Back-of-the-envelope calculations are one of our chief tools as scientists. When you spend most of your time wondering if your latest measurement is correct, having a tool to check if the numbers make sense is simply priceless. If you are lucky, a good estimate might just avoid a costly or laborious measurement — this is very common in disciplines like chemical engineering, which a friend described as “the art of estimating numbers and plugging them into some variation of Bernoulli’s continuity equation”. Unsurprisingly, these Fermi problems are now common interview questions at major consultancy and tech companies, and have even started to go viral.
Last week, I thought I would ask my biochemistry students to solve a back-of-the-envelope problem as part of their tutorial work. Disguised as an enzyme catalysis problem, I asked them to estimate the energy of a single hydrogen bond. Needless to say, they were puzzled. Some of them asked if I had forgotten to include some information in the problem sheet. For some reason, Fermi problems seem to be less common in chemistry and biology that they are in physics of engineering. Of course, estimating the energy of a hydrogen bond is in many ways much harder than guessing the number of ping pong balls that fit a Boeing 747. Nobody has seen a hydrogen bond in the flesh. And our minds struggle to grasp the vast numbers present at the molecular level. Nevertheless, guesstimates are incredibly useful
In this post, I will show two very simple guesstimates of the energy of a hydrogen bond, using either readily accessible measurements (nothing harder than a Google search) or a physical understanding of the hydrogen bond. Many other possibilities exist — if you can think of one, feel free to share!
Heat of vaporization
The easiest way to approach this problem is with a thermodynamic approach. The abstract formulation of thermodynamics allows to define easily measurable properties without barely referring to the microscopic world. In this case, if we can define two thermodynamic states that differ only in the formation of hydrogen bonds, we can measure the amount of energy needed to transfer these two states. We then need only to calculate the amount of hydrogen bonds we have broken, and divide the total energy by this number, to have a decent estimate of the energy of a hydrogen bond.
If this sounds too abstract, I am just talking about boiling water. In its liquid state, water molecules form hydrogen bonds with each other. What happens is that (partially) negatively charged oxygens get in touch with (also partially) positively charged hydrogens and form a stabilising interaction. You don’t even need to know how this works to use thermodynamics, though. You just need to reason that, in the gas state, molecules are too far apart to form hydrogen bonds. Therefore, to a very good approximation, boiling water amounts to removing the hydrogen bonds in liquid water.
The energy required to transform a mol of liquid water into a mol of gaseous water is the enthalpy of vaporization. A quick Google search reveals that the enthalpy of vaporization of water at 100 C is 40.66 kJ/mol.
Note: the enthalpy is the energy required at constant pressure. Vaporisation will incur in expansion work when the water increases in volume. We can argue, based on “chemical intuition” or whatever, that this PV work is negligible in comparison to hydrogen bond cleavage — in practice, we just use whatever measurement we have at hand.
How many hydrogen bonds do we form per mol? This is slightly harder. Every molecule has three hydrogen-bonding-forming atoms, so the number must be somewhat smaller than 3. It is easy to get confused about the numbers, and frankly, to get the right answer you need to either think a bit about it or have seen a crystal structure of ice. But we are doing napkin math, so we will just take it as a guess. And, in this case, assuming that we have two donor hydrogen atoms, it is an okay guess that every molecule contributes two hydrogen bonds. If you crunch the numbers, this leads to an estimate of 20.3 kJ/mol for an O-H⋅⋅⋅H hydrogen bond. Wikipedia suggests the “official value” is somewhere around 21 kJ/mol — so we have a very fair estimation, if you ask me!
Molecular geometry and electrostatics
The previous example borders in cheating, for two reasons. First, because, well, we haven’t exactly “estimated” the value, but rather calculated it to some good accuracy. Second, as you will have heard if you have ever taught undergraduate courses, “how would we do that in an exam?”. Those are both fair points, so there is a much simpler approach: let’s consider the geometry and apply some basics electrostatics.
The physical background for a hydrogen bond is as follows. Oxygen is a more electronegative atom than hydrogen, hence this atom will “pull” the electron density of the molecule, acquiring a slight negative charge — and likewise, the hydrogen bond, having lost some electron density, will be positively charged. When two water molecules approach each other, moieties with opposite charges will be attracted and form a stable structure. Thus, if we assume that all energy is electrostatic*, we can use this to calculate the electrostatic energy of an oxygen atom. Since we know that the electrostatic energy is given by Coulomb’s law, we just need to have an estimate for the charges on both atoms and the distance between them.
*Many theoretical chemists would rightly object to this view, arguing that the hydrogen bond does indeed have covalent behaviour, but for the purposes here we will assume hydrogen bonds are totally electrostatic.
The distance between the atoms is the easy part. I don’t remember off the top of my head the O-H bond length, so let’s assume it is about 1.2 A — in fact, to make a point, let’s assume it is exactly 1 A. If we assume that a hydrogen bond is about twice as long as a covalent bond, this leads to an intermolecular separation of about 2 A.
Atomic charges are slightly more complicated. One may argue that the formation of charges is a continuum, where apolar bonds (e.g. H-H) sit on one end with no charge transfer, and ionic compounds (e.g. NaCl) are at the other one, where the atoms have transmitted a full electron. Exactly where you place water is a matter of choice.
I am going to assume that the partial positive charge on the hydrogen is 0.1e, and of course this means the oxygen must have a partial negative charge of -0.2e to maintain electroneutrality. Using Coulomb’s law, this leads to:
This leads to an initial estimate of 2.31e-20 J. Multiplying by Avogadro’s number, however, we get to a final estimate of 14 kJ/mol. Slightly worse estimate than before — but definitely within range (and only worse by a factor of ~2).
Sensitivity analysis
At this point you may want to complain. You may claim that I am a charlatan, and that I have just “estimated” by choosing values that more or less lead to the right answer. If you tried to “guess” values for these problems, you may find something completely different. You are right, I already have an intuition for the scales that we are dealing with, and hence my parameter choices are more than lucky. Yet, as I hope I can convince you, these estimates are rather insensitive to errors.
In our example in thermodynamics, the only parameter we had direct influence on was the number of hydrogen bonds per water molecule. We assumed a number of ~2, because every molecule has two hydrogen atoms that can bind to another molecule. However, the result does not change much if we assume that this number is 3 (13.6 kJ/mol) or 0.5 (81.3 kJ/mol). Both of these numbers are not quite that close to the real value of 21 kJ/mol, but they are within an order of magnitude. And, if you wanted to check if your value of 1e-3 kJ/mol is correct, you would already have your answer. A similar story occurs for the electrostatic measurement.
What is more interesting: once we have built this story, we can go the other way around. Since we know the energy of a hydrogen bond, we can now go back to our estimates, and determine the number of hydrogen bonds per molecule. Even more interesting: we can use the right answer, together with accurate water-water distances, to estimate the atomic charges on the atoms, which is much harder to measure!
Guesstimates are a powerful tool in scientific research, but also a great way to foster creativity and lateral thinking about problems we care about. There are many interesting stories about using them for science — what is yours?
