Concepts

Quantity

In science and other fields of research, quantities are used to describe some "attribute of a phenomenon, body or substance" [McNaught/Wilkinson_1997](p. 330) in a quantitative way. A quantity (sometimes also physical quantity) is an entity which is defined by a (numerical) value and a unit, and related to other quantities by a dimension. For example, time is a particular quantity. A quantity can be interpreted in an abstract or a concrete way (for details, see the discussions in [deBoer_1995][Emerson_2008]). An abstract quantity does not refer to a particular object or measurement Fleischmann_1959/60 "[Fleischmann_1959/60]". Consequently, it describes a certain concept, e.g. the concept of a "time". On the other hand, a concrete quantity represents a real physical situation (for example, a particular time, measured in an experiment)i [deBoer_1995]. Furthermore, the kind of quantity (or quantity class Fleischmann_1959/60 "[Fleischmann_1959/60]") is an abstraction distinguished from the quantity itself [deBoer_1995].

Apart from the numerical value, the unit in which the value is expressed, and the dimension, a scientist will often additionally qualify a quantity by a name and/or a symbol, which denote its use in a given context. For example, in electroanalytical chemistry, when defining a cyclic voltammetric experiment, the time needed to scan the potential of the electrode from its starting value to the value when its direction is reversed is called the "switching time" (name), and, conventionally, the symbol $t_\lambda$ is attached to this time quantity.

The SI (Système International d'Unités) defines seven basic quantities: length, mass, time, electric current, thermodynamic temperature , amount of substance and luminous intensity. In the context of the Quantities library these are termed the SI base quantities (not to be confused with the "base quantities" used in the implementation of Quantities). Quantities that are not SI base quantities are SI derived quantities [Cohen/Cvitas/etal_2007](p. 4).

All quantities together ideally form a consistent set of terms and descriptions of physical, chemical, etc. systems and processes. This requires that only a single definition of a particular kind of quantity exists [Fleischmann_1951].

Unit

The quantity's unit is the standard of measurement for values of that quantity. For example, time is measured in seconds (s). Units are also identified by a name and a symbol. In contrast to a quantities' name and symbol, unit names and symbols are defined by the SI or by convention, and do not vary with the context. The units' symbols are mandatory [Cohen/Cvitas/etal_2007](p. 5).

The SI is based on seven SI base units: metre, kilogram, second, ampere, kelvin, mole, and candela, corresponding to the SI base quantities, respectively.

Although the SI provides only a single unit for a particular quantity, quite often several units are associated for practical use. For example, time is not only measured in seconds but also in minutes (min), hours (h) and many more non-SI units.

Some units, e.g. the second, may contain a prefix, producing multiples of 10 of the underlying unit. For example, the millisecond (ms) corresponds to 1/1000 of a second.

There is a list of 20 prefixes in the SI. For other units, such as e.g. minute, the use of a prefix is unusual. A unit can be composed of powers of more simple units, e.g. the metre per second (m s^{-1}) for quantity velocity or the square metre (m^2) for quantity area. Such composed units should be not be regarded as being generated by multiplication or division, but rather constitute units of their own right [Emerson_2008].

With respect to the SI we distinguish three types of units:

SI base units, e.g. the second (s) for quantity time (see above), the metre (m) for quantity length, or the Ampere (A) for the quantity electrical current
SI derived units with (sometimes) special names, e.g. the Coulomb (C; corresponding to ampere second, As) for the quantity electrical charge
units used together with the SI, e.g. the minute (min) for quantity time.

Dimension

A SI derived quantity is related to one or more of the base quantities by an algebraic expression devised by multiplying and/or dividing [Cohen/Cvitas/etal_2007](p. 4). or example, an electric charge quantity results from the multiplication of an electric current and a time quantity: $Q=i \cdot t$ . The dimension of a (SI base or derived) quantity is given by the dimensional exponents [deBoer_1995](p. 420) of the seven SI base quantities in the expression, ordered with respect to the list of SI base quantities given above.

We call the seven rational numbers the components of the quantity's dimension. Consequently, the dimension is an ordered septuple of the components.

For example, the SI derived quantity area (symbol $A$ ) is related to the SI base quantity length (symbol $l$ ) by $A = l^2$ . Thus, the length component of area's dimension is 2, while all other six components are equal to 0. The dimension of the area quantity will thus be $(2, 0, 0, 0, 0, 0, 0)$ .

It has been noted [Fleischmann_1951] that the exponents would only be integer numbers. However, in general, also rational exponents will be encountered. For example, a particular normalization of an electric current in cyclic voltammetry is given by

$i^{\rm norm} = i/\sqrt{v}c, v = {\rm d}E/{\rm d}t$

where $v$ is the potential scan rate used in the cyclic voltammetric experiment, $E$ being the electric potential (dimension: $(2, 1, -3, -1, 0, 0, 0)$ ), and $c$ is the concentration of the electroactive substrate (both $v$ and $c$ are derived quantities). The dimension if this normalized current is $(2, -1/2, 2, 3/2, 0, 1, 0)$ ).

The SI base quantities' dimensions consist of septuples with exactly one component equal to 1 and all other components equal to 0. If all seven components of the dimension are zero, we call the quantity dimensionless.

Quantity Calculus

Quantity calculus is defined as the (calculational) manipulation of (physical) quantities in the form of mathematical relations between quantities, [deBoer_1995][Copley_1960][Emerson_2008]. for example:

$Q = i\cdots t$

The quantities $Q$ , $i$ , and $t$ used in this quantity equation are essentially "independent of the choice of units". The unit merely defines the numerical value. This is in accordance with our notion to use the quantity as the basic concept in the present context. Again, this equation may be regarded as abstract (presenting a general definition of electric charge on the basis of electric current and time), or concrete, as an equation relating the particular quantities

$Q = 5\ {\rm A} \cdot 3\ {\rm s} = 15\ {\rm C}$

where both the values and the units are well defined. In analogy, essentially arbitrarily complex expressions can be formed.

In quantity calculus, the quantity represents the product of the value and the attached unit [Mills_1997]. For example, a time of 5 s may be written as

$t = 5\ {\rm s}$

On the other hand, the quantity's value is given by the quantity divided by the unit:

$t/{\rm s} = 5$

This notation is advantageously used e.g. in tables or graphs of scientific data [White_1998]. The importance of the correct use of dimensionless quantities whenn using transcendental functions in quantity calculus has been emphasized [Matta/Massa/etal_2010].

Dimensional analysis [Matta/Massa/etal_2010][BoostMPL][BartonNackman_1994](section 16.5) deals with the relation of dimensions of quantities in an expression based on quantity calculus. In particular, the dimensions on the left and right hand sides of an `=' sign in an abstract quantity equation (e.g., $Q = i\cdot t$ ) must be equal (dimensional homogeneity [Matta/Massa/etal_2010]).

Commensurability, Convertibility and Coherent Unit Systems

The value of a quantity can be converted to be measured in any of its associated units. In this case, the value is multiplied by a factor that is determined from both units between which the conversion takes place. For example, if a time quantity $t = 5\ {\rm s}$ is to be converted such that the value is represented in milliseconds, the value 5 has to be multiplied by a factor of 1000: $t = 5\ {\rm s} = 5000\ {\rm ms}$ . This is not only possible for units distinguished by a prefix, but also other units associated with the same quantity: $t = 5\ {\rm min} = 300\ {\rm s}$ . Note, that only the number "5" changes into "300" during conversion

the underlying quantity itself remains the same.

Such conversions are indispensable in quantity calculus as soon as two quantities are involved, e.g. when comparing quantities. Thus, conversion has to be considered if we compare two time quantities:

$t_1 = 5\ {\rm s},\ \ \ \ t_2 = 1\ {\rm min},\ \ \ \ \ \ t_2 > t_1$

We call two quantities commensurable if they share a common set of associated units. The word commensurablity means "having a common measure" [Webster_1993]. In contrast, in the context of dimensional analysis (see above) the term is used in the sense of "having the same dimension" [Brown_2003][Commensurability_Wikipedia]. However, this definition may lead to ambiguities, as is exemplified by the two quantities torque (moment of force [Cohen/Cvitas/etal_2007](p. 14); measured in unit newton metres (N m), or joule per radian, J/rad [BIPM_SI_Units]) and energy (measured in Joule) sharing the same dimension $(2, 1, -2, 0, 0, 0, 0)$ [Commensurability_Wikipedia]. Still, a torque and an energy can not be converted into each other. Another example is given by the two quantities frequency (measured coventionally in Hertz, symbol Hz, and corresponding to 1/s) and first order chemical rate constant (measured in 1/s), where conversion does not make sense, although the dimension is $(0, 0, -1, 0, 0, 0, 0)$ in both cases. These quantities are incommensurable according to our definition above, but not by that used in dimensional analysis.

On the other hand, in some cases, more complex conversions are possible, even if the units are different. For example, a thermodynamic temperature given in kelvin can be converted to a Celsius temperature as

$T = 298.15\ {\rm K} = 25\ ^{\rm o}{\rm C}$

by numerically subtracting a constant value of 273.15, although the two types of temperature are not commensurable in the strict sense defined above. Converting from unit millikelvin (mK) into degrees Celsius would require to divide by 1000 and then subtracting 273.15.

We use the term convertibility to designate the fact that two quantities can be converted. It is a necessary (but not sufficient [Emerson_2008]) requirement that the two quantities involved in the conversion have the same dimension. Moreover, however, the quantities must either be commensurable in our strict sense or a special conversion, as e.g. in the case of the temperatures, must exist. Since convertibility is central to quantity calculus, it becomes again apparent that relying on units or dimensions alone is not sufficient. Rather, the quantity concept including units and dimensions as properties is essential.

Apart from being dimensionally homogeneous, only commensurable quantities can be assigned or compared to each other in a calculation. For example, a product of two lengths has the dimension of an area and can be assigned to an area quantity. However, it can not be assigned to e.g., a time quantity. Similarly, a frequency can not be compared to a first-order chemical rate constant. For non-commensurable quantities also some other operations are illegal (see below).

If the numerical factors used for conversions between units is unity, we call the resulting set of units coherent [BIPM_SI_coherent]. For example, the SI units (base and derived) are coherent. The use of prefixes breaks coherence in this narrow sense [Cohen/Cvitas/etal_2007](p. 93).

Quantity Aggregates

A quantity aggregate is a collection of values, quantities, or a quantity aggregate itself, the elements. A collection of values (see, e.g., a variable vector) shares a quantity, whith the unit of measure for one or more numerical values. A collection of quantities can be homogeneous (when the elements are of the same type of quantity) or non-homogeneous (then, the elements may be of diffent type). A collection of quantity aggregates handles for example several variable vectors.

variable vector

variable tuple

variable vector tuple

Physical Quantities

A physical quantity is a quantity which can be used in the context of physical, chemical, or physico-chemical calculations. These calculations follow the rules of quantity calculus. Thus, in particular situations, quantities must be commensurable in order to make a calculation meaningful. Moreover, the units attached to the respective quantities must be taken into account.

Mathematical Quantities

General Quantities

References

[McNaught/Wilkinson_1997] A.D. McNaught and A. Wilkinson, Compendium of Chemical Technology, IUPAC Recommendations, 2nd edition, Blackwell Science, Oxford, 1997.

[deBoer_1995] J. de Boer, Metrologia 31, 405 - 429 (1994/95).

[Emerson_2008] W.H. Emerson, Metrologia 45, 134 - 138 (2008).

[Fleischmann_1959/60] R. Fleischmann, MNU 12, i386 - 399, 443 - 458 (1959/60). 377 - 400 (1951).

[Cohen/Cvitas/etal_2007] E.R. Cohen, T. Cvitas, J.G. Frey, B. Holmström, K. Kuchitsu, R. Marquardt, I. Mills, F. Pavese, M. Quack, J. Stohner, H.L. Strauss, M. Takami, and A.J. Thor, Quantities, Units and Symbols in Physical Chemistry, 3rd ed., Blackwell Science, Oxford, 2007.

[Fleischmann_1951] R. Fleischmann, Z. Physik 129, 377 - 400 (1951).

[Copley_1960] G.N. Copley, Nature 188, 254 (1960).

[Mills_1997] I.M. Mills, Metrologia 34, 101 - 109 (1997).

[White_1998] M.A. White, J. Chem. Educ. 75, 607 - 609 (1998).

[BoostMPL] http://www.boost.org/libs/mpl/doc/tutorial/dimensional-analysis.html

[Barton/Nackman_1994] J.J. Barton and L.R. Nackman, Scientific and engineering C++: an introduction with advanced techniques and examples, Addison-Wesley, Reading, 1994.

[Webster_1993] Webster's Third New International Dictionary, Merriam-Webster, Springfield, 1993.

[Brown_2003] W.E. Brown, A Case for Template Aliasing, Document WG21/N1451 = J16/03-0034; the document can be downloaded here.

"[BIPM_SI_Units]" http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html

"[BIPM_SI_coherent]" http://www.bipm.org/en/si/si_brochure/chapter1/1-4.html

"[Commensurability_Wikipedia]" http://en.wikipedia.org/wiki/Unit_commensurability

[CODATA] http://physics.nist.gov/constants

[Matta/Massa/etal_2010] C.F. Matta, L. Massa, A.V. Gubskaya and E. Knoll, J. Chem. Educ. 88, 67 - 70 (2010).

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