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My broad question is how do you measure entropy change? I was doing a bit of digging into how thermodynamic tables are developed at I got this from NIST. Basically it seems like you need to indirectly determine the entropy of a substance from other experimental parameters. What NIST says is

  1. You use heat capacities and phase transition enthalpies to do a ground up calculation
  2. Statistical mechanics methods once you have a partition function
  3. Measure temperature dependence of equilibrium constants from van't hoft plot
  4. Calculate it from previously determined enthalpies and free energies.

The thing that's bothering me is that all of these measurements depend on other experimental parameters and it just seems to be a raw calculation of entropy after determining these other quantities.

My primary question then is it possible to do a calorimetry experiment to calculate $\Delta S_{reaction}$? After all, how did people measure entropy experimentally in the olden days before we had copious information on heat capacities?

I tried to think about this for a constant pressure apparatus. A reaction happens and I have a thermometer to measure $\Delta T$. I can correlate this to $\Delta H$ using the heat capacity of the calorimeter. If I have a resistive circuit running through the calorimeter, I can supply heat externally to lower $\Delta T$ so now I can say that $\Delta H$ occurred at the same initial and final temperatures

However, this only tells me about the $\Delta S_{surroundings} = -\frac{\Delta H}{T}$. How do I calculate the change in the reaction system? I tried to imagine the reversible path but if I start and end at the same point, then isn't $\Delta S_{system} = 0$? Where is the flaw in my reasoning or is it just something we cannot end up measuring?

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  • $\begingroup$ You have the right idea in devising a reversible path and measuring the reversible heat from the surroundings for that path. But, it isn't that easy to devise a reversible path for a chemical reaction, starting with pure reactants in stoichiometric proportions at T and P and going to pure products in corresponding stoichiometric proportions at T and P. Are you familiar with the concept of the van't Hoff equilibrium box? $\endgroup$
    – Chet Miller
    Commented Mar 31, 2021 at 10:54
  • $\begingroup$ (1) I don't understand what you mean by a "raw" calculation of entropy. I would have just written "a calculation of entropy". Raw means "unprocessed", as in raw data. Given that the methods you list determine entropy by going through the process of a calculation, rather than by direct measurement, it's the opposite of raw. (2) It sounds like you are trying to measure entropy directly, rather that by calculating it from other measurable quantities and, in so doing, you are presenting it by contrast with $\Delta H$, which you seem to think can be directly measured... $\endgroup$
    – theorist
    Commented Apr 1, 2021 at 1:24
  • $\begingroup$ ....but no state function can be directly measured. They all need to be calculated from measurable quantities. E.g., we can't directly measure $\Delta H$. Rather, we can only determine $\Delta H$ by measuring the heat flow (a path function) at constant pressure. So I don't see a significant qualitative difference between calculating $\Delta H$ from $q_p$ at a single temp., vs. calculating $\Delta S$ from the temp-dependence of $q_p$ via the van't Hoff eqn.... $\endgroup$
    – theorist
    Commented Apr 1, 2021 at 1:33
  • $\begingroup$ ...I.e., we can determine $\Delta S$ calorimetrically the same way we determine $\Delta H$ calorimetrically—by measuring $q_p$; the only diff. is that, for $\Delta S$, we need to measure at multiple temps. (3) You wrote "in the olden days before we had copious information on heat capacities". You don't need copious info. on heat capacities to determine $\Delta H$. You just use a small quantity of reactants, such that the heat capacity of the rxn mixture is small (and can thus be ignored) relative to that of your calorimeter. Thus the only heat capacity you need to know is your calorimeter's. $\endgroup$
    – theorist
    Commented Apr 1, 2021 at 1:41

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You can do a titration monitored by isothermal calorimetry. Once the titration is done, you can figure out the enthalpy of the completed reaction. Then, you know much reacted at each point (from the enthalpy fraction at that point), so you can figure out the equilibrium constant.

Taking the equilibrium constant and the enthalpy, you can calculate the Gibbs energy and the entropy. This requires that the reaction is complete at the end of the titration, but at equilibrium in the first part of the titration.

Below is an illustration of the (perfect) results of such an experiment:

enter image description here Source: https://upload.wikimedia.org/wikipedia/commons/5/59/ITC_THERMOGRAM.png

In general, measuring equilibrium constant and reaction enthalpy is a common path to calculate the entropy of reaction. The equilibrium constant can be measured in many different ways (or you can use electrochemistry to directly measure the Gibbs energy of reaction).

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