The Problem of “Chemistry” Units and its Solution in Mathcad

Valery Ochkov

Russian Version of the article


System SI is legitimated almost everywhere. Chemical engineers cannot express hardness of water in meq/L as this unit is outlawed. Demands of SI system forbid us to say, “concentration of calcium in the solution is 1 eq/L”. But it is also forbidden to say, “Concentration of calcium in the solution is 1 mole/L”. One must define what concentration, molar or demal, is meant; i.e. define the base unit in the solution, calcium ions or charges of calcium ions. Preterition can result in serious mistake: charges of calcium ions are twice times as large as calcium ions themselves. For example, if molar concentration of calcium in solution is 1 mole/L, demal (equivalent) – 2 mole/L. Old method (out of date but holding their positions) of uncovering is to use two groups of units, similar in their physical - chemical essence but different (as a rule) in value, mole (or, rather, g-mol, mg-mol, gram-molecule and others) and equivalents (g.eq., mg.eq., and so on).

System SI abolished units with equivalent ending and upholds moles and another way to point a base unit in concentration defining. Its demands that permit moles and exclude equivalents transfer concrete definition of the base unit from units to the name of physical (or, rather, chemical, physical – chemical) value.

The author wants to show that up-to-date computer mathematical programs supporting physical quantities and their units, Mathcad, Maple, and Derive, cause a new impulse to the problem of “moles – equivalents” and finally solve it.

As shown in Figure 1 variable Cm holding molar concentration is entered into a Worksheet. One can choose Unit from Insert menu to enter the variables with assigned built-in units of volume (L, liter) and quantity of substance (moles).

Figure 1 Insert Unit Window in Mathcad

User units can be also defined in Mathcad including russified (see Figure 3, 4, 5, and 6). Figure 2 shows Maple Help page with description of substance quantity unit. One can assume that here is a mistake (after Freud): under the title Units only one unit is listed. Probably, the equivalents were meant to be (g.eq. and others) but only mole is kept recalling SI demands.

Figure 2 Help page in Maple

But Mathcad is the most advanced in the field having the features that let us solve the inveterate “chemical – metrological” problem easily connected with “banishment” the equivalents from calculations. Further Mathcad worksheets with characteristic “concentration” problems show these features. Two variables, different (mmole/L, meq/L) but with similar name, are defined in the worksheet based on condition that the same unit must express both the molar and the equivalent concentration. The method, which is impossible from conventional programming viewpoint, is applied using style of variables (see Figure 3).

Figure 3 Variables style in Mathcad

After defining the variables one can solve the following water – chemical problem.

Example 1

It is given composition of natural water: concentration of basic ions expressed in mg/L. Find hardness (total, carbonate, noncarbonate) and alcalinity.

Figure 4 Calculation of hardness and alcalinity of natural water

To recur to the problem of concentration units, the number of ions (molarity) and ion charges (normality) must be expressed in one unit of substance quantity, mole, not in two different units (in spelling not in essence), mole and equivalent, as before. Mathcad worksheet has one feature to solve the problem easily; it is possible to have different variables but with similar name for concentration units (see Figure 3). There are two variables in our calculation: mmole/L for the univalent ions and mmole/L for bivalent ions. That let us type equivalent concentrations (expressed in meq/L earlier, see area 5), in mmole/L too using only molar concentrations (Figure 4 shows area 6 where variables with default units are typed). The kernel of the calculation is that a user changes default unit of the result, mole/m3, for mmole/L in the area 5, different for alcalinity (the valence of bicarbonate ion is 1) and hardness (the valence of calcium and magnesium is 2). The ‘2’ is present evidently in the reaction when water being heated scales with precipitation of calcium carbonate and magnesium hydroxide and simultaneous removing of carbon dioxide.

Ña2+ + 2 HCO3- → CaCO3↓ + CO2↑+ H2O              (1)

Mg2+ + 2 HCO3- → Mg(OH)2↓ + 2 CO2                    (2)

Comment. Anions of the weak acids (usually carbonic acid in natural water) give alcalinity that is not alkali yet but so to speak future alkali produced when the water is boiled.

Na+ + HCO3- → NaOH + CO2                          (3)

Alkalinity of natural water is an important protective factor, which define buffering of a solution; if an acid or an alkali is added into water, pH of a solution is nearly constant due to displacement of carbonate equilibrium. Very pure water (for example, in Baikal Lake or rainwater) without buffering is very sensitive to the contaminations (sulfur or/and nitric oxides) that result to sharp decreasing of pH value and serious environmental consequences.

Computer calculating systems supporting physical quantities let us reconsider strict requirements of SI concerning units. Moreover, Maple has not only built-in units allowed temporarily in calculations but even out-of-date and peculiar units settled down in some scientific and engineering fields and rejected hard by SI.

Specialists are believed to grasp themselves what units are better to use in calculations. Dictates are not appropriate; only recommendations are taken into consideration.

Figure 5 Converting of concentrations in Mathcad

Figure 5 shows converting of concentrations with “illegal”, but as with meq/L, widely used equivalent unit – normality. A reader can notice also that the result shown in Figure 4 was duplicated in different units (legal and out-of-date).

Three advices on using concentration units in Mathcad

  1. Variables should contain only molar concentration, which can be typed (or printed) by request in equivalents too. Nearly the same rule is applied for temperature: one can enter and type (or print) in any convenient scale but should use only absolute temperature scales (Kelvin or Rankine) in calculations. We can notice certain absoluteness in molarity and relativity in normality. The valence of many chemical elements changes (is relative) depending on the reactions, which we consider. For example, iron can be bivalent and trivalent. Of course, we can to have two variables one of which contains molarity and another normality of a solution. It is quite illogical: we store not radius and diameter of a circle but only one quantity keeping in mind that the second is twice as much (half as much) as the first one. Thus, Figure 4 shows that the variable Hk (carbonate hardness) in area 5 contains only half as much as alkalinity of water because variables Hk è Al store molar not equivalent values of concentration.
  2. One can have two or more variables of the same name in Mathcad to enter or type (print) concentrations in molar or normal format but not to use in formulae (see item 1). Such variables have different styles and vary considerably (the quantity of equivalent see area 1 in Figure 3 and 4).
  3. The worksheet can contain several dissimilar variables connected with different units of physical quantities – legal, allowed for temporarily use and even out-of-date (age-old). If the velocity can be expressed in hitches or versts per hour (of course, that is not SI units) why we cannot express concentration in milligram-equivalents per liter. Earlier there were well-founded suspicions to make mistakes with different units at hand computations or in programs without unit software tools (calculator, computer languages, electronic worksheets and others). Now computer automatically makes necessary converting that increases reliability of calculations sharply; at that the widest choice let us type every possible units and ask a reader delete useless one. Figure 6 shows converting of hardness in units used (in Germany are in use nowadays) in different countries (see the beginning of the article).

Figure 6 Working with age-old units of hardness

The author works at Department of Technology of Water and Fuel ( of Moscow Power Engineering Institute ( The department often receives requisitions from different places of Russia. The problem is: if the boiler-house use water with certain hardness will the boilers operate. If hardness is expressed in meq/L it is clear but some doubts are cast if it is in mmole/L (that is seldom). The right value can be twice as much as written. An alternative version can have serious consequences – a frozen settlement in winter.