Atomic Structure

Definitions

• Element: a substance that cannot be broken down into simpler substances by a chemical reaction.
• Compound: a substance made by chemically combining two or more elements. It has different properties from its constituent elements.
• Mass number($A$): the number of protons plus the number of neutrons in an atom. Give the total number of nucleons, and often referred to as nucleon number.
• Isotopes: atoms of the same element with different mass numbers.
• Ions: when an atom loses or gains an electron, an ion is formed. Positive ions are known as cations and negative ions are known as anions.
• Relative atomic mass($A_r$): average mass of an atom of an element, taking into account all the isotopes and their relative abundance, compared to one atom of carbon-12.
• Wavelength($\lambda$): the distance between two successive crests or troughs.
  • Measured in metres(m).
• Frequency($\nu$): the number of waves produced in one second.
  • Measured in Hertz(Hz).
• Ground state: when an electron is in its lowest energy level.
• Absorption spectrum: electron absorbs light photons and jumps to an excited state of higher energy.
  • This spectrum would be black lines, representing absorbed colours, on a coloured background.
• Emission spectrum: electron emits light photon as it jumps down to a state of lower energy.
  • This spectrum would consist of coloured lines, representing emitted colours, on a black background.
• Ionisation energy: the energy needed to remove an electron from the ground state of each atom in a mole of gaseous atoms, ions, or molecules.

Equations

Energy, frequency and wavelength.

$c = \lambda \nu$

• $c$: Speed of light, approximately $3.00 \times 10^8 ms^{-1}$.
• $\lambda$: Wavelength of light, measured in metres($\text{m}$).
• $\nu$: Frequency, measured in Hertz($\text{Hz}$).
  $E = h \nu$
• $E$: Energy, measured in joules($J$).
• $h$: Planck’s constant.
• $\nu$: Frequency, measured in Hertz($\text{Hz}$).
  $E = h(\frac{c}{\lambda})$
• $E$: Energy, measured in joules($J$).
• $h$: Planck’s constant.
• $c$: Speed of light, approximately $3.00 \times 10^8 ms^{-1}$.
• $\lambda$: Wavelength of light, measured in metres($\text{m}$).
  $E \propto \nu$
  $E \propto \frac{1}{\lambda}$

Photons

$\Delta{E_{electron}} = E_{photon}$

• The energy change when an electron drops a level is equal to the energy of the photon emitted due to the drop.
  $E_{photon} = h \nu \\ E_{photon} \propto \nu$
• The energy of the photon is proportional to the frequency of light emitted. High energy photons emit UV or violet light(high frequency) and low energy photons emit IR or red light(low frequency).

Orbital configuration

• The number of sub-shells at the $n$th energy level: $n$.
• The number of orbitals at the $n$th energy level: $n^2$.
• The number of electrons at the $n$th energy level: $2n^2$.
• The number of orbitals at the $l$th sub-level: $2l + 1$, where $l = 0 \text{ to } (n-1)$

Isotopes

• All isotopes of the same element have the same chemical properties, but different physical properties.

Atomic and mass number

$^{A}_{Z}\text{X}$

• $A$: mass number.
• $X$: symbol of element.
• $Z$: atomic number.

Mass spectrum

• The results of the analysis of an element’s isotopes by a mass spectrometer are presented in the form of a mass spectrum.
• It is a bar graph where mass/charge is plotted on the $x$ axis and % abundance is plotted on the $y$ axis.

Spectra

Continuous spectrum versus discontinuous spectrum

Continuous spectrum	Discontinuous spectrum
Continuous energy levels	Discrete energy levels
Unbroken sequence of frequencies/colour	Only certain frequencies/colours are produced/displayed
Spectrum of visible light	Emission/absorption spectra

Emission Spectrum

• It is produced by excited atoms and ions as they fall back to a lower energy level.
• Line emission spectrum is characteristic to the element. Therefore, different elements have different line spectra, and can be used to identify unknown elements.
• Emission spectra are not continuous, but consist of separate lines.
• These lines converge towards the higher energy end of the spectrum.

Hydrogen Emission Spectrum

Series	Initial orbit	Range	Wave emitted
Lyman	$n_i = 2, 3, 4 ,\dots ,\infty$	$n_f = 1$	Ultraviolet
Balmer	$n_i = 3, 4 ,5\dots ,\infty$	$n_ f= 2$	Visible
Paschen	$n_i = 4 ,5,6\dots ,\infty$	$n_f = 3$	Infrared

• As we go from Lyman to Balmer to Paschen:
  • Energy decreases.
  • Frequency decreases.
  • Wavelength increases.
• The colour emitted (VIBGYOR) corresponds to energy:
  • V ⇒ in highest visible energy region, as it has the highest frequency.
  • R ⇒ in lowest visible energy region, as it has the lowest frequency.
• The highest possible energy emitted would fall in the UV region.
• The lowest possible energy emitted would fall in the infrared region.

Hydrogen Absorption Spectrum

Series	Initial orbit	Range	Wave absorbed
Lyman	$n_i = 1$	$n_f = 2, 3, 4 ,\dots ,\infty$	Ultraviolet
Balmer	$n_i = 2$	$n_f = 3, 4 ,5\dots ,\infty$	Visible
Paschen	$n_i = 3$	$n_f = 4 ,5,6\dots ,\infty$	Infrared

• The amount of light absorbed at a particular frequency depends on the identity and concentration of the atoms present. Atomic absorption spectroscopy is used to measure the concentration of metallic elements.

Atomic models

Bohr’s atomic model

• According to Bohr’s atomic model:
  • electrons move around the nucleus in a fixed path known as an orbit.
  • electrons can occupy orbits of discrete energy.
  • when electrons drop energy levels, they release energy in discrete, or quantised amounts, which makes the emission spectrum discontinuous. Therefore, the emission spectrum is vital evidence for quantisation, because if energy were not quantised, the emission spectrum would be continuous.

Merits and demerits of Bohr’s model

Merits	Demerits
Explained hydrogen emission spectrum	Failed to explain multi-electronic system
Explained orbits (2D)	Failed to explain wave-particle duality and concept of orbitals (3D)
Energy and radii of orbits were explained.	Failed to explain quantum mechanics and splitting of spectral lines of elements with more than one electron.

De Broglie’s wave equation

$\begin{aligned} E &= mc^2\\ mc^2 &= \frac{hc}{\lambda}\\ mc &= \frac{h}{\lambda}\\ \lambda &= \frac{h}{mc}\\[1em] \implies \lambda &\propto \frac{1}{m} \end{aligned}$

Wave-Particle Duality

• The difference between the particle and wave nature of EM radiation:

Particle	Wave nature
Occupies space	Spread out in a space
No two particles can occupy the same space at the same time. (ie, no interference)	Two or more waves can co-exist. (there exists interference)
Explained by photo electric effect and black body radiation.	Explained by electromagnetic wave theory.

Heisenberg’s Uncertainty Principle

• It is impossible to know both the momentum and position of a moving particle simultaneously.

Significance of uncertainty principle

Orbit	Orbitals
Well defined circular path around the nucleus in which the electron revolves.	The 3D space around the nucleus of the atom having maximum probability of finding the electrons.
Represents the planar (2D) motion of electron.	Represents 3D motion of electrons.
Fails to explain De-Broglie principle and Heisenberg’s principle.	Explains both the principles.
Circular shape.	Has different shapes.
Non-directional in nature	Directional in nature.

Electronic Configuration

• A more detailed model of the atom describes the division of the main energy level into $s, p, d$, and $f$ sub-levels of successively higher energies.
  $1s<2s<2p<3s<3p<4s<3d<4p<5s$
• Orbitals ranked in ascending order of their energy:
  $s < p<d<f$

$s$ orbitals

• Spherical in shape.
• Has only 1 shape.
• Holds maximum 2 electrons with opposite spins.
• Lowest energy orbital.

$p$ orbitals

• Dumbbell shaped.
• Has 3 orbital shapes, which hold 2 electrons each, for a total of 6 electrons.
  • The orientations are: $p_x$, $p_y$, $p_z$.
• Higher in energy than $s$.

$d$ orbitals

• Double dumbbell shaped.
• Has 5 orbital shapes, which hold 2 electrons each, for a total of 10 electrons.
• Higher in energy than $p$.

$f$ orbitals

• Has 7 possible shapes, for a total of 14 electrons.
• Highest in energy, out of the 4.

Aufbau Principle

• When electrons fill up the orbitals, the orbitals with the lowest energy level get filled up with electrons first.
• The order of energy levels of orbitals is:
  $1s<2s<2p<3s<3p<4s<3d<4p<5s$

Hund’s Rule

• In orbitals of the same energy level, electron pairing will not take place until each orbital has got 1 electron each in parallel spin.
• This is mainly because electrons are negatively charged will repel each other and create distance between each other and try to remain unpaired.
• This means that when drawing electron configurations, you must fill each orbital with an electron before pairing them up.

Periods

Period 3

Quantum Numbers

Principal Quantum Number ($n$)

• Designates the the principal electron shell.
  $n = 1, 2, 3, 4...$
• As value of $n$ increases, the distance between the furthest electron and the nucleus increases.
• The attraction between the electron and the nucleus decreases.
• It is easier for an atom to expel that electron, reducing ionisation energy.

Azimuthal Quantum Number($l$)

• Also known as orbital angular momentum number.
• The largest value of $l$ is always 1 less than the principal quantum number.
  $l = 0, 1, 2, 3, 4...(n-1)$
• For example:
  • When $n = 1$, $l = 0$:
    • $l$ takes only one value, which means there is only 1 subshell in this principal shell: $s$.
  • When $n = 2$, $l = 0, 1$:
    • $l$ takes 2 values, which means there are 2 subshells in this principal shell: $s$ and $p$.
  • When $n = 3$, $l = 0, 1, 2$:
    • $l$ takes 3 values, which means there are 3 subshells in this principal shell: $s$, $p$, and $d$.
• In general:

Value of $l$	Name of subshell
0	$s$
1	$p$
2	$d$
3	$f$

Magnetic Quantum Number($m_l$)

• determines the orientation of orbitals and the number of orbitals in a sub-shell.
• $m_l$ is an interval ranging from $-l$ to $+l$.
  $m_l = -l, (-l + 1), (-l + 2)...0...(+l-2)(+l-1)(+l)$
• Each value of the $m_l$ represents a specific orientation of a sub-shell in a principal shell.
• The number of values that $m_l$ can take represents the number of orbitals in a sub-shell.
• For example:
  • For $l = 1$:

Value of $m_l$	Orientation
$-1$	$p_x$
$0$	$p_y$
$+1$	$p_z$

Electron spin quantum number($m_s$)

• Designates the direction of electron spin and is represented by either $+\frac{1}{2}$ or $-\frac{1}{2}$.
  $m_s = \pm\frac{1}{2}$
• If $m_s$ is positive, it means the electron has an upward spin($\uparrow$).
• If $m_s$ is negative, it means the electron has a downward spin($\downarrow$).

Patterns in Successive Ionisation Energies (HL)

• Patterns in successive ionisation energies give evidence for shells and subshells.
• Following is a graph showing the successive ionisation energies of Aluminium:

• The general, continuous increase in successive ionisation energies is explained by the fact that it becomes harder and harder to remove electrons from more and more positively charged ions.
• This is because, for the same nuclear charge there are lesser and lesser electrons, making each one harder to remove than the previous one as a result of greater attraction.
• The above graph also shows evidence for shells - there are steep increases between ionisation numbers 3-4 and 11-12, which signify that electrons are now being removed from a shell that is closer to the nucleus - since there is more attraction as we move closer to the nucleus, more energy is required.
• The next graph shows a zoomed-in version:

• This graph, which depicts more closely the successive ionisation energies from 4-12, shows evidence for sub shells - the slightly larger increase between ionisation numbers 9-10 indicates that there is a change in sub-shell, since slightly more energy is required to remove the 10th electron.
• Thus we can deduce that the 2p sub-shell can hold 6 electrons and the 2s sub-shell can hold 2.
• More evidence for sub-shells can be observed if the graph on page 90 in the textbook is observed.
• The explanations for the reasons for the fluctuations in the patterns are given clearly under ‘Solution’ on Page 91. This information is concise and so has not been mentioned here.