# Calorimeter calculations with aqueous electrolytes

In her article “2016 Canadian Chemistry Contest: Worst discriminators” (March 2017, Chem 13 News) Jennifer Pitt-Lainsbury described her analysis of a CCC question with a low discriminating index and asked for comments (see boxes, below). She challenged us to look more critically at all of our tests and evaluations and ask whether they evaluate what we think they are evaluating. She analyzed what it was about the question on the CCC that produced such a poor discriminating index.

Question 14: A student places 0.750 g of solid sodium hydroxide (NaOH) into 20.00 mL of water at 25.0  C inside a coffee cup calorimeter. The final temperature of the calorimeter contents is 34.6  C. The density of water is 1.0 g mL-1. Assume the specific heat capacity of the solution approximates that of water at 4.184 J g-1  C-1 and the calorimeter has 100% efficiency. What is the ΔHsol'n for the dissolution reaction below?

NaOH(s) → Na+(aq) + OH-(aq)

It was a coffee cup calorimeter question that only 15% of the students answered correctly (difficulty 0.15) and there was no correlation to overall success on the test (discriminating level 0.02). What is it about the teaching of this topic or the question that produced this result? Having encountered this type of calorimetry problem when I was a college professor, I reflected on how I approached this question and it made me even more curious about the result. I would like to share my personal exploration of this intriguing problem.

 Question 14 student responses 1 2 3 4 A - 42.8 kJ mol-1 mwcΔT/n* 32% B - 44.4 kJ mol-1 mscΔT/n* 15% C - 803 J mol-1 mwcΔT 26% D - 833 J mol-1 mscΔT 22% E - 1070 J mol-1 mwcΔT/m* 5%

In 20 years of teaching this topic I found that students have little difficulty with the theory after they have been shown example calculations. Students did well on this material on my in-house tests. Why then were so many students unable to get to the correct answer on the CCC test?

Columns 1 and 2 show the five answer choices (correct answer B). Column 3 is the calculation formula for each answer, where mw and ms are the mass of the water only and the mass of the total solution, respectively, and n* and m* are the moles of NaOH and the mass of NaOH, respectively. Column 4 is the test response percentage of each answer choice. The question was not intentionally difficult, but it proved to be extremely tricky.

The 53% of students who chose C, D or E did not divide by the number of moles of NaOH. They did not seem to realize that special ΔH values such as those of formation, vaporization, fusion, combustion, etc., are always per mole values. And they missed the obvious clue that all of the answers were stated as per mole values.

The 58% of students who chose A or C used the mass of water instead of the mass of the solution. They may have never seen a heat of solution problem or used a solution mass in a calculation. Some may have had a teacher who ignored solute mass in all calculations. There is a YouTube video1 by a teacher who ignores solute mass in a heat of solution example.

Even for students who have seen a correct heat of solution calculation, the use of the specific heat of water as the specific heat of the solution may act as a powerful behavioural nudge to use the mass of the water, as they would have done in most other calorimeter calculations.

## Calorimeter calculations

Calorimeter calculations are done for bomb calorimeter experiments and coffee cup calorimeter experiments. In a bomb calorimeter, heat flows into a water jacket around the bomb: Q = mwcwΔT. Coffee cup calorimeter experiments come in three types: hot metal in water; heat of reaction (e.g., acid-base neutralization); and heat of solution. For a hot metal in water, the calculation is the same as for a bomb calorimeter. For a heat of

reaction or a heat of solution experiment there is an electrolyte solution in the calorimeter; logically the mass of the solution and the specific heat capacity of the solution in the calorimeter should be used in the calculation: Q = mscsΔT.
Secondary school chemistry textbooks make two simplifying assumptions for coffee cup calorimeter calculations where electrolyte solutions occur: all solution densities are 1.000 g mL-1 and the specific heat capacities are as for water at 4.184 J g-1  C-1. Neither assumption is very good, but at this level this is acceptable. The important point is, that by making these assumptions, students are being trained to ignore the solute mass in most instances. It does not surprise me that they would ignore the solute mass in the heat of solution calculation on a test.

In teaching this topic I followed what was in my course textbook without critical thought about the method. My last textbook was Fundamentals of Chemistry, 3rd Edition, 1988, by Brady and Holum. In this text there is a worked example of a neutralization reaction in a coffee cup calorimeter, 50 mL of 1.0 M HCl added to 50 mL of 1.0 M NaOH. In the calculation, the mass of the solute (about 3.0 g of NaCl) is totally ignored. Is this typical of current textbooks? This is then followed by an exercise posed for the students that is a heat of solution experiment, the final line of which states: "Don't forget to use the total mass of the solution." In fact, there is a higher mass percent of solute in the neutralization question, where it is ignored, than there is in the heat of solution exercise, where it is included. I never noticed this inconsistency while teaching with the text.

## Specific heat capacity of electrolyte solutions

Experiment-derived values of the specific heat capacities of most common aqueous electrolytes are available in a CRC Press reference volume, accessible via Google Books,2 containing 125 pages of tabled values for all common electrolytes. Specific heat capacity values of aqueous solutions of electrolytes DECREASE rapidly at first, then more slowly as concentration increases. Values of the specific heat capacity of aqueous NaOH solutions at 20 °C from 0% to 20% by mass have been plotted. A similar plot for NaCl solutions and an explanation of the effect in terms of the structure of solvated aqueous ions is available online.3
Specific heat capacity of aqueous NaOH solutions at 20 °C
The specific heat capacity of the NaOH solution in the calorimeter of CCC Question 14 is about 3.980 J g-1  C-1. Using the value for water (4.184 J g-1  C-1) introduces an error of + 5.1%. Using the mass of the water (20.00 g) rather than the mass of the solution (20.75 g) introduces an error of - 3.6%.

My introduction to this topic occurred serendipitously in the mid-1990s. Ayaaz Pirani, who was then teaching our first year classes, adopted the text Chemistry by Raymond Chang after the publishers of Brady and Holum decided not to issue a new edition. It was my task to revise our laboratory manual references for the new textbook. When I got to our coffee cup calorimeter experiment, I found that actual values of specific heat capacity were given in Chang in worked examples and questions for every electrolyte solution. Our experiment had to be extensively revised. I could not find the source of the data and had to remove all of our own examples from the manual because of this lack. This must have frustrated other teachers as well. A check of the 10th edition of Chang (2010),4 which is available online, shows that it reverted to a standard presentation in which such data are not given. Not only that, Chang 10th edition does not give an example of a heat of solution calculation, although the topic has an entire section of text and a table of the values of ΔHsol'n for several substances. Is this omission deliberate, in order to avoid the difficulty encountered by the CCC and AP Chemistry examiners?

Using Google and Google Books, I was able to locate and view the data source, wrapping up some unfinished business. It is certainly not necessary to include specific values of experiment-based specific heat capacities of electrolyte solutions in examples, but secondary chemistry teachers and students might benefit from being aware that the specific heat capacity of an electrolyte solution is significantly lower than that of water. It could be introduced as an additional property of electrolyte solutions, alongside conductivity, and it could be introduced when discussing the aqueous solvation of ions in water, which causes this effect.

## Conclusions

Chemistry teachers should explicitly inform students that special enthalpy changes such as those of formation, combustion, vapourization, fusion and solution are always referred to a particular substance and are per mole quantities.

In my opinion multi-school exams such as the Canadian Chemistry Contest and AP Chemistry should avoid having heat of solution calorimetry as the subject of a question without taking into account that some teachers may not have done a heat of solution example, or may have ignored the mass of the solute in their calculation. Or may not have covered calorimetry in their course at all.

A beneficial tweak to coffee cup calorimetry would be to use a non-water value, e.g., 4.0 J g-1  C-1, as the approximate specific heat capacity of all dilute electrolyte solutions.

Looking at the statistics of exams, be they just school-wide or national, can offer valuable insight into how we are teaching high school chemistry.

## Acknowledgement

Thanks to Professor Ernest Zinck (retired) of Acadia University, who read a draft of the article. In particular, he noted that answer E of Question 14 is derivable from the given information in the question.

## References

1. The Organic Chemistry Tutor, YouTube (relevant problem begins at 12 min 55 seconds elapsed on video): www.youtube.com/watch?v=HlvllF6Ml9c.
2. Properties of Aqueous Solutions of Electrolytes, Zaytsev and Aseyev Editors, CRC Press, 1992, Section 1, Chapter 3, Experimental Values of Specific Heat Capacity: https://books.google.ca/books?id=XJkALTxffGUC&printsec=frontcover#v=onepage&q&f=false.
3. Socratic Chemistry: https://socratic.org/questions/how-does-salt-change-the-specific-heat-capacity-of-water.
4. Chemistry, 10th Edition, Raymond Chang, 2010: https://ia801309.us.archive.org/13/items/Chemistry_10th_Edition_Raymond_Chang/Chemistry_10th_Edition_Raymond_Chang.pdf.