# Physical Chemistry—how to get the math across?

Which is harder, chemistry of physics? It is generally accepted that physics is harder because of the math.

In fact I am pretty sure that many students opt for chemistry over physics because of the harder math in physics; and life sciences over chemistry for the same reason. Ok, it might be only one of many reasons, and life sciences is an excellent option for interesting material and jobs, but I know that many students are also happy that there is less math in life sciences than physics or chemistry.

In teaching physical chemistry to large classes of life science students, 150 or so, one of the problems is getting to know all the students. Of course there is a large cross section of abilities and interest, and some just cannot get the hang of it. Until the first midterm it is difficult to know who are struggling.

After the first midterm, I order the grades in Excel and look for those with 60% or less. Usually there are less than a dozen or so, and I invite them for a powwow one by one. I try to figure out what is wrong, and then I take the group and invite them to tutorials in my office, like the British system. Usually only two or three take advantage of this.

An aversion to math is the common complaint. I find it curious when a 20 year old student says,

“I have not studied math for a long time!”

Or

“I cannot memorize all the equations.”

I object to the word “memorize” because it is not possible to “memorize” logic: you have to work through it. So I tell them the math derivations are like reading words, except they are equations. Once the words are understood they will feel comfortable with the result.

It does little good to tell them math is the language of science because they see very little of it in the core courses. A look, however, at their math background and prerequisites reveals that they should be well prepared to handle the math in my class. They have been exposed to far more than I use in introductory Physical Chemistry. All they need is simple algebra, logarithms, and integration and differentiation of things like 1/*x* and *x ^{n}*. The only aspect they have not met is a function of two variables, like f(

*x,y*), or more, which is not much of an issue, except for notation.

I find that there are two big stumbling blocks to overcome. The first is they have a hard time assigning meaning to variables and functions. For example they have no trouble with,

(*c* a constant) They can take derivatives and integrate wrt both variables. However if I write down,

they are lost. They find it hard to understand what such an equation describes as the temperature or volume is varied, which is why they are called “variables”. So what I often do now (especially at the beginning of the course) is to write such equations with generic variables, like *T=x *and* V=y*, do any mathematical manipulations, then put back the variables. They then seem to get it.

The second big issue they have is applying conditions. Thermodynamics is a general theory and gives the relationships between measureable quantities (but not the numbers). Hence there are different conditions, like constant *T *and* P*, or constant *T *and* V*, and reversible or irreversible—that lead to different relationships. Later on in the course I try to insist they understand how to do this by showing that the Gibb’s energy gives the maximum (non-PV) work for a reversible process at constant temperature and pressure.

Can you imagine the problems with trying to memorize the following. However the derivation tells a story. (This is standard for thermo, but you can skip the following if you are not up in thermo),

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“Always start with the definitions.” I say along with, “This will likely be on the final exam.”

Take the total derivative and plug in the first law:

Use constant *T *and* P*:

Now recognize that work is d*w* = -*P*d*V* +d*w*_{non-PV}, which defines non-PV work. Explain that if the volume increases work must be done to push back the atmosphere at constant pressure (and vice versa for volume decrease). Finally we get,

Summary: at constant temperature and pressure for a reversible process the maximum work obtained is given by the change in Gibbs energy.

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Then do examples.

What happens under different conditions like when temperature and pressure are not constant and the only work is *PV *work? Applying these conditions to the same starting point gives an entirely different result, which is also a useful starting point to study the effects of changing temperature and pressure,

These derivations are not easy when first studied but are essential to have. That is what thermodynamics is: generally formulated but at the same time, flexible. I try to choose life science examples as much as possible.

Getting the hang of thermo takes time and I try many ways to get these ideas over. I think that it helps some when I tell them that when I was a student taking thermo, I failed my first midterm.