Introduction

I gave a lecture on transition metal chemistry in 2018 where I tried to plot out the Frost diagram for titanium as an example. I plotted it perfectly, except that it was upside-down.

This kind of mistake is something I’m confident seasoned teachers will sympathise with. The grinding tiredness of a semester. The pressure of doing a sequence of calculations in front of students who are studiously copying down everything you say. The rush of a busy day making your 3pm lecture a surprise, somehow. You may not have made this exact mistake, but I bet you’ve made one like it.

To make amends to my students, I made a 10min video of me constructing the correct plot and uploaded it to YouTube. I can’t bear to watch it, but the video has now racked up 15k views. 15k! Clearly, the mistake I made was not some Michael-specific quirk: it was a really common problem. I changed my teaching the following year to a ‘flipped’ lecture for Frost Diagrams - it was one of my earliest blog posts.

It is good, I think, to share stories of failures from time to time. But the main reason I wanted to open with this anecdote is because it made me very sensitive to the specific errors students make when they plot a Frost Diagram. I’ve been looking out for them ever since, and have so far found four.

Four Common Errors in Plotting Frost Diagrams

1. Plotting the whole diagram upside-down

2. Plotting most of the diagram correctly, but upside-down for the negative oxidation states

3. Plotting the “steps” as absolute values rather than as cumulative ones

4. “The Blunder” - a simple arithmetic error or mis-transcription of a digit.

I’ve shown what each of the first 3 looks like graphically below in the specific case of sulphur, and will discuss each in turn.

Plotting Upside-Down

This error is - in my experience - the most common. Plotting the whole diagram upside down is an error which emerges from one of two sources: declarative knowledge from textbooks and the incorrect use of Latimer diagrams.

Textbooks

The y-axis on textbook representations of the Frost diagram is represented in two different ways: +nE and representations equivalent to -nE (pictured below). It is fair to say that this contradiction in the secondary literature is confusing for students. If a student flips the sign of the y-axis, they will plot the Frost Diagram upside-down.

Latimer diagrams

Latimer diagrams represent the reduction potentials of couples, and therefore give the negative number required when the Frost diagram codes oxidation as the positive x-direction. Using the reduction potentials instead of the oxidation potentials will result in a numerically accurate Frost diagram plotted upside-down. This problem is particularly acute when the lines linking substances are not arrows but lines, which is common in older texts (pictured below).

Potting the negative oxidation states upside-down

It’s worth making the obvious point that this error will only be visible in contexts where the element has negative oxidation states; if you’re asking about the Frost diagram for manganese, you’ll never see this mistake. If you’re asking about sulphur or bromine, you might.

This error seems to arise because students adopt an inflexible algorithm based on the Frost diagram for metals. If all the oxidation states are positive, then flipping all of the reduction potentials is the correct first step. If there is a negative oxidation state, though, this manipulation will cause the y-trace of the resulting diagram to be the wrong sign for any negative states. This is because the reduction direction is the correct direction when you are reducing the (0) oxidation state (which is defined as having a y-value of zero).

Plotting the steps as absolute values

This mistake seems to arise from associating the cell potential with the formation of a substance rather than the formation of a substance from a specific starting material. The mistake could be seen as like confusing the height of a step with the height of a staircase.

This manifests through students constructing a Frost diagram with no sense of a cumulative energy on the y-axis (i.e. the trace reflects the energy change associated with each step, rather than the total energy change associated with all the steps from the zero oxidation state).

An important source of confusion for this error is about what “n” is. In a one-step electrochemical reaction, n is often used as a symbol for the number of electrons transferred in the reaction. In the context of a Frost diagram y-axis, n instead refers to the oxidation state. The two ideas are identical only for the first step of a diagram. 

The presentation of Frost diagrams in Greenwood and Earnshaw gives the left y-axis legend as a sum to try and arrest students’ scope to form this misconception. This seems wise to me, though the effect is slightly spoiled by a right-hand y-axis which gives a misleading sign of the Gibbs energy. Books like Shriver use a capital N to try and distinguish from the Greek nu used in the maths of the Gibbs energy, which should do the same thing (though this notation somehow seems slightly less clear to me than the Greenwood & Earnshaw summation).

The Blunder

This is an important mistake, but it is not a conceptual mistake. It’s often very hard to systematically address blunders because they are typically idiosyncratic. Writing down 2.14V as 21.4V or 2.41V is not an expression of a deep misunderstanding of a topic, it’s a slip-up.

Some of the errors can be caught by sense-checking answers (21.4V will give you a Frost Diagram with a massive jump in it), but others won’t (2.41V will probably slip through unnoticed). I’d love to read more about blunders but I’ve never stumbled across any deep writing on them.

My responses to these errors

All my responses to these errors relate to a general feature of expert problem solving: checking that an answer is reasonable. I think this is a useful thing to emphasise to first year students, because the act of asking “does this make sense?” is a very useful skill in almost every situation. It’s worth getting into the habit of asking this of yourself in most Chemistry settings.

The specific way I tie this to the Frost diagram is to interpret the y-axis as a proxy for energy (the Gibbs energy in units of the Faraday constant relative to the zero oxidation state). This lets me make two sense-check suggestions:

1. Metals slope downwards (at first), non-metals slope upwards

Having a general sense of the gradient is chemically significant because it indicates whether the act of oxidising an element stabilises it or destabilises it. This check can catch the upside-down error, as well as the negative states inverted error. Depending on the numbers, it usually catches the steps-as-absolute values error, too. It’s a good sense-check.

This advice can become difficult to apply cleanly in unusual pH regimes, or for metalloid elements (the Frost diagram for phosphorus is unhelpfully flat, for example).

2. “Famous” ions (like bromide, Mg(II), oxide) should be low down

The merit of this check is that it helps to focus students on a small area of the graph. This check can catch the upside-down error, as well as the negative states inverted error.

It is a weaker piece of guidance, and I’m considering abandoning it. I’ve found it helpful for elements like bromine (“you’ve heard of bromide, but how many times have you seen Br(VII)?”), but there are times when the most stable species is not as famous as higher-energy species. Sulphur is an important case of this: sulphide is fairly unusual in a first-year course (it’s not ionic enough to feature in many Kapustinskii questions), while sulphate is pretty common (as the deprotonated form of sulphuric acid). Fame is not a chemical idea!

Conclusion

I am sure that my catalogue of common errors is not particularly novel; I am confident that all the points I’ve tried to make here have been discovered and rediscovered by educators for decades. I don’t claim to have completely polished a way to teach this topic, either, but I hope this reflection helps someone to reflect on their practice quicker than I could.

I think these student errors have important consequences for assessment as well as teaching. A question which involves negative oxidation states is harder than one which doesn’t, for example. This might be what you want to test, or it might not - either way, it seems valuable to understand how students typically construct their answers.

In this vein, it is perhaps worth considering whether plotting a Frost diagram is an over-used exam question. I realise they are an incredibly attractive object to mark by pattern recognition (right shape? seven marks!), but it might be that analysing a diagram is a type of question which might bring out a more holistic appreciation of the object when students crunch through past papers.

I’d be really interested to hear about anything you disagree with, or any other observations you’ve made about Frost diagram errors. Get in touch!

Endnote: publication venues for reflective work; a Frost diagram lab

In writing this post, I’ve thought about what kinds of venues would publish a reflection like this. On their standards of rigour, I don’t think the RSC would in CERP. EiC - which I think used to be a natural venue for pieces like this - has moved very deliberately to supporting school teachers (I actually think this is a good thing - secondary education is probably a more urgent policy focus than tertiary for the future of the discipline). The RSC doesn’t have an obvious space for practical reflection on HE teaching.

In contrast, J Chem Ed - the ACS journal - published a paper in 2021 which addressed the ‘upside down’ error (I only discovered the paper last week! I enjoyed reading it!), describing how interactive Excel spreadsheets could be used in teaching Frost diagrams. Similarly, David Smith’s 2023 J Chem Ed paper on the design of an Organic lecture course in the UK is the kind of practical discussion which I think is valuable to professional educators. I think it is a really good thing that J Chem Ed makes space in its portfolio for work like this.

More idly, I was also thinking that a lab practical in which students construct a Frost diagram (or even a Pourbaix diagram) would maybe be useful for students. The time students spend with the data in a lab setting seems like it might be an interesting chance to develop a much deeper appreciation of what all the numbers mean. I’ve never heard of a practical like this. Does one exist? Would it be very hard to make one?