DP IB Physics: SL

E: Nuclear and quantum physics

E.1 Structure of the atom

DP IB Physics: SL E: Nuclear and quantum physics E.1 Structure of the atom Linking questions:
a)	How can emission spectra allow for the properties of stars to be deduced?
b)	How is the distance of closest approach calculated using conservation of energy?
c)	How can emission spectra be used to calculate the distances and velocities of celestial bodies?
d)	Under what circumstances does the Bohr model fail? (NOS)
e)	How have observations led to developments in the model of the atom? (NOS)

a) How can emission spectra allow for the properties of stars to be deduced?
Solution:
Since each element has a distinct emission spectrum that may be used to identify its existence in a star, much like a fingerprint, emission spectra enable the derivation of stellar attributes.
By examining the distinct light wavelengths that a star emits, astronomers may ascertain its temperature, chemical makeup, and even whether magnetic fields are present.
When an element’s atoms become excited and then return to lower energy levels, they produce distinct patterns of light known as emission spectra.
Astronomers can create a spectrum by breaking down the light from a star into its individual wavelengths using a spectroscope.
Figure 1 Star spectra simulation
The emission and absorption lines so reveal important details about the characteristics of the star:
⇒ Chemical composition:
When the atoms in each element are stimulated, light at particular wavelengths is released.
Astronomers can identify the components that make up a star’s atmosphere by analysing the wavelengths that are present in the star’s emission spectrum.
Every element has spectral lines, or distinct wavelengths at which it emits light.
Scientists can identify the components that make up a star’s atmosphere by looking for these lines in the star’s spectrum.
For instance, helium, sodium, calcium, and hydrogen each have their own distinctive lines, while hydrogen exhibits the Balmer series.
⇒ Temperature:
The star’s surface temperature may be inferred from the spectrum’s intensity and distribution throughout its many wavelengths (colours).
Wien’s Law states that colder stars produce more light at longer (redder) wavelengths, whereas hotter stars emit more light at shorter (bluer) wavelengths.
For instance, blue stars have a higher temperature than red stars.
⇒ Motion (Doppler effect):
If the star is moving towards or away from us, its spectral lines will shift:
– Redshifted → moving away.
– Blueshifted → moving toward us.
This allows astronomers to measure the radial velocity of stars and galaxies.
⇒ Age and evolutionary stage:
Certain elements or ionization states (e.g., ionized calcium, molecular bands) are more common in younger or older stars.
The presence or absence of certain lines can help determine if a star is on the main sequence, a giant, or a white dwarf, etc.
b) How is the distance of closest approach calculated using conservation of energy?
Solution:
By relating the initial kinetic energy of a particle (such as an alpha particle) to the electrostatic potential energy at the point where the particle momentarily pauses and reverses course, the distance of closest approach is determined.
This is predicated on the idea that at the nearest point of approach, all of the original kinetic energy is transformed into potential energy. This energy-saving approach serves as the basis for the formula.
The smallest separation between a travelling charged particle (such as an alpha particle) and a nucleus before it briefly halts because of electrostatic repulsion is known as the distance of closest approach.
Rutherford scattering is one example of an experiment in nuclear physics where this happens.
⇒ Conservation of energy principle:
At the instant of closest approach, the particle’s initial kinetic energy (KE) is entirely transformed into electric potential energy (PE):
[math]\text{Initial KE} = \text{Electrostatic PE at closest approach}[/math]
– Kinetic energy (initial):
[math]K.E = \frac{1}{2}mv^2 [/math]
m is mass of the incoming particle (e.g. alpha particle)
v = initial velocity of the particle
– Electric Potential Energy:
[math]PE = \frac{1}{4\pi \varepsilon_0} \cdot \frac{Q_1 Q_2}{r}[/math]
[math]Q_1[/math] = charge of the incoming particle
[math]Q_2[/math] = charge of the nucleus
[math]\varepsilon_0[/math] = permittivity of free space
[math]r[/math] = distance of closed approach
⇒ Application:
The alpha particle’s kinetic energy (KE), which is often expressed in electron volts (eV) or mega-electron volts (MeV), must be multiplied by the conversion factor [math]1.602 × 10^{-19} J/eV[/math] in order to be converted to Joules (J).
The target nucleus’s atomic number (Z), such as 79 for gold, is utilised.
The estimated value of Coulomb’s constant (k) is [math] 8.98755 × 10^9 N⋅m^2⋅C^{-2}[/math].
At [math]1.602 × 10^{-19} C [/math], the elementary charge (e) is
c)How can emission spectra be used to calculate the distances and velocities of celestial bodies?
Solution:
By using redshift and the Doppler effect, emission spectra may be utilised to determine the velocities and distances of celestial bodies.
Astronomers can ascertain whether these things are travelling towards or away from us, as well as how quickly, by examining the spectral lines of light that they emit.
Figure 2 Using spectra to measure stellar radius, composition and motion
This is made feasible by the fact that spectral line wavelengths vary according on the relative velocity of the source and the observer.
⇒ Redshift and distance:
Doppler effect:
When light waves are subject to the Doppler effect, the relative speed of the source and the observer produces a change in the wavelength of light that is viewed.
Redshift:
A shift towards the red end of the spectrum results from the stretching of the light waves emitted by astronomical objects as they move away from us.
Hubble’s Law:
According to Hubble’s Law, a galaxy’s redshift and, consequently, its velocity, are closely correlated with its distance.
⇒ Spectral line analysis and velocity:
Unique spectral signatures:
Every element has a distinct collection of spectral lines, or particular light wavelengths that are absorbed or emitted.
Identifying elements:
Astronomers can determine the elements and relative abundances of a celestial object by examining its emission or absorption spectra.
Doppler shift and velocity:
The Doppler effect will cause an element’s spectral lines to change whether it is travelling in our direction or away from us.
Radial velocity:
The object’s radial velocity, or velocity along the line of sight, is indicated by the degree of shift in the spectral lines.
d) Under what circumstances does the Bohr model fail? (NOS)
Solution:
Although a major breakthrough in the knowledge of atomic structure, the Bohr model of the atom is unable to explain the wave-particle duality of electrons, the Zeeman and Stark phenomena, or the intensities of spectral lines when applied to atoms containing multiple electrons.
A significant development in atomic theory was the Bohr model of the atom, which was first forward in 1913. It effectively anticipated the light wavelengths emitted by hydrogen atoms and introduced quantized energy levels to explain the hydrogen spectrum.
But as knowledge in science grew, the Bohr model’s shortcomings were exposed.
Figure 3 Bohr model
Multi-electron atoms:
The single-electron hydrogen is the subject of the Bohr model. Because the model does not take into account the complexity of electron-electron interactions, it is unable to predict the behaviour or spectral lines of atoms with multiple electrons.
Spectral line intensities:
Although spectral lines are predicted by the Bohr model, it is unable to explain why certain lines are more intense than others.
Zeeman and stark effects:
The splitting of spectral lines in response to electric (Stark effect) or magnetic (Zeeman effect) fields is not explained by the Bohr model.
Wave – particle duality:
Although electrons are treated as particles in fixed orbits in the Bohr model, they may also behave as waves. The Bohr model does not take this wave-particle duality into consideration.
Heisenberg uncertainty principle:
By giving electrons precise, well-defined orbits and locations, the Bohr model defies the Heisenberg uncertainty principle, which is impossible given quantum physics.
e) How have observations led to developments in the model of the atom? (NOS)
Solution:
Atomic models have been developed as a result of observations that have played a significant role in forming our understanding of the atom.
Through their experiments, scientists such as Thomson, Rutherford, and Bohr improved upon Dalton’s solid sphere model, exposing subatomic particles, a nucleus, and electron orbits.
The self-correcting and forward-thinking nature of science is reflected in the evolution of the atomic model through methodical observations and experimental data.
The need to explain novel experimental findings that previous models were unable to account for propelled each stage of the atomic model’s development.
Figure 4 Rutherford’s observations that led to an atomic view of matter
Dalton’s atomic theory:
Although it was a fundamental idea, John Dalton’s theory that all matter is made up of indivisible atoms was later improved.
Thomson’s plum pudding model:
– Observation: Cathode ray experiments led to the discovery of the electron.
– Model: Atoms have embedded negative electrons and are a positive “soup.”
– Limitation: Unable to explain alpha particle scattering.
Rutherford’s Nuclear Model (1911):
– Observation: While some alpha particles were deflected in the gold foil experiment, the majority of them passed through.
– Model: Electrons orbit an atom’s tiny, dense, positively charged nucleus.
– Limitation: Unable to explain why attraction does not cause electrons to spiral into the nucleus.
Bohr’s Quantized orbit model (1913):
– Observation: There were distinct lines visible in the hydrogen emission spectrum.
– Model: Energy is released and absorbed in jumps, and electrons travel in fixed orbits with quantized energy.
– Limitation: Only worked with hydrogen; fine structure and multi-electron atoms were ineffective.