather than a continuous spectrum.

Using $E_n=-13.6/n^2\ \text{eV}$ for hydrogen, with $h=6.63\times10^{-34}\ \text{J s}$ and $c=3.0\times10^{8}\ \text{m s}^{-1}$:

1. a) Calculate the wavelength of the photon emitted in the $n=4\to n=2$ transition.
2. b) Calculate the ionisation energy from the ground state in eV and the longest wavelength capable of ionising it.
3. c) Explain why photons of energy $11.0\ \text{eV}$ cannot be absorbed by a ground-state hydrogen atom, while an electron of $11.0\ \text{eV}$ can excite it.
4. d) Explain the difference between an emission spectrum and an absorption spectrum.

The Bohr model assumes an electron of mass $m$ moves in a circular orbit of radius $r$ with angular momentum $mvr=nh/2\pi$.

1. a) Show that the electrostatic force provides the centripetal force: $\dfrac{ke^2}{r^2}=\dfrac{mv^2}{r}$.
2. b) Use this to show that the kinetic energy is $\dfrac{ke^2}{2r}$ and the total energy is $-\dfrac{ke^2}{2r}$.
3. c) Calculate the radius for $n=1$, given $r_n = n^2 \times 5.3\times10^{-11}\ \text{m}$ and hence the total energy in eV for $n=1$ using $k=8.99\times10^{9}\ \text{N m}^2\text{C}^{-2}$ and $e=1.6\times10^{-19}\ \text{C}$.
4. d) Comment on the result compared with $-13.6\ \text{eV}$.

A discharge tube of hydrogen gas is excited so that atoms reach $n=4$. Use $E_n=-13.6/n^2\ \text{eV}$.

1. a) Determine the number of different spectral lines possible and list the transitions.
2. b) Calculate the shortest and longest wavelength emitted among these lines.
3. c) Identify which lines are in the visible region (approximately $400$ to $700\ \text{nm}$).
4. d) Explain how a diffraction grating of $600\ \text{lines per mm}$ would separate them, and calculate the first-order angle for the $n=3\to2$ line, which has wavelength $656\ \text{nm}$.