Practice Questions

Name the model of the atom that is picturesquely called the 'plum pudding model'.

Identify the key conclusion drawn from the observation that most alpha-particles passed undeviated through the gold foil in the Geiger-Marsden experiment.

Name the type of spectrum that consists of a series of bright lines on a dark background, emitted by excited atomic gases.

Recall Bohr's first postulate regarding electron orbits in an atom.

Define the term 'ground state' of a hydrogen atom according to Bohr's model.

Compare the distribution of positive charge and mass in Thomson's atomic model with that in Rutherford's atomic model.

Evaluate the impact of de Broglie's hypothesis on the Bohr model. What specific postulate did it provide a physical justification for?

Based on the Bohr model, the total energy of an electron in a hydrogen atom is negative. Propose an explanation for the physical significance of this negative energy and what would be implied if the total energy were positive.

The radius of the innermost electron orbit (ground state, $n=1$) of a hydrogen atom is $5.3 \times 10^{-11} \text{ m}$. Calculate the radius of the orbit corresponding to the second excited state ($n=3$).

Justify why an atom emits a line spectrum rather than a continuous spectrum, according to Bohr's model.

Apply Bohr's second postulate of quantization to calculate the angular momentum of an electron in the third stationary orbit ($n=3$) of a hydrogen atom. (Planck's constant $h = 6.63 \times 10^{-34} \text{ J s}$)

Solve for the energy (in eV) of a photon emitted when an electron in a hydrogen atom makes a transition from the $n=4$ state to the $n=2$ state.

Compare and contrast the appearance and formation of an emission line spectrum and an absorption line spectrum for a gas like hydrogen.

Examine the physical significance of the total energy of an electron being negative in the Bohr model for the hydrogen atom.

Describe Bohr's second postulate, also known as the quantisation condition.

Analyze one major limitation of Bohr's atomic model, specifically addressing its failure when applied to multi-electron atoms like helium.

Formulate a quantitative prediction for the radius of the first Bohr orbit ($n=1$) for a singly ionized helium atom ($\text{He}^+$). Justify your formula based on the principles of the Bohr model.

Recall the formula for the energy of an electron in a hydrogen atom and calculate the energy required to excite a hydrogen atom from its ground state ($n=1$) to its first excited state ($n=2$).

Explain Bohr's third postulate concerning the emission of radiation by an atom.

Summarize the key features of Rutherford's nuclear model of the atom and explain why it is also called the planetary model.

Explain the concept of 'impact parameter' in Rutherford's alpha-particle scattering experiment.

The radius of the innermost electron orbit of a hydrogen atom (Bohr radius) is $5.3 \times 10^{-11} \text{ m}$. Recall the relationship between the radius of the nth orbit and the Bohr radius, and calculate the radius of the $n=3$ orbit.

In a Geiger-Marsden experiment, an $\alpha$-particle with kinetic energy of $5.0 \text{ MeV}$ is aimed at a gold nucleus ($Z=79$). Calculate the distance of closest approach.

Analyze why Rutherford's planetary model of the atom is considered unstable according to classical electromagnetic theory.

Using the data from Example 12.3 in the source text, calculate the frequency of revolution of the electron in the first Bohr orbit of a hydrogen atom.

Calculate the minimum energy in eV required to ionize a hydrogen atom that is in its first excited state ($n=2$).

The radius of a hydrogen nucleus is approximately $1.2 \times 10^{-15} \text{ m}$ and the radius of the electron's orbit in the ground state is $5.3 \times 10^{-11} \text{ m}$. Calculate the ratio of the atomic radius to the nuclear radius and analyze what this implies about the structure of an atom.

Critique the Rutherford nuclear model of the atom. Evaluate its major successes in explaining experimental observations and justify why it was ultimately deemed insufficient, leading to the development of the Bohr model.

Formulate an argument to defend Niels Bohr's decision to postulate the existence of stationary orbits, a concept that directly contradicted classical electromagnetic theory. Your argument should highlight the experimental evidence that necessitated such a radical departure from classical physics.

Critique the assumption made in the Rutherford scattering analysis that the gold nucleus remains stationary. Is this assumption completely valid?

Critique the planetary analogy used in Rutherford's model (electrons orbiting the nucleus like planets around the sun). Identify at least two fundamental physical differences between the atom and the solar system that make this analogy flawed, beyond the issue of electromagnetic radiation.

Evaluate the statement: "The Bohr model is merely a semi-classical stepping stone and holds no real value in modern quantum mechanics." Justify your position by discussing both the profound successes of the model and its significant limitations.

In the Geiger-Marsden experiment, gold foil was used. Evaluate the choice of gold as the target material. Justify why using a much lighter element, like solid hydrogen (a single proton nucleus), would produce significantly different and less conclusive results.

Design a method to determine the ionization energy of hydrogen gas experimentally using a beam of electrons with variable kinetic energy. Describe the setup, the procedure, and how you would interpret the results. Propose what you would observe if the electron beam energy was exactly $10.2 \text{ eV}$.

Explain the two main difficulties of Rutherford's nuclear model of the atom.

Demonstrate how de Broglie's hypothesis of matter waves provides an explanation for Bohr's quantization of angular momentum. Analyze the condition for a stable orbit in terms of standing waves.

Examine the expected results if the Geiger-Marsden alpha-particle scattering experiment were repeated using a thin sheet of solid hydrogen instead of gold foil. Contrast this with the original results.

Solve for the wavelength of the photon that must be absorbed by a hydrogen atom to excite its electron from the ground state ($n=1$) to the second excited state ($n=3$). Use $h = 6.63 \times 10^{-34} \text{ J s}$ and $c = 3 \times 10^8 \text{ m/s}$.

Recall the formula for the total energy ($E_n$) of an electron in the nth stationary orbit of a hydrogen atom and explain the significance of the negative sign.

Explain how de Broglie's hypothesis of matter waves provided a justification for Bohr's second postulate of quantisation of angular momentum.

A scientist proposes a new atomic model where the angular momentum of an electron is quantized as $L = (n + 1/2)h/(2\pi)$ instead of $L = nh/(2\pi)$. Justify, using de Broglie's standing wave hypothesis, why this proposed quantization condition is physically untenable for stable electron orbits.

Imagine a universe where Planck's constant, $h$, is significantly larger, say $h = 1.0 \text{ J·s}$ (instead of $6.626 \times 10^{-34} \text{ J·s}$). Create a description of how the properties of a hydrogen atom in this universe would differ from ours. Specifically, evaluate the consequences for the size of the ground state orbit, the ionization energy, and the appearance of its emission spectrum.

Design a hypothetical experiment to differentiate between Thomson's atomic model and Rutherford's atomic model without using alpha-particle scattering. Propose the experimental setup, the expected observations for each model, and justify how these observations would lead to a conclusive result.

Describe the experimental setup of the Geiger-Marsden experiment and summarize its three main observations.

Formulate a mathematical derivation to show that for large quantum numbers (e.g., a transition from $n$ to $n-1$ where $n \gg 1$), the frequency of the emitted photon in the Bohr model approaches the classical frequency of the electron's revolution in its orbit.