A. C. Norris Portsmouth Polytechnic Portsmouth, Hampshire, England
51 Units in Physico-Chemical Calculations
A
thorough understanding of the result of any chemical calculation depends on an appreciation of how unit,s enter the problem and influence the answer. The need for such an appreciation is particularly evident at the present time when SI units are being introduced throughout all branches of chemistry. This article demonstrates how the adoption of SI units affects some of the more important physico-chemical calculations which arc met at an undergraduate level. The readcr is referred elsewhere for articles on the basic principles of SI units and for t,hc convent,ions relating to the writing and printing of the symbols for units (1-3). Similarly, t,he specialized topic of SI units in electricity and magnetism has also received recent coverage (44).
Examples of S I Units in Physico-Chemical Calculations
Examples are given here of the changes produced by S I units together with some general points on the use of units in physical chemistry. Additional examples can be found in references (5) and (8). Units Associated with the Properties of Gases
Pressure By definition, the standard atmosphere is the pressure exerted by a column 0.760 m high of a fluid (mercury) of density exactly 13 595.1 kg m-3 in a place where the acceleration of free fall is 9.806 65 m s-%. Thus, 1 tltm = density X length
Relationship between Physical Quantities and Units
I n practice it is only possible t,o measure the value of a physical quant,ity relative to a second value of that quantity, for example, the ratio 1/1" of a length 1 and a length l o . Hence, the result of any such measurement is always a pure number. However, if thc length lo is sufficiently well defined and reproducible it mav be acce~tedas a unit of leneth and so in ecncral physical qnantity/unit of physical quantity = numerical value
or more concisely physical qnantity/unit
=
numerical value
from which physical quantity
=
numerical vdue X unit
This last equat,ionl makes it clear that the unit of a physical quantity should always be quot,ed. Thus it is correct to write, pressure P = 100 kN m-% or P/ (kN m-l) = 100 but incorrect to write P = 100. However, the value of a physical quantity does not, imply the choice of a particular unit since different combinations of numerical values and unit,s give t,he same value of the physical quantity, e.g., a volume, v = 1 X m30rv = lo3 X dm30rv = 10% cm3. When the relationship between two physical quantities is represent,ed graphically, only the numerical values of the quantities are plotted. For this reason it is desirable to label each axis with the quotient. physical quantity/unit. For example, typical axes for a plot of vapor pressure-temperature data would be labeled P/(BN m-l) and T/R, or log (P/atm) and lo3R/T. 'The result also shows that it is only possible to take the logarithm of s pure number. If the IognriLhm is taken of the physical qusnt,ity then log(physieal quantity) = log(numerieal lag(unit) and the last term is meaningless because a. value) logarithm must be dimensionless since d log a = ds/z (7).
+
X acceleration
=
kg m-3 X m
X m s-=
=
13 ,595.1 X 0.760
X 9.806 65
=
101 32.5 kg rn sea m-$
= 101
323 N m-2
sinre N
=
kg m s 9
The comparable calculation in CGS units gives 1 atm = 1.01325 X 106 dyn em-%. However, t,he dyn is not cohercnt with the SI and is now a rcdundant unit. While the atm is similarly non-coherent, t,he reference state, standard atmosphere pressure, is likely to be retained because of its convenience. Root Mean Square Velocity of a Molecule The SI usage of a single unit for energy, the joule, can produce changes in the treatment of equations which involve the gas constant, R. For example, the root mean square velocity, Z, of a gas molecule is defined by the equation F = (3RT/M)'/n
where T is the thermodynamic temperature and M is the molecular weight. When the SI value of R = 8.314 J I
