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PHYSICS, M8 2024 HSC 23

Development of models of the atom has resulted from both experimental investigations and hypotheses based on theoretical considerations.

  1. A key piece of experimental evidence supporting the nuclear model of the atom was a discovery by Chadwick in 1932.
  2. An aspect of the experimental design is shown.
     

    1. What was the role of paraffin in Chadwick's experiment?   (2 marks)

    2. How did Chadwick's experiment change the model of the atom?  (3 marks)

  2. Explain how de Broglie's hypothesis regarding the nature of electrons addressed limitations in the Bohr-Rutherford model of the atom.   (4 marks)

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a.i.  Role of paraffin wax:

→ Paraffin wax is a rich source of protons.

→ When the paraffin was placed in front of the unknown radiation, the transfer of momentum from the radiation caused protons to be emitted from the paraffin wax.

→ The emitted protons could then be detected and analysed.

→ From studying the protons ejected from the paraffin wax, Chadwick proposed the existence of the neutron.
 

a.ii. Changes to the model of the atom:

→ Previous to Chadwick’s experiment, the model of the atom proposed by Rutherford consisted of a dense positive charge in the nucleus which was orbited by electrons.

→ In this model however, the protons did not account for the total mass of the nucleus.

→ Through using the conservation of momentum and energy in his experiment, Chadwick proposed the existence of the neutron particle which was slightly heavier than the proton.

→ The model of the atom was updated to include both protons and neutrons in the nucleus which then fully accounted for the mass of the nucleus.
 

b.   Limitations in the Bohr-Rutherford model:

→ Rutherford’s model of the atom stated that electrons orbited the nucleus and were electrostatically attracted to the positive nucleus. This meant that the electrons were in circular motion and were constantly under centripetal acceleration.

→ However, Maxwell predicted that an accelerating charge would emit electro-magnetic radiation and in Rutherford’s model, all atoms should have been unstable as the electrons would emit EMR, lose energy and spiral into the nucleus.

→ Bohr proposed that electrons orbited the nucleus in stationary states at fixed energies with no intermediate states possible where they would not emit EMR but provided no theoretical explanation for this.

De Broglie’s hypothesis:

→ De Broglie proposed that electrons could exhibit a wave nature and could act as matter-waves. The electrons would form standing waves around the nucleus and would no longer be an accelerating particle which addressed the limitation of all atoms being unstable.

→ Further, De Broglie proposed that the standing waves would occur at integer wavelengths where the circumference of the electron orbit would be equal to an integer electron wavelength, \(2\pi r=n\lambda\)  where  \(\lambda = \dfrac{h}{mv}\). At any other radii other than this, deconstructive interference would occur and a standing electron wave would not form. This addressed why electrons could only be present at fixed radii/energy levels in the atom.

Show Worked Solution

a.i.  Role of paraffin wax:

→ Paraffin wax is a rich source of protons.

→ When the paraffin was placed in front of the unknown radiation, the transfer of momentum from the radiation caused protons to be emitted from the paraffin wax.

→ The emitted protons could then be detected and analysed.

→ From studying the protons ejected from the paraffin wax, Chadwick proposed the existence of the neutron.
 

Mean mark (a)(i) 52%.

a.ii. Changes to the model of the atom:

→ Previous to Chadwick’s experiment, the model of the atom proposed by Rutherford consisted of a dense positive charge in the nucleus which was orbited by electrons.

→ In this model however, the protons did not account for the total mass of the nucleus.

→ Through using the conservation of momentum and energy in his experiment, Chadwick proposed the existence of the neutron particle which was slightly heavier than the proton.

→ The model of the atom was updated to include both protons and neutrons in the nucleus which then fully accounted for the mass of the nucleus.
 

b.   Limitations in the Bohr-Rutherford model:

→ Rutherford’s model of the atom stated that electrons orbited the nucleus and were electrostatically attracted to the positive nucleus. This meant that the electrons were in circular motion and were constantly under centripetal acceleration.

→ However, Maxwell predicted that an accelerating charge would emit electro-magnetic radiation and in Rutherford’s model, all atoms should have been unstable as the electrons would emit EMR, lose energy and spiral into the nucleus.

→ Bohr proposed that electrons orbited the nucleus in stationary states at fixed energies with no intermediate states possible where they would not emit EMR but provided no theoretical explanation for this.

De Broglie’s hypothesis:

→ De Broglie proposed that electrons could exhibit a wave nature and could act as matter-waves. The electrons would form standing waves around the nucleus and would no longer be an accelerating particle which addressed the limitation of all atoms being unstable.

→ Further, De Broglie proposed that the standing waves would occur at integer wavelengths where the circumference of the electron orbit would be equal to an integer electron wavelength, \(2\pi r=n\lambda\)  where  \(\lambda = \dfrac{h}{mv}\). At any other radii other than this, deconstructive interference would occur and a standing electron wave would not form. This addressed why electrons could only be present at fixed radii/energy levels in the atom.

♦ Mean mark (b) 44%.

PHYSICS, M8 2019 VCE 18

The energy level diagram for a hydrogen atom is shown in Figure 19.
 

  1. A hydrogen atom in the ground state is excited to the \(n=4\) state.
  2. Explain how the hydrogen atom could be excited to the \(n=4\) state in one step.   (2 marks)
  1. List the possible photon energies that could be emitted as the atom goes from the \(n=4\) state to the \(n=2\) state.   (3 marks) 

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a.    → The hydrogen atom would need to absorb exactly 12.8 eV of energy.

→ The energy source could be either an incident photon or electron.
 

b.    From \(n=4\) to \(n=3\ \ \Rightarrow \ 0.7\ \text{eV}\)

From \(n=3\) to \(n=2\ \ \Rightarrow \ 1.9\ \text{eV}\)

From \(n=4\) to \(n=2\ \ \Rightarrow \ 2.6\ \text{eV}\)

Show Worked Solution

a.    → The hydrogen atom would need to absorb exactly 12.8 eV of energy.

→ The energy source could be either an incident photon or electron.
 

b.    From \(n=4\) to \(n=3\ \ \Rightarrow \ 0.7\ \text{eV}\)

From \(n=3\) to \(n=2\ \ \Rightarrow \ 1.9\ \text{eV}\)

From \(n=4\) to \(n=2\ \ \Rightarrow \ 2.6\ \text{eV}\)

PHYSICS, M8 EQ-Bank 26

Observations and mathematical ideas are critical to the improvement of scientific models.

Discuss this statement with reference to scientific discoveries that have contributed to our understanding of the atom.   (8 marks)

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If scientific models are not consistent with observations and mathematical ideas, then they need to be changed. The historical progression of models of the atom is an example of this scientific method in action.

→ The Geiger-Marsden experiment involved shooting a beam of alpha particles at a thin sheet of gold foil. It was observed that most alpha particles passed through the gold however some bounced back.

→ Rutherford explained these observations in his model. He postulated that a small, dense positively charged nucleus caused the alpha particles to bounce back and that this nucleus was surrounded by orbiting electrons. The atom was mostly empty space so the majority of alpha particles passed through.

→ This model was limited however as an electron orbiting the nucleus would be accelerating and would radiate electromagnetic radiation, losing energy and spiralling into the nucleus.

→ Bohr overcame this limitation and improved the atomic model by analysing observations of the hydrogen spectrum. He postulated that electrons occupied specific energy orbits and could jump from one energy level to another and electrons travelling from a higher to lower energy orbit would emit a photon, corresponding to emission lines on hydrogen’s spectrum. 

→ Bohr developed a mathematical model explaining the specific wavelengths emitted which relies on Rydberg’s equation: `(1)/(lambda)=R((1)/(n_(f)^2)-(1)/(n_(i)^2))`.

→ This model was limited as it was only able to explain the hydrogen atom and wasn’t able to explain why electrons were quantised in such a way.

→ De Broglie built upon this model by postulating that electrons behaved as a wave. He stated that electrons behaved as standing waves with an integer number of wavelengths fitting the circumference of an orbit.

→ De Broglie’s contribution was critically supported by his mathematical model for calculating the associated wavelength `lambda=(h)/(mv)`

→ From these examples, it is clear that our understanding of the atom has been increased following observations and improved mathematical models.
 

Answers could also reference:

→ Thomson’s discovery of the electron.

→ Further contributions from Davisson-Germer as well as Schrodinger and Heisenberg.

Show Worked Solution

If scientific models are not consistent with observations and mathematical ideas, then they need to be changed. The historical progression of models of the atom is an example of this scientific method in action.

→ The Geiger-Marsden experiment involved shooting a beam of alpha particles at a thin sheet of gold foil. It was observed that most alpha particles passed through the gold however some bounced back.

→ Rutherford explained these observations in his model. He postulated that a small, dense positively charged nucleus caused the alpha particles to bounce back and that this nucleus was surrounded by orbiting electrons. The atom was mostly empty space so the majority of alpha particles passed through.

→ This model was limited however as an electron orbiting the nucleus would be accelerating and would radiate electromagnetic radiation, losing energy and spiralling into the nucleus.

→ Bohr overcame this limitation and improved the atomic model by analysing observations of the hydrogen spectrum. He postulated that electrons occupied specific energy orbits and could jump from one energy level to another and electrons travelling from a higher to lower energy orbit would emit a photon, corresponding to emission lines on hydrogen’s spectrum. 

→ Bohr developed a mathematical model explaining the specific wavelengths emitted which relies on Rydberg’s equation: `(1)/(lambda)=R((1)/(n_(f)^2)-(1)/(n_(i)^2))`.

→ This model was limited as it was only able to explain the hydrogen atom and wasn’t able to explain why electrons were quantised in such a way.

→ De Broglie built upon this model by postulating that electrons behaved as a wave. He stated that electrons behaved as standing waves with an integer number of wavelengths fitting the circumference of an orbit.

→ De Broglie’s contribution was critically supported by his mathematical model for calculating the associated wavelength `lambda=(h)/(mv)`

→ From these examples, it is clear that our understanding of the atom has been increased following observations and improved mathematical models.
 

Answers could also reference:

→ Thomson’s discovery of the electron.

→ Further contributions from Davisson-Germer as well as Schrodinger and Heisenberg.

PHYSICS, M8 2015 HSC 34d

In 1927, Davisson and Germer reported the results of an experiment in which they fired electrons at a crystal of nickel and observed how the electrons were scattered.

  1. State their conclusion, with reference to the results they obtained.   (2 marks)
  1. Explain the significance of this experiment to the Rutherford-Bohr model of the atom.   (3 marks)
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i.    Experiment Results:

→ Davisson and Germer found that the electrons were not scattered in a random pattern, but formed an interference pattern after passage through the crystal.

→ This pattern is formed by constructive and destructive wave interference.

Conclusion:

→ Since interference is a phenomenon only observed with waves they concluded that electrons were also waves.
 

ii.   The Rutherford-Bohr model:

→ Postulated that electrons existed in fixed orbits. The model however, was unable to explain why only these orbits were stable.

→ The knowledge that electrons are waves provides a plausible explanation for this stability.

→ If electrons act as waves, as indicated by de Broglie, the electron can only exist in orbits where it experiences constructive interference.

Show Worked Solution

i.    Experiment Results:

→ Davisson and Germer found that the electrons were not scattered in a random pattern, but formed an interference pattern after passage through the crystal.

→ This pattern is formed by constructive and destructive wave interference.

Conclusion:

→ Since interference is a phenomenon only observed with waves they concluded that electrons were also waves.
 

ii.   The Rutherford-Bohr model:

→ Postulated that electrons existed in fixed orbits. The model however, was unable to explain why only these orbits were stable.

→ The knowledge that electrons are waves provides a plausible explanation for this stability.

→ If electrons act as waves, as indicated by de Broglie and experimentally shown by Davisson and Germer, the electron can only exist in orbits where it experiences constructive interference.


Mean mark (ii) 54%.

PHYSICS, M8 2016 HSC 34bi

Outline features of the hydrogen spectrum that Bohr's model could not explain.   (3 marks)

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→ Bohr’s model could not provide an adequate explanation of why some spectral lines were observed to be more intense than others.

→ This result suggested that certain transitions occurred much more frequently than others which Bohr’s model could not explain.

→ With more sensitive instruments, hyperfine lines were discovered to also exist in the spectrum.

→ This indicates there is some splitting of the energy level that Bohr’s model cannot explain.

Show Worked Solution

→ Bohr’s model could not provide an adequate explanation of why some spectral lines were observed to be more intense than others.

→ This result suggested that certain transitions occurred much more frequently than others which Bohr’s model could not explain.

→ With more sensitive instruments, hyperfine lines were discovered to also exist in the spectrum.

→ This indicates there is some splitting of the energy level that Bohr’s model cannot explain.

PHYSICS, M8 2017 HSC 34c

The diagrams show features of the hydrogen emission spectrum.
 

With reference to Bohr's postulates, explain how the line at 434.0 nm in the hydrogen emission spectrum is produced. Support your answer with calculations.   (4 marks)

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→ Bohr’s postulates state that an electron orbits an atom’s nucleus at fixed energy levels.

→ If an electron jumps from a higher energy level to a lower energy level, it will emit a photon that contains an exact amount of energy.

→ The Rydberg equation can be used to calculate a photon’s wavelength.

→ The blue light is part of the visible Balmer series and therefore `n_f=2`.

→ Using `n_f=2`, `lambda` = 434.0 nm: 

`1/lambda` `=R(1/((n_f)^2)-1/((n_i)^2))`  
  `=R(1/(2^2)-1/((n_i)^2))`  
`1/(n_i)^2` `=1/4-1/(lambdaR)`  
`(n_i)^2` `=1/(1/4-1/(1.097 xx 10^7 xx 434 xx 10^9))`  
`n_i` `=5`  

 
→ The equation shows that a 434 nm wavelength for the emitted photon corresponds to a jump from the 5th energy level to the 2nd energy level.

Show Worked Solution

→ Bohr’s postulates state that an electron orbits an atom’s nucleus at fixed energy levels.

→ If an electron jumps from a higher energy level to a lower energy level, it will emit a photon that contains an exact amount of energy.

→ The Rydberg equation can be used to calculate a photon’s wavelength.

→ The blue light is part of the visible Balmer series and therefore `n_f=2`.

→ Using `n_f=2`, `lambda` = 434.0 nm: 

`1/lambda` `=R(1/((n_f)^2)-1/((n_i)^2))`  
  `=R(1/(2^2)-1/((n_i)^2))`  
`1/(n_i)^2` `=1/4-1/(lambdaR)`  
`(n_i)^2` `=1/(1/4-1/(1.097 xx 10^7 xx 434 xx 10^9))`  
`n_i` `=5`  

 
→ The equation shows that a 434 nm wavelength for the emitted photon corresponds to a jump from the 5th energy level to the 2nd energy level.


Mean mark 56%.

PHYSICS M8 2022 HSC 31

Following the Geiger-Marsden experiment, Rutherford proposed a model of the atom.
 


 

Bohr modified this model to explain the spectrum of hydrogen observed in experiments.
 


 

The Bohr-Rutherford model of the atom consists of electrons in energy levels around a positive nucleus.

How do features of this model account for all the experimental evidence above? Support your answer with a sample calculation and a diagram, and refer to energy, forces and photons.  (9 marks)

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The Geiger-Marsden experiment, which involved firing alpha particles at a thin sheet of gold foil produced results which can be explained by the Bohr-Rutherford model:

→ The majority of fired alpha particles passed through the gold foil undeflected. Rutherford concluded from this that the atom had a small, central nucleus.

→ Some alpha particles were deflected and some of these were deflected at very large angles. Rutherford concluded from this that the nucleus was dense and positively charged exerting a repulsive electromagnetic force on the fired alpha particles.

→ The model accounts for Rutherford’s conclusions, placing electrons in orbits around a small positive nucleus.
 

Rutherford’s model alone could not explain the emission spectra of elements such as hydrogen. Bohr’s contribution to the Bohr-Rutherford model amended this:
 


 

→ Bohr proposed that electrons orbited the atomic nucleus in quantised orbits at fixed energies. He proposed that electrons could move from a higher energy orbit (eg. n=1) to a lower energy orbit (n=3) by emitting a photon with energy  `E=hf`  equal to the energy difference between the two orbits.

→ Additionally, he proposed that electrons could move from a lower energy orbit to a higher energy orbit by absorbing a photon with energy  `E=hf`  equal to the energy difference between the two orbits. 

→ This is able to account for the given emission spectra of hydrogen, where emission lines correspond to electron transitions from higher energy orbits to the second energy orbit which produce photons within the spectrum of visible light.
 

Using Rydberg’s equation it is possible to predict the emission lines of hydrogen, using an electron moving from the sixth to the second Bohr energy orbit as an example:
 

`(1)/(lambda)` `=R((1)/(n_(f)^(2))-(1)/(n_(i)^(2)))`  
  `=(1.097 xx10^7)((1)/(2^(2))-(1)/(6^(2)))`  
  `=(2 xx1.097 xx10^7)/(9)`  
  `=2.438 xx10^6`  
`lambda` `=410  text{nm}`  

 
→ This value corresponds to the leftmost line on the given spectrum, reflecting how Bohr’s model can account for the emission spectra of hydrogen.

 

Show Worked Solution

The Geiger-Marsden experiment, which involved firing alpha particles at a thin sheet of gold foil produced results which can be explained by the Bohr-Rutherford model:

→ The majority of fired alpha particles passed through the gold foil undeflected. Rutherford concluded from this that the atom had a small, central nucleus.

→ Some alpha particles were deflected and some of these were deflected at very large angles. Rutherford concluded from this that the nucleus was dense and positively charged exerting a repulsive electromagnetic force on the fired alpha particles.

→ The model accounts for Rutherford’s conclusions, placing electrons in orbits around a small positive nucleus.
 

Rutherford’s model alone could not explain the emission spectra of elements such as hydrogen. Bohr’s contribution to the Bohr-Rutherford model amended this:
 


 

→ Bohr proposed that electrons orbited the atomic nucleus in quantised orbits at fixed energies. He proposed that electrons could move from a higher energy orbit (eg. n=1) to a lower energy orbit (n=3) by emitting a photon with energy  `E=hf`  equal to the energy difference between the two orbits.

→ Additionally, he proposed that electrons could move from a lower energy orbit to a higher energy orbit by absorbing a photon with energy  `E=hf`  equal to the energy difference between the two orbits. 

→ This is able to account for the given emission spectra of hydrogen, where emission lines correspond to electron transitions from higher energy orbits to the second energy orbit which produce photons within the spectrum of visible light.
 

Using Rydberg’s equation it is possible to predict the emission lines of hydrogen, using an electron moving from the sixth to the second Bohr energy orbit as an example:
 

`(1)/(lambda)` `=R((1)/(n_(f)^(2))-(1)/(n_(i)^(2)))`  
  `=(1.097 xx10^7)((1)/(2^(2))-(1)/(6^(2)))`  
  `=(2 xx1.097 xx10^7)/(9)`  
  `=2.438 xx10^6`  
`lambda` `=410  text{nm}`  

 
→ This value corresponds to the leftmost line on the given spectrum, reflecting how Bohr’s model can account for the emission spectra of hydrogen.


♦ Mean mark 51%.

PHYSICS, M8 2020 HSC 21

  1. Calculate the wavelength of light emitted by an electron moving from energy level 3 to 2 in a Bohr model hydrogen atom.   (2 marks)
  1. Describe the behaviour of electrons in the Bohr model of the atom with reference to the law of conservation of energy.   (3 marks)
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a.   `6.563 xx10^(-7) text{m}`

b.   Behaviour of electrons in the Bohr model:

→ Bohr’s model describes electrons orbiting the atomic nucleus in discrete energy levels.

→ When these electrons absorb a photon they gain energy and move from lower to higher energy levels.

→ When they move from higher to lower energy levels, they emit energy in the form of a photon.

→ This is consistent with the law of conservation of energy as the energy of absorbed or emitted photons is equal to the difference in energy of the discrete levels between which electrons move.

Show Worked Solution
a.
`(1)/(lambda)` `=R((1)/(n_(f^(2)))-(1)/(n_(i^(2))))`
    `=1.097 xx10^(7)((1)/(2^(2))-(1)/(3^(2)))`
    `=1.524 xx10^(6)`
  `lambda` `=6.563 xx10^(-7) text{m}`

 

b.   Behaviour of electrons in the Bohr model

→ Bohr’s model describes electrons orbiting the atomic nucleus in discrete energy levels.

→ When these electrons absorb a photon they gain energy and move from lower to higher energy levels.

→ When they move from higher to lower energy levels, they emit energy in the form of a photon.

→ This is consistent with the law of conservation of energy as the energy of absorbed or emitted photons is equal to the difference in energy of the discrete levels between which electrons move.

PHYSICS, M8 2020 HSC 9 MC

Bohr improved on Rutherford's model of the atom.

Which observation by Bohr provided evidence supporting the improvement?

  1. Elements produced unique emission spectra consisting of discrete wavelengths.
  2. The collision of an electron and a positron produced two photons that travelled in opposite directions.
  3. A small percentage of alpha particles fired at a gold foil target were deflected by angles of more than 90 degrees.
  4. A beam of electrons reflected from a nickel crystal produced a pattern of intensity at different angles, consistent with their wave properties.
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`A`

Show Worked Solution

Bohr’s model placed electrons in discrete energy levels explaining why emission spectra consisted of discrete wavelengths.

`=>A`

PHYSICS, M8 2021 HSC 29

Bohr, de Broglie and Schrödinger EACH proposed a model for the structure of the atom.

How does the nature of the electron proposed in each of the three models differ?   (5 marks)

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Bohr’s model of the atom proposed that the electron was a negatively charged particle which orbits the nucleus in a circular path at discrete, quantised energy levels.

De Broglie’s model proposed that electrons had a dual wave/particle nature existing as stable standing waves around the nucleus.

Schrödinger’s quantum mechanical model described electrons as having a wave nature and existing as orbitals. His equation described a cloud surrounding the nucleus in which electrons had a high probability to be found.

Show Worked Solution

Bohr’s model of the atom proposed that the electron was a negatively charged particle which orbits the nucleus in a circular path at discrete, quantised energy levels.

De Broglie’s model proposed that electrons had a dual wave-particle nature existing as stable standing waves around the nucleus.

Schrödinger’s quantum mechanical model described electrons as having a wave nature and existing as orbitals. His equation described a cloud surrounding the nucleus in which electrons had a high probability to be found.


♦ Mean mark 49%.