Quantity calculus

This section gives examples of how we may manipulate physical quantities hy the rules of algebra. The method is called quantity calculus, although a better term might be quantity algebra.  

Quantity calculus is based on the concept that a physical quantity, unless it is dimensionless, has a value equal to the product of a numerical value (a pme number) and one or more units  

A simple example illustrates the use of quantity calculus. We may express the density of water at 25 °C to four significant digits in SI base units by the equation 

We may divide both sides of the last equation by 1 g cm to obtain a new equation 

Now the pure number 0.9970 appearing in this equation is the number of grams in one cubic centimeter of water, so we may call the ratio p/gcm the number of grams per cubic centimeter. By the same reasoning, p/kg m is the number of kilograms per cubic meter. In general, a physical quantity divided by particular units for the physical quantity is a pure number representing the number of those units. 

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A related concept to dimensional analysis is quantity calculus, a method we find particularly useful when it comes to setting out table header rows and graph axes. Quantity calculus is the handling of physical quantities and their units using the normal rules of algebra. A physical quantity is defined by a numerical value and a unit ... 

Most of the time, we try to avoid writing tables in this way, by incorporating the common factor of xlO4 into the header. We accomplish this by making use of the quantity calculus concept (see p. 13). Consider the second value of K. The table says that, at 36 °C, K = 15.1 x 104. If we divide both sides of this little equation by 104, we obtain, K 104 = 15.1. This equation is completely correct, but is more usually written as 10 4 T = 15.1. 

We could have achieved this conversion with quantity calculus knowing there are 1000 cm3 per dm3 (so 10-3 dm3 cm-3). In SI units, we write the volume as 34.2 cm3 x 10-3dm3cm-3. The units of cm3 and cm-3 cancel to yield V = 0.0342 dm3. 

Throughout this work SI units have been used, with the exception of pressure, which is given in Torr. For formulae, table headings etc. the Quantity Calculus has been used, as advised by M. L. McGlashan, Physicochemical Quantities and Units, Royal Society of Chemistry Sales and Promotions Department, Burlington House, London, WIV OBN. 

Section 7.1 gives examples illustrating the use of quantity calculus for converting the values of physical quantities between different units. The table in section 7.2 lists a variety of non-SI units used in chemistry, with the conversion factors to the corresponding SI units. Conversion factors for energy and energy-related units (wavenumber, frequency, temperature and molar energy), and for pressure units, are also presented in tables inside the back cover. 

Quantity calculus is a system of algebra in which symbols are consistently used to represent physical quantities rather, than their measures, i.e. numerical values in certain units. Thus we always take the values of physical quantities to be the product of a numerical value and a unit (see section 1.1), and we manipulate the symbols for physical quantities, numerical values, and units by the ordinary rules of algebra.1 This system is recommended for general use in science. Quantity calculus has particular advantages in facilitating the problems of converting between different units and different systems of units, as illustrated by the examples below. In all of these examples the numerical values are approximate. 

A more appropriate name for quantity calculus might be algebra of quantities , because it is the principles of algebra rather than calculus that are involved. 

However, in this form the symbols do not represent physical quantities, but the numerical values of physical quantities in certain units. Specifically, the last equation is true only if A is the molar conductivity in S mol"1 cm2, k is the conductivity in Scm"1, and c is the concentration in mol dm"3. This form does not follow the rules of quantity calculus, and should be avoided. The equation A = k/c, in which the symbols represent physical quantities, is true in any units. If it is desired to write the relationship between numerical values it should be written in the form... 

The equations used here are sometimes quoted in the form mB = 1000nB/ms, and xB mBMs/1000. However, this is not a correct use of quantity calculus because in this form the symbols denote the numerical values of the physical quantities in particular units specifically it is assumed that mB, ms and Ms denote numerical values in mol kg x, g, and g mol -1 respectively. A correct way of writing the second equation would, for example, be... 

The objective of this manual is to improve the international exchange of scientific information. The recommendations made to achieve this end come under three general headings. The first is the use of quantity calculus for handling physical quantities, and the general rules for the symbolism of quantities and units, described in chapter 1. The second is the use of internationally agreed symbols for the most frequently used quantities, described in chapter 2. The third is the use of SI units wherever possible for the expression of the values of physical quantities the SI units are described in chapter 3. 

The method described here for handling physical quantities and their units is known as quantity calculus. It is recommended for use throughout science and technology. The use of quantity calculus does not imply any particular choice of units indeed one of the advantages of quantity calculus is that it makes changes between units particularly easy to follow. Further examples of the use of quantity calculus are given in chapter 7, which is concerned with the problems of transforming from one set of units to another. 

Equations between quantities do not depend on the choice of units however, equations between numerical values do depend on the choice of units. Physical quantities, numerical values and units may be manipulated by algebraic rules ( quantity calculus ). The wavelength A of one of the yellow sodium lines, for example, can be written in various equivalent ways... 

Conversion between different units proportional to energy E can be achieved by using quantity calculus ... 

One should avoid using non-SI units from the following unit systems the esu (electrostatic unit system), the emu (electromagnetic unit system) and the Gaussian unit system. However, equations relating these still widely used unit systems to the SI are listed in Chapter 7 of [1] which also makes extensive use of quantity calculus to help converting between those systems of units. 

In handling physical quantities and their units we have used the method of quantity calculus [3] (also called dimensional analysis). The value of a physical quantity is expressed as the product of a numerical value and a unit... 

Quantity calculus, the manipulation of numerical values, physical quantities and units, obeys the ordinary rules of algebra.11 Combined units are separated by a space, e.g. J K 1 mol 1. The ratio of a physical quantity and its unit, Q/ Q, is a pure number. Functions of physical quantities must be expressed as functions of pure numbers, e.g. log(k/s 1) or sinfoj/). The scaled quantities Q/ Q are particularly useful for headings in tables and axis labels in graphs, where pure numbers appear in the table entries or on the axes of the graph. We will also make extensive use of scaled quantities Q/ Q in practical engineering equations. Such equations are very convenient for repeated use and it is immediately clear, which units must be used in applying them. For example, a practical form of the ideal gas equation is shown in Equation 1.2. 

The relation between the Celsius temperature, t, and the thermodynamic temperature, T, provides an illustration of the quantity calculus. Heretofore we have commonly written... 

We shall not use the method of quantity calculus which had a period of popularity in the experimental chemistry community writing, for example,... 

Thermodynamics is a quantitative subject. It allows us to derive relations between the values of numerous physical quantities. Some physical quantities, such as a mole fraction, are dimensionless the value of one of these quantities is a pure number. Most quantities, however, are not dimensionless and their values must include one or more units. This chapter reviews the SI system of units, which are the preferred units in science applications. The chapter then discusses some useful mathematical manipulations of physical quantities using quantity calculus, and certain general aspects of dimensional analysis. 

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