Chemistry and 'Dimensional Analysis'

This Web Page has been written in the conviction that the current method for teaching stoichiometric calculations, far from making these calculations easier for students, actually hinders their understanding of the subject. The technique known as the "factor-label method" or"dimensional analysis" has two major deficiencies. Firstly, as shown on the next page, this technique muddies the distinction a physical quantity and the units used to measure that quantity. Secondly, it also confuses the difference between equality and proportionality.

I: Confusing the quantity and the units used to measure it

A good example of confusing a quantity with its units is the equation:

M = 0.137 M

Most chemists will have no difficulty in interpreting this equation to mean that the concentration of a solution is 0.137 mole per liter. Though it may make chemical sense, this equation makes no algebraic sense at all. If we divide both sides by M we get the nonsensical result:

1 = 0.137

The mistake made here was to use the symbol M to represent the molarity (a quantity)on the left-hand side, while using the same symbol to represent the units of molarity, namely mole / liter, on the right hand side of the equation.

II: Confusing equality and proportionality In textbooks, we often encounter "equations" which make this mistake. An example is:
1 mol H₂SO₄ = 2 mol NaOH sometimes written:

Again, these equations seem to make chemical sense, but make no mathematical sense. One mole can simply not be set equal to two moles. The experimental law which these equations are trying to describe, is as follows: when this acid-base reaction occurs, for every mole of H₂SO₄ consumed two moles of NaOH are consumed. In other words, the amounts consumed are proportional to each other, not equal. Furthermore, in a specific sense of the word amount (to be explored later), this proportionality is the whole number ratio of one to two. As we shall presently see, this proportional relationship is easily described mathematically but not in the form of either of the above equations.

Taking it Further The above two examples suggest that the way we currently teach stoichiometry has serious mathematical deficiencies. It is surely no accident that textbooks use almost no algebra is dealing with stoichiometry, but have no hesitation in applying algebra to the Gas Laws. In this presentation this situation will be addressed head on. By making careful distinctions and by giving a name and a symbol to everything, it is quite possible to make algebra compatible with stoichiometry.

You should be warned that this is quite a long discussion. Also, it may rub some of your prejudices up the wrong way. It will ask you to think in terms of quantities rather than units, an also to abandon terms like "number of moles", and "molarity." But persevere! It will be worth it in the long run! You will have acquired an exact and concise language to describe the logic of stoichiometric problems which is much superior to the cumbersome ambiguities of "dimensional analysis"

The next page features a main menu which will give the reader some idea of what is to come. At the same time it will make it possible to navigate back and forth through the ensuing discussion.

Contact me at davieswi@esumail.emporia.edu

Main Menu

Example I: Changing units only

We will start this discussion by contrasting two calculations using "dimensional analysis". On paper, these two calculations look very similar in appearance. On close inspection, though, there are crucial differences between them. These need to be understood before we can see what is wrong with this method

The first example involves finding the density of aluminum in Anglo units from the value given in International units.

It is instructive to write out the result at each successive stage in this calculation. We find:

These partial results show us that at each stage in the calculation, the result is the same quantity, namely the density of aluminum. Only the units have been changed. Indeed, we can write the three partial results as equal to each other and also to the original 2.70 g cm^-3 without fear of ambiguity. Algebraically also, each partial result can be equated to the symbol d_Al:

Another feature of this calculation deserves attention. Each of the factors in the calculation is algebraically equivalent to one, i.e.,

This is quite in order since in all three cases the numerator and denominator not only have the same dimensions, (mass or length), they are also different descriptions of the same quantity. Thus, for example 12 in. and 1 ft. are different names for the same length and so can be considered algebraically as equal to each other.

A suitable term for describing this kind of factor is "unity factor," since its use is algebraically equivalent to multiplying by unity.

Example II: Changing Quantities as well as units

Our second example is a fairly typical stoichiometric calculation in which we calculate the volume of 0.200 M Na₂CO₃ needed to neutralize a 0.500 mL sample of pure formic acid obtained from a pipet of this volume.

The difference between this second example and the first is readily apparent if we again write out the partial answer after each stage in the calculation. We then find:

Here we are not just changing units. In the first stage, for instance, the logical purpose of the calculation is not to convert the units of 0.500 mL from milliliters to grams. Rather, it is to calculate the mass of the acid from its volume. Equivalent remarks apply to the three remaining steps in this calculation. In each step a new quantity is calculated: first 0.0133 mol, then 0.00663 mol, and finally 0.0331 liters. We cannot understand the logic of these calculations if we concentrate purely on the units. Much more important than the units, is the logical relationship between the various quantities. In particular, our understanding of the first stage in the calculation is incomplete until we realize that the mass and the volume of a sample are logically and algebraically connected through the density. If we neglect to point this out to our students, by pretending that we are merely changing units, we are depriving our students of a crucial insight. Similar remarks apply to the other three stages in the calculation, as we will presently see.

Another point to observe is the nature of the conversion factors in this second example. By contrast to Example I, none of the factors in Example II is equal to one. The first factor, 1.022 g / mL, for example cannot be equal to one since 1.022 g and 1 mL cannot be equal to each other. The numerator is a mass and the denominator is a volume. They have different dimensions! Far from being equal to one, the factor 1.022 g / mL is a measurable physical property, the density of formic acid.

In what follows we will use the term "proportionality factor" to describe the factors of the kind used in this example.

Numbers, Units, and Quantities

Up to this point we have used the term "quantity" without defining it exactly. The time has now come to rectify that omission. A quantity is any property of matter which can be expressed in the form "number times unit(s)." For example, we consider the density of aluminum to be a quantity because we describe it in terms of a number (2.70) multiplied by a set of units (g/cm³). Algebraically we would write this:

Any other property of matter that we can describe in this way, as a product of a number and a set of units, can be called a quantity. Examples of quantities are the mass, the volume, the temperature, and the pressure of a sample of matter. Note also that the term quantity can be applied to an intensive property (like density, pressure, or temperature) as well as an extensive property (like mass or volume). It is also conventional to use a unit-neutral term to describe a quantity. Thus we use the term "mass" to describe that quantity which the gram measures. We avoid terms like "number of grams" or "grammage." This is also true when we define one quantity in terms of other quantities. We do not define the density as the "number of grams per cubic centimeter" but in unit-neutral terms as the "mass per unit volume." Equally, we avoid reference to units in naming this quantity. We call it the "density" rather than some term like "gram-packedness." This unit-neutral convention emphasizes the fact that we are not restricted to certain units like grams or cubic centimeters and prohibited from using other units like kilograms, cubic millimeters or cubic meters in dealing with the density.

The Algebra of Quantities

The unit neutral convention for the names of quantities also extends to the corresponding algebraic symbols. When we write a formula like:

all three symbols should be interpreted in a unit-neutral sense. Thus m represents the mass no matter what its units. Similarly, V represents the volume in any units, and the same is true of d. It is therefore entirely correct to write as we did previously:

We have four different descriptions of the same quantity, using four different units. Because they are all different descriptions of the same quantity, we are allowed to equate them by this convention.

There is a second rule that we need to apply to an algebraic treatment of quantities. It is this: when we write an algebraic equation involving quantities, both sides of the equation must have the same dimensions. A good example of this rule is the equation:

to describe the distance fallen by a body after being dropped at time zero. In this equation, the acceleration, a, has the dimensions length x time^-2 while t² has the dimensions time². Thus the right hand side has the dimensions (length x time^-2) x (time²) = length. This is the same dimension as the distance s on the left hand side. By adopting this convention we ensure that all the equations we write are unit-neutral. The above equation can just as easily be used to handle Anglo units (e.g., feet and seconds) as it can be used to handle International units (e.g., meters and seconds.)

It cannot be emphasized too strongly that this convention can never be taken for granted. Textbooks contain numerous examples of equations which are unit-dependent. In other words, these equations only work for a specific set of units. For instance, we sometimes find the previous equation written in the form:

This equation also describes the distance fallen by a body after being dropped, but only if we measure the distance in feet. Moreover, if we include the units they do not cancel correctly. Consider the case when we substitute t = 2 seconds. We then have:

The truth is that the above equation is really about numbers and not about quantities at all. The symbol t represents the number of seconds while the symbol s represents the number of feet. In other words we should exclude the units and write:

Though this convention can occasionally be useful (but seldom at a freshman level) it has two disadvantages. In the first place we need to carefully spell out which units are intended. In the second place if we want to change the units, we also need to change the formula: In International units, for instance, we would have to rewrite the above formula in the form:

Here again t stands for a pure number, namely the number of seconds, while s represent the number of meters.

In unit-neutral form, we would write the formula for a falling body as:

Here the symbol, "a", represents the gravitational acceleration. Note how this unit-neutral formulation makes the equation planet-neutral as well. It could just as well be applied to bodies falling on the surface of the Moon or Mars as well as of Earth. For Earth-bound calculations, we would substitute the value a = 32 ft/s², or a = 9.80 m/s² and obtain:

In either case, the units behave correctly. Substituting t = 2 seconds yields the answer s = 64 ft. and s = 19.6 m. Of course, these two distances are equal to each other.

The Chemical Quantity

In chemistry the most important quantity is the one that the mole measures. In the older convention used in most textbooks, this quantity is called the "number of moles". Two ambiguities arise when this term is used. First, by using "number of moles" we suggest that we are referring to a number, when in fact a quantity is intended. Secondly, this older term is unit-specific. It suggests that this quantity can only be measured in moles. In practice millimoles and micromoles, even nanomoles, are often more appropriate.

The official term for this quantity, given by the International Committee on Weights and Measures, is the "amount of substance". In practice, this term is often abbreviated to 'amount'. There is a distinct prejudice among Anglophone chemists against the term "amount" since the word is often used, even in chemistry, in a more general way. The "amount of H₂O" for instance, might well refer to a volume of water. A way out of this difficulty has recently been suggested by Gorin (J.Chem.Educ. 1994, 71, 114-116) that the term "chemical amount" be used in any context in which the term "amount" could be considered ambiguous. In this presentation Gorin's suggestion will be used.

A unit-neutral term like "chemical amount" is much to be preferred to the older unit-specific term "number of moles". We can easily deal with the chemical amount algebraically by using the symbol n. Suppose we have a sample of NaCl with a mass of 0.500 g. The chemical amount of NaCl of this sample can be described in algebraic terms:

There is no need to favor any particular unit in this algebraic description. By contrast, if we use the symbol n_NaCl to indicate the "number of moles", the situation immediately becomes ambiguous. Does the same symbol indicate the "number of millimoles of NaCl"? Indeed, are the "number of moles" and the "number of millimoles" the same thing or not? Finally, the ambiguity mentioned above still remains. Is the term "number of moles" to be considered a number or a quantity? Asked to find the number of moles of NaCl in 0.500 g should we reply 0.00856 or 0.00856 mole? Use of the unit-neutral term "chemical amount" avoids all these difficulties.

There are two points about the chemical amount which it is important to appreciate. The first is that the chemical amount is not a number and the second is closely related to the first. It is that the chemical amount is a macroscopic property.

The chemical amount cannot be a number if we want it to behave like a regular quantity. It must have the form number unit(s) and it cannot have this form if it is itself a number. Likewise, the base unit of chemical amount, the mole, cannot be regarded as a number if it is to behave like a regular unit. It cannot be both a unit and a number at the same time.

This point is best made by comparison to another quantity, the electrical charge. We have no difficulty in thinking of the electrical charge as a macroscopic property. Nevertheless, each coulomb of negative charge contains 6.2415 10¹⁸ electrons. This fact does not allow us to write the 'equation':

1 coulomb = 6.2145 10¹⁸ electrons
or to say that we measure electrical charge in order to count electrons. Even worse would be to describe the coulomb as the "physicists' dozen".

The best way of seeing that the mole measures a macroscopic quantity is to look at chemical history. The mole and its antecedents (like gram-molecular weight) were developed for stoichiometric purposes. These terms made it easier to calculate "how much" of one substance would react with "how much" of another. Throughout the second half of the 19th. century, chemists did stoichiometric experiments on the macroscopic level to discover the formulas of compounds on the molecular level. They did all this without knowing even approximately what numbers of molecules were actually involved. If anything, they thought of a gram-molecular weight as a mass rather than as a number -- certainly as a macroscopic property. Of course, they realized that a gram-molecular weight of H₂O contained the same number of molecules as a gram-molecular weight of HCl.

Even today, most chemistry instructors try to present the mole in some sense at least as a macroscopic property. They haul out a fistful or two of a simple salt or the usual black cube to show how "big" a mole of solid or a mole of gas actually is. This is a good idea, but it would be better if they showed the students a few millimoles instead. In the freshman laboratory, students seldom encounter as much as a mole of any chemical except water. A millimole of gas at room temperature and pressure occupies about 25mL which is only a little less than the volume of a ping-pong ball.

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