Models: simple pictures that still predict things

A model is a simplified picture that is still good enough to predict

Intuition

A street map leaves out every tree and every shop. That is why you can read it. A model works the same way. It drops detail on purpose so you can think with it.

Lecture 1 defines a model as a representation of some aspect of the natural world, simple enough to be understood while still capturing some aspect of reality. A map is a model. You judge a map by the task it was made for, not by how much it shows.

In chemistry the same molecule can be modelled at three levels. The lecture shows methane, ethane and pentane as connectivity only, then connectivity plus bond angles, then connectivity plus bond angles plus filled space. A model does not have to be a physical object. It can live on paper or only as an idea.

Models also lead to discoveries. In 1948 Linus Pauling folded a paper chain along creases to copy the tetrahedral geometry of carbon, and it predicted the $\alpha$-helix of proteins. Watson and Crick used physical models of the bases to show that paired bases fit inside the double helix. Lord Kelvin (1884) said he could not feel he understood a thing until he could make a mechanical model of it.

Maths is the exact way to write a model down

Intuition

A drawing of wheels and vortices can be read in many ways. An equation can be read only one way. That is why scientists swap the picture for the equation once the idea works.

Maxwell (1863) first explained how a current gives a magnetic field using a mechanical model of wheels and hexagonal vortices. He soon replaced it with equations that captured the same behaviour without the extra pieces. The lecture’s point: mathematical expressions are the most compact and unambiguous way to state the relations between quantities, and they also give quantitative predictions. Lecture 1 shows the four Maxwell equations, listed in the formulas for this page.

The lecture then gives a general strategy. Physical phenomena have length scales and energy scales, so scaling and back-of-the-envelope estimates are the best place to start. Two examples follow.

Try it: worked example

Is Giant-Man (Ant-Man scaled up) plausible? Use scaling.

Show answer and steps

AnswerNo. Mass grows as L cubed but bone area only as L squared, so his mass would crush his bones.

1. Let $L$ be the size of the object. Mass is proportional to volume, so $m \propto L^3$.
2. Bone strength is proportional to the cross-section area $A$.
3. To avoid breaking under its own mass, $A$ must grow with the mass, so $A \propto L^3$ is required.
4. If every dimension is scaled by the same factor, as for Giant-Man, the area only grows as $A \propto L^2$.
5. $L^2$ grows more slowly than $L^3$, so the bones are too thin. The lecture’s conclusion: his mass would crush his bones.

The second example is how surface tension $\sigma$ changes with temperature. Surface tension has units of energy per area. The energy to create a surface should be proportional to $\Delta H_{vap}$, the enthalpy difference between liquid and vapour. The area per molecule should scale with the volume $v$ per molecule. Since $A \propto L^2$ and $v \propto L^3$, we get $A \propto v^{2/3}$. So $\Delta H_{vap} \propto \sigma v^{2/3}$, and a plot of $\Delta H_{vap}$ against $\sigma v^{2/3}$ is a straight line. The lecture’s plot (Keeney and Heicklen, 1979) shows lines for Cs, Rb, K, Na and Li, with Li highest and Cs lowest.

Classical or quantum depends on the de Broglie length

Intuition

Everything has a wave side. For a heavy, slow thing the wave is tiny compared with the thing, so you can treat it like a ball. For a light thing, like an electron in a molecule, the wave is bigger than the molecule, so you cannot.

Classical mechanics describes the world as trajectories (positions and velocities), as when tracking planes. Quantum mechanics describes it as wave functions, as for an electron in an orbital. Trajectories are far easier and cheaper to compute. The size of a wave function is the de Broglie length $\lambda = h/p$, with $p = mv$ the momentum and $h = 6.626 \times 10^{-34}$ J s. Classical mechanics is valid when $\lambda$ is much smaller than the characteristic length of the phenomenon.

Exam detail

Lecture 1 table. Electron in H$_2$: $\lambda = 3.3 \times 10^{-10}$ m, larger than the $7.4 \times 10^{-11}$ m H-H bond length, so quantum. C$_{60}$ at 273 K: $\lambda = 4.0 \times 10^{-12}$ m, much smaller than the $0.7 \times 10^{-9}$ m diameter, so classical. Liquid argon at 80 K: $\lambda = 0.9 \times 10^{-10}$ m against an atomic radius of $1.8 \times 10^{-10}$ m. The slide gives this one a question mark, so it does not state an answer.

Classical mechanics can generally be used except when momenta are small, meaning low-mass particles (electrons, photons, protons) or very low temperatures.

The lecture ends by calling computer modelling a litmus test of understanding: could you tell a computer how to model the phenomenon? Boiling of a simple liquid can be simulated and predicted by theory. Liquid turbulence can be simulated although there is no theory for complex flows. High-temperature superconductivity cannot yet be reproduced. A battery case study (the solid electrolyte interface, SEI, on graphite) shows a layer that lets Li$^+$ through but blocks electrons, and is hard to model because it is uneven at the nanoscale and changes over time.

Optional aside

Unit map from the course overview. Lecture 1 is models. Lectures 2 to 6 cover stability, symmetry and structure of matter. Lectures 7 to 8 cover bonding and electron delocalisation. Lectures 9 to 11 cover electronic, dielectric and magnetic properties. Lectures 12 to 17 are statistical mechanics. Lectures 18 to 19 are defects, 20 to 23 thermodynamics, 24 kinetics of phase transitions, 25 glasses, 26 to 29 soft matter, 30 to 31 liquids, 32 to 36 surfaces and 37 to 39 metal-organic frameworks.