Can We Measure a Molecule's Cross-Sectional Area?
Author: delta
Reviewed by: Guanfu · Juntian
No. That concludes this explainer.
But if you learned about Cavendish’s torsion-balance experiment in middle school, you are probably not willing to stop there. Physics has a technique known as two-stage amplification, so could the same or a similar idea be used to measure dimensions in the microscopic world?
It certainly can. This article explains, in accessible language, one method for measuring the cross-sectional area of a molecule: the surface-tension method.
What is surface tension?
Really, do we have to start all the way back here?
You have seen surface tension countless times: raindrops on a window, dew on leaves in the early morning, and large soap bubbles in the laundry. You may also have encountered, or will encounter, the topic in high-school physics. At the microscopic level, a liquid consists of vast numbers of molecules. They are neither as far apart as gas molecules nor as tightly packed as molecules in a solid, so they can move freely while still attracting one another. If circles represent molecules, the microscopic state near a liquid surface exposed to a gas looks roughly like the diagram below.
You may have noticed that the circles, representing molecules, are more sparsely distributed near the surface. At the gas-liquid interface, thermal motion lets liquid molecules escape into the air while gas molecules also enter the liquid. For equal volumes, however, the liquid contains more molecules, so more leave the liquid than enter it. The surface layer is therefore less densely populated. To keep the surface continuous at constant volume, the liquid must form it with as few surface molecules as possible. Only a spherical surface satisfies this condition; we will omit the proof that a sphere has the smallest surface area for a given volume. In other words, a force among molecules at the liquid’s surface makes that surface spherical when gravity is neglected. The force maintaining this state is called surface tension.
What gives rise to surface tension? Let us return to Figure 1 and add a few more strokes.
Let the arrows represent intermolecular forces. Because there are far more liquid molecules, their attraction dominates; the comparatively sparse gas molecules contribute little. This imbalance in attraction gives rise to surface tension, though it is not surface tension itself. It tends to draw surface molecules into the liquid and thus keeps the surface taut. This is only an intuitive model, not a literal account of the phenomenon; the real situation is considerably more complicated. Surface tension, for example, acts tangent to the liquid surface. How can an imbalance in attraction produce a force in that direction? One rough way to picture it is that the imbalance pulls a surface molecule downward while its neighbors on either side resist letting it leave the surface, as shown below.
The angle θ is extremely small; at the molecular scale, how large could it be? We therefore neglect it. The resulting pull is surface tension, tangent to the liquid surface.
Everything above assumes a liquid containing only one substance. What happens if we add a solute, say, to water?
That night, I was pure no more
The water, that is: adding a solute makes it impure. Nothing scandalous here; the experiment simply happened to be done at night.
Adding the solute lowers water’s surface tension. According to the principle of minimum energy, the system favors the lower surface tension. The solute concentration at the surface will therefore be slightly higher than in the bulk; if the solute raises surface tension, the reverse occurs. A difference between the surface and bulk concentrations is called surface adsorption. Once a phenomenon exists, some supposedly idle (just kidding) scientist will look for the rule behind it. Consider Szyszkowski: I could not even find biographical information about him on Baidu, yet from his experimental data he obtained the following empirical equation:
σ＝σ₀-σ₀αln（1+c/β）
Here, σ is the surface tension of the solution, σ₀ is that of the solvent, α and β are empirical coefficients, and c is the solution concentration.
Is this empirical equation valid? In 1916, physical chemist Irving Langmuir proposed that atoms or molecules at a solid surface possess outward residual valence forces capable of capturing gas molecules. Because these forces act over a distance comparable to a molecular diameter, only a single molecular layer can be adsorbed on the surface. He accordingly derived the Langmuir monolayer adsorption isotherm, whose form for a gas-liquid interface is shown below:
Here, Γ is the amount adsorbed, and Γ∞ is the limiting adsorption: the amount present when the entire surface is covered by adsorbed molecules. It can be approximated as the amount of solute at saturation. Under this model, the liquid surface likewise supports only a monolayer. If Γ∞ can be determined, the molecular cross-sectional area is 
, bringing us back to the question we started with.
Chemical thermodynamics gives the following result; the lengthy derivation is omitted here: the Gibbs adsorption equation:
Substituting the Langmuir equation gives:
Integration yields:
σ＝σ₀-RTΓ∞ln（1+kc）
This has the same form as Szyszkowski’s empirical equation. It supports that equation and gives us a route back to our original goal: measuring molecular cross-sectional area. Measure the surface tension and solution concentration, plot the resulting curve, and fit its functional form; Γ∞ can then be obtained, and from it the molecular cross-sectional area.
Solution concentration can be measured accurately by photometry and several other methods. Surface tension can be found by forming a small bubble in a capillary and measuring the pressure difference across it. This gives the bubble’s excess pressure. At its smallest, the bubble has the same diameter as the capillary. The capillary’s internal diameter can first be calibrated from the maximum excess pressure of a liquid with known surface tension. The Laplace equation then relates the maximum excess pressure to surface tension; its derivation is not central to this article.
In practice: determining the cross-sectional area of an n-butanol molecule
I used n-butanol here simply because that was the substance in my experiment. If you have access to the equipment, you can apply the same principle to measure the molecular cross-sectional area of other substances.
What follows is data processing rather than popular-science exposition. Read on if you are interested; it also shows that the method is reasonably reliable.
Code:
1 | data = {{0, 72.75}, {0.022, 65.91}, {0.066, 58.15}, {0.109,52.94}, {0.153, 49.27}, {0.262, 42.94}, {0.372, 38.53}, {0.590, 32.99}, {0.809, 28.86}} |
Published values for the cross-sectional area of an n-butanol molecule lie between 
, so the error is small.
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