Exercises: Classical atomic models

When John Dalton proposed his atomic model in $1808$, he imagined atoms as tiny solid spheres. Explain in your own words the two main ideas of his model: what an atom is according to Dalton and how atoms of one element differ from those of a different element.

In $1897$, J. J. Thomson experimented with cathode ray tubes, which are nearly empty glass tubes with two electrodes connected to a very high voltage. A diagram of one of these tubes is shown in the figure.

What particle did Thomson discover through these experiments? What electric charge does this particle have, and where did it come from, according to his interpretation?

In $1911$, Rutherford and his collaborators bombarded an extremely thin sheet of gold foil with $\alpha$ particles, which have a positive charge. What happened to the vast majority of the particles when they reached the foil? What conclusion did Rutherford draw about the inside of the atom?

In $1913$, Niels Bohr applied the new quantum ideas to the hydrogen atom. Explain what Bohr postulated about electron orbits and what happens, according to his model, when an electron jumps from a permitted orbit to a lower one.

On a timeline in the physics classroom, the following milestones appear out of order: Rutherford's model ($1911$), Dalton's model ($1808$), Bohr's model ($1913$) and Thomson's model ($1897$). Order the four models from oldest to most recent and write the popular name by which Thomson's model is known.

Fireworks owe their colors to the salts they contain: strontium produces red light, sodium yellow, and copper blue-green. When heated, each element emits light of characteristic colors that form its emission spectrum. Explain what an element's spectrum is and why it is said to be like its fingerprint.

A popular science magazine publishes four quotes without mentioning who wrote them. Attribute each one to Dalton, Thomson, Rutherford or Bohr:

1. a) "The atom is a solid, indivisible sphere."
2. b) "The atom is a sphere of positive charge with electrons embedded in it."
3. c) "The positive charge and almost all the mass are concentrated in a tiny nucleus."
4. d) "Electrons can only occupy certain orbits, with well-defined energies."

Old cathode-ray tube televisions worked using a gun that fired a beam of electrons against the screen.

1. a) What are cathode rays?
2. b) What experimental observation proves that they are composed of charged particles rather than a form of light?

In the gold foil experiment, three things were observed: most of the $\alpha$ particles passed through the foil without deflecting, a small fraction deflected at large angles, and a very few bounced straight back. The figure summarizes these three observations.

1. a) Write down the conclusion Rutherford drew from each of the three observations.
2. b) What would Thomson's model have predicted for this experiment? Why was this prediction not fulfilled?

The electron discovered by Thomson is a particle that comes from inside atoms, regardless of the element.

1. a) Explain what fundamental idea of Dalton's model was disproved by this discovery.
2. b) If atoms contain electrons, which are negatively charged, but are electrically neutral, what did Thomson have to add to his model to explain this neutrality?

A discharge tube filled with hydrogen glows with a pinkish light which, when viewed through a spectroscope, separates into a few well-defined colored lines. Observe the setup in the figure:

1. a) Explain, using Bohr's model, why hydrogen emits only certain colors and not a continuous rainbow.
2. b) What is the relationship between the energy of the electron jump and the color of the emitted line?

Thomson placed two electrically charged plates on either side of the cathode ray tube and observed that the beam deflected towards the positive plate. The set-up is shown in the figure.

1. a) What did Thomson conclude about the sign of the charge of the particles in the beam? Explain your reasoning.
2. b) He repeated the experiment with tubes containing different gases and always obtained the same particles, with the same charge-to-mass ratio. What does this result imply about the electron as a constituent of matter?

The diameter of an atom is of the order of $10^{-10}\ \text{m}$ and that of its nucleus is of the order of $10^{-15}\ \text{m}$.

1. a) Calculate how many times larger the diameter of the atom is than that of the nucleus.
2. b) If the nucleus were the size of a marble with a diameter of $1\ \text{cm}$, what diameter would the atom have at that scale? Express the result in kilometers.

In $1868$, while studying the spectrum of sunlight during an eclipse, a yellow line was observed that did not correspond to any element known on Earth. It was proposed that it belonged to a new element, named helium, from the Greek helios, "Sun".

1. a) Explain how it is possible to know the chemical composition of a star without bringing back samples from it.
2. b) How does this discovery relate to the idea that each element possesses a characteristic spectrum?

In the gold foil experiment, for every $20000$ $\alpha$ particles fired, about $19980$ passed through the foil without deflecting, $19$ deflected at large angles and $1$ bounced straight back.

1. a) Calculate what percentage of the particles passed through the foil without deflecting.
2. b) Calculate what percentage bounced back.
3. c) Interpret these two percentages: what do they say about the size of the nucleus compared to that of the atom?

Dalton's model was not just a whim: it explained the chemical laws of his time.

1. a) Explain how the idea that atoms are neither created nor destroyed, but only rearranged, justifies the law of conservation of mass in a chemical reaction.
2. b) How does Dalton's model explain why a compound like water always contains hydrogen and oxygen in the same ratio?

$\alpha$ particles have a positive charge and a mass about $7000$ times greater than that of the electron.

1. a) Explain, using forces between electric charges, why an $\alpha$ particle passing very close to the nucleus deflects significantly, but barely changes its trajectory when passing near an electron.
2. b) Why do the particles that get closest to the nucleus bounce straight back?
3. c) If the gold foil were twice as thick, how would you expect the proportion of particles bouncing back to change? Explain your answer.

According to Bohr's model, the first three energy levels of hydrogen are $E_1 = -13.6\ \text{eV}$, $E_2 = -3.4\ \text{eV}$ and $E_3 = -1.5\ \text{eV}$. When an electron falls from one level to a lower one, it emits a photon whose energy is exactly the difference between the two levels. The figure shows the three levels and the two transitions.

1. a) Calculate the energy of the photon emitted in transitions $E_3 \rightarrow E_2$ and $E_2 \rightarrow E_1$.
2. b) Which of the two photons has higher energy? Knowing that higher photon energy corresponds to a higher light frequency, which of the two emitted lights has a higher frequency?
3. c) Explain, based on these results, why the spectrum of hydrogen consists of discrete lines and is not continuous.

Imagine that Thomson's model were correct and the atom were a soft sphere of positive charge with electrons embedded in it.

1. a) What would happen to $\alpha$ particles as they passed through the gold foil according to that model? Could they ever bounce back? Explain your reasoning thinking about how charge and mass are distributed.
2. b) What observation from the actual experiment proves that model to be false?
3. c) Explain why, for an $\alpha$ particle to be able to bounce back, the positive charge and almost all the mass must be concentrated in a very small space.

In $1932$, Chadwick bombarded beryllium with $\alpha$ particles and obtained highly penetrating radiation that was not deflected by electric fields and was capable of knocking protons out of a paraffin block: he had discovered the neutron. The experimental set-up is shown in the figure.

1. a) Why is uncharged radiation not deflected by electric fields?
2. b) Explain why the neutron was much harder to detect than the electron or the $\alpha$ particle.
3. c) Known nuclei weighed more than twice as much as they would if they only contained protons: explain how the discovery of the neutron resolved this problem with the nuclear model.

According to classical physics, an electron orbiting the nucleus should continuously lose energy by emitting light and end up spiralling into the nucleus in a fraction of a second: Rutherford's model rendered atoms unstable.

1. a) What did Bohr postulate about "stationary orbits" to prevent this catastrophe?
2. b) What does it mean that Bohr introduced a "postulate" instead of deducing it from known laws? What justifies accepting a new postulate in science?
3. c) What experimental fact, inexplicable by Rutherford's model, was indeed explained by Bohr's model thanks to its energy levels?

The Balmer series of hydrogen satisfies the formula

$\frac{1}{\lambda} = R_{\text{H}}\times\left(\frac{1}{2^2} - \frac{1}{n^2}\right)$, with $R_{\text{H}} = 1.097\times10^{7}\ \text{m}^{-1}$ and $n = 3, 4, 5\dots$

1. a) Calculate the wavelength $\lambda$ for $n = 3$ and express it in $\text{nm}$. What color of the visible spectrum does it approximately correspond to?
2. b) Repeat the calculation for $n = 4$.
3. c) What happens to the wavelengths as $n$ increases? Calculate the limiting value for $n \rightarrow \infty$ and explain why the lines in the series get closer and closer together.

The visible spectrum of hydrogen shows lines at $656\ \text{nm}$ and $486\ \text{nm}$; that of helium, at $588\ \text{nm}$ and $447\ \text{nm}$. The light from a discharge tube containing an unknown gas is analysed and lines appear at $656\ \text{nm}$, $486\ \text{nm}$, $588\ \text{nm}$ and $447\ \text{nm}$. The figure shows the observed spectrum.

1. a) Which gas or gases does the tube contain? Justify your answer.
2. b) Would observing a single line, for example $588\ \text{nm}$, be enough to state with certainty that helium is present? Explain what role the full set of lines plays in identification.
3. c) This method made it possible to discover helium in the Sun before finding it on Earth: explain why spectral analysis is such a powerful tool in astronomy.

Write a short popular science article entitled "From Billiards to the Quantum Leap: A Century of Atomic Models".

1. a) For each model (Dalton, Thomson, Rutherford and Bohr), summarise its image of the atom and the experiment or observation that motivated it.
2. b) Indicate what each model retained from the previous one and what it had to change.
3. c) Conclude by explaining the role that experiments play in the evolution of scientific models.