A Critical Look at Units

Introduction

The proper and consistent use of units throughout calculations is of paramount importance in physical science. This should be inculcated in students from the first day of class. As chemistry educators, we must—without fail—model the proper, consistent use of units and significant figures.1

Proper units are logical units. Units describe. Units help us understand what something means. For example, when students realize that the mol is a unit of quantity², their understanding will deepen.

Looking at, for example, entropy, S, which has units of J · mol^–1 · K^–1. These units represent energy (J) per quantity of particles (mol) per degree. Logically, then, entropy refers to the dispersion of the energy of particles, NOT to the dispersion of the particles themselves.³ This goes a long way in dispelling the unfortunate—and common—misconception that entropy represents disorder of particles.

Consider the units of R, the Ideal Gas constant. Starting from the ideal gas law,

PV=nRT,

R=PVnT;

R has units kPa · Lmol · K = kPa · L·mol-1 · K-1.

These units are purely algebraic; they have no meaning for students, in the sense that a meter or a second has. That’s okay—not all units are intuitive.

But R also has units of J · mol^–1 · K^–1.

Logically, it must be that J = kPa · L. How is this so?

kPa = Pa x 10³; L = m³ x 10^–3 and so kPa · L = Pa · m³.⁴

Pa is a unit of pressure such that Pa =N m2; and so

N m2 · m³   =   N · m   =   Force · distance   =   work   =   energy   =   J.

This means that R, in units of J · mol^–1 · K^–1, describes the energy of one mol of any gas, 6.02 x 10²³ particles (atoms or molecules) per Kelvin.

Taking this one step further, I’ll dredge memories of physical chemistry:

Boltzmann constant, k= RAvogadro constant

Inserting units gives, k = 8.314 J · mol-1 · K-16.02 x 1023 molecules · mol-1 = 1.38 x 10-23 J · molecule-1 · K-1

This tells us that the Boltzmann constant represents the average thermal energy, per degree, of a molecule or atom in the gas phase.

The Best Way to Express Units

I prefer that units be written on one line, rather than as a rational expression. For example, the units of the ideal gas constant, R,  can be written as kPa · L / mol · K.

This is not wrong.

But a better way to express these units is to put them all on one line: kPa · L · mol^–1 · K^–1.

This pays homage to what students learned previously in mathematics.

For example:

x2y= x2·y-1.

For students in the physical science game for the long haul, this will pay dividends—it makes unit simplification easier.⁶

That said, if a student wishes to express units in rational form, which is 100% correct, who are we to stop them?

Illogical Units

1. “M” to represent molarity, rather than simply “mol · L^–1”. I have also seen M used to represent molar mass⁷. So, which is it? M = molarity or M = molar mass? Using M to represent molarity, rather than telling it like it is mol · L^–1, demands needless and pointless memorization.

The use of a derived unit, such as M, makes it more difficult for students to reason things out. This will convince you:

mol = (mol · L^–1) · L

Cancelling “L” leaves mol on the right-hand side, which makes sense.

How, logically, can M · L = mol?

This forced memorization is frequently perpetuated by the use of an image mnemonic⁸, a triangle⁹.

Figure 1. A mnemonic triangle to calculate molarity, etc. Cover up the desired quantity; multiply or divide as shown: M =mol/L; mol = M·L; L =mol/M.

2. The use of dm³ to represent liters (L). This is, in my humble opinion, ridiculously pedantic for a high school student. I understand the reasoning: it pays homage to the fact that a L is derived from the metric system: 1 L = (1 dm)³ = 1 dm³; since 1 dm = 10 cm, 1 dm³ = (10 cm)³ = 1000 cm³. None of this is wrong. But it is not intuitive. Most students, especially in the United States, where the metric system is not widely used, don’t have a “feel” for a cubic decimeter. A litre is roughly equal to a quart; a large bottle of soda contains 2 L.¹⁰ This is relatable for most.

Let’s take this one step further: If you were to walk into a Chemistry Research Lab, academic or industrial, and ask for ½ dm³ of hexane, I guarantee that you would be made fun of.

3. Corollary: the presentation of mol · L^–1 as kmol · m^–3. Again, this isn’t wrong. Who, outside of a factory or an oil refinery, works with 1000 moles or with 1000 litres of anything? Think about it: a m³ = 1000 L. 1000 litres of soda looks like 500 2-L bottles. That is a LOT of soda.

4. Consider the Arrhenius equation:

k=Ae- EaRT

The units in the exponential term cancel.¹¹ Therefore, the constant A, the pre-exponential term, has the same units as k, which depend on the total reaction order.

Why pH—and logarithm-derived items—have no units

pH has no units. It’s a logarithmic scale to describe acidity. pH is calculated from hydronium ion concentration, which does have units.

pH= -log⁡[H3O+]

How can we explain that units of mol · L^–1 disappear when their logarithm is taken?

Think of it as taking the log of the ratio of [H₃O⁺], in mol · L^–1, to 1 mol · L^–1 of hydronium ions . That is,

pH=-log⁡(H3O+mol·L–11 mol·L-1)

The units on the right-hand side cancel, and so pH has no units.

The Arrhenius equation can be expressed in logarithmic form:

lnk=lnA-EaRT

Using the above reasoning, all terms in this equation are dimensionless. If you call this mathematical sleight-of-hand, go ahead. But it informs students.

Finally . . .

This is all part of the bigger picture: Units must be used consistently—in EVERY STEP OF EVERY CALCULATION.

I am not a fan of calculations that do not include units in the intermediate steps, only to have them magically appear in the final answer–this is Calculator Ballet.¹²

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Footnotes

1. Confidence Level: Measurement and Significant Figures Simplified | Chemical Education Xchange
2. The Soft Introduction | Chemical Education Xchange
3. See for example: Frank L Lambert, J Chem Ed, 79, 2, p 187
4. 1 L = (1 dm)³ = (0.1 m)³ = 0.001 m³, and so 1 m³ = 1000 L
5. PV = nRT; P in kPa; V in L; n represents mol; T is the absolute temperature in K.
6. Any student who plays, or who has played a musical instrument, understands that the fingering, at least to the novice, may seem uncomfortable initially. But to play anything more than “Twinkle-Twinkle, Little Star”, proper finger-position is required. And so for units . . .
7. Molar mass - Wikipedia
8. Mnemonic - Wikipedia
9. This was introduced to me as a “dumb-boy” triangle by a Grade 11 Physics student at Crescent School, an all-boys independent school, shortly after my arrival on the faculty in 1988. The boy meant it as it was: a way to memorize something that one doesn’t necessarily understand
10. Two-liter bottle - Wikipedia
11. J·mol-1J·mol-1·K-1* K  ➜ no units
12. This term was coined by a former Physics-teacher colleague, whose name escapes me