Two rogue units of the SI: the kelvin and the candela W H. Emerson Quantities and the expressions of their cannot be assigned a unit, for it has in magnitudes and their units, if any, are its derivation two extensive quantities; effectively defined by three it is a mass per unit of volume. documents: the Vocabulaire Thermodynamic temperature is called Interntaionale de Métrologie 3rd Edition a SI base quantity; it is not ‘per’ any (VIM3) the SI Brochure 8th Edition other quantity. In classical, equilibrium (SI8) and the International System of thermodynamics it is defined by Quantities (ISQ). Until the publication reference to an ideal heat-engine such of VIM3 every kind of quantity was as that operating on a Carnot cycle. A required to have an associated unit, quantity of heat Q1 is input isothermally now defined as ‘a real scalar quantity, to the engine at temperature T1 and a defined and adopted by convention, quantity Q2 is withdrawn isothermally with which any quantity of the same at temperature T2 The amount of work kind can be compared to express the done by the cycle is Q1 - Q2, and ratio of the two quantities as a number T1/T2= Q1/ Q2 . The work done is a (VIM3)’. The current ISQ names five maximum when Q2 = 0 and when kinds of quantity as ‘base quantities’ therefore T2 also equals zero: its which are considered not to require lowest possible value. definition because of their familiarity, and all other quantities requiring units If a Carnot cycle operating between are defined as algebraic functions of temperatures T1 and T2 receives heat base quantities as determined by Q1 from another Carnot cycle observed physical laws. This long operating between temperatures T3 preamble is needed because I intend and T1, the upper temperature T3 to show that the SI’s ‘unit of cannot be said to be a thermodynamic thermodynamic temperature’, called temperature. Its lowest possible kelvin, is not a unit of a base quantity, temperature is not zero; it is T1. It is and therefore not a base unit, with not possible to increase a important consequences for physics thermodynamic temperature by and engineering; and indeed what the addition SI calls thermodynamic temperature is T1 + δT is not a thermodynamic not a base kind of quantity. temperature. An increment of Thermodynamic temperature is an temperature δT cannot be adopted as intensive quantity and, like all such a unit of thermodynamic temperature; quantities, it has an absolute zero, with every thermodynamic temperature has no negative values; and it is not its foot on absolute zero as its datum additive. For example a density point. Its foot cannot be detached cannot be added to another density to from zero. If it is, the temperature produce a new density that is their ceases to be thermodynamic. sum. That is not to say that a density Thermodynamic temperatures cannot be piled one on top of another and be called units of thermodynamic Kelvin Scale or the Celsius Scale. If it temperature. A temperature of 2 K is were possible to realise temperatures not 1 K + 1 K; it is 2 on the Kelvin on the Kelvin Scale the symbol K scale of temperature. The SI is wrong might be used, not to identify a unit but in assigning a unit (kelvin) to the Kelvin the scale itself. scale, which has no unit. That unit is of temperature difference. None of this has much relevance to the measurement of temperature. The The SI assigns the same unit to two current definition of the Kelvin scale different scalar quantities: heat has one realisable fixed point: the capacity and entropy. By definition of triple point of water of a specified the term unit the units of the quantities isotopic composition, and one that is cannot be identical or the two not realisable: zero. All other quantities cannot differ. Heat capacity temperatures remain undefined. Or is defined as rather they are defined theoretically by C = dQ/dT the properties of non-existent ideal gases to interpolate between the fixed at constant pressure, where dT is an points, and practically by reference to increment of temperature: a difference the International Temperature Scale of temperature. Entropy is defined by ITS-90. The latter has eighteen fixed the equation: points and complicated mathematical equations to interpolate between them. dS = dQ/T ITS–90 is not a thermodynamic scale where T is the thermodynamic at all; it is an equipment calibration temperature at which dQ is added. T standard. However, all measurements is not a temperature difference and of temperature are made with has no unit; it is very different from the reference to that standard and are quantity called temperature difference. generally expressed in ‘degrees kelvin’ with no reference to the SI definitions. A value of a temperature is stated as a In the proposed redefinition of the number and a reference: ITS-90, in the kelvin for the New SI no distinction is manner of hardness or octane number. made between thermodynamic The measurement and evaluation of temperature and interval of cryogenic temperatures extremely temperature. Temperatures on the close to absolute zero is a specialised Kelvin Scale and the Celsius Scale are science outside the scope of the SI. both called ‘thermodynamic’. If that were acceptable the adjective would In the proposed New SI the triple point serve no purpose and might as well be of water is abandoned as the fixed dropped. What the New SI defines as point on the Kelvin Scale in favour of a the unit of thermodynamic temperature fixed (exact) value of the Boltzmann is in fact a unit of temperature constant, expressed as 1.380 65 × 10- difference: the difference between 23 joule per kelvin (JK-1). Here again temperatures separated by one on the we find the SI using kelvin as though it 2 were a unit of absolute temperature, wavelength: a derived quantity. The and the same objections must apply. candela is a non-coherent unit of power divided by solid angle. Despite Thermodynamic temperature should their widespread use by lighting cease to be a base quantity of the SI engineers the lumen and all the units and the kelvin should not be derived from the concept, including the recognised as a unit . candela, should not be described as base quantities and units of the SI. A parallel might be drawn between the The concept of luminous intensity is of concepts of luminous intensity and the intensity of light perceived by a terrestrial weight. The weight of a human eye, as opposed to that particular mass, say a standard actually received by the eye. It is the kilogram, is the force required to intensity of light visible to that eye. counter the force of gravity on it. That That is not, however, a concept that is force varies according to where the consistent with the SI unit, the mass is, because the acceleration due candela. Every human eye is different, to gravity varies from place to place. so the SI abandons the physiological The weight can be made a property of definition and defines the unit, the mass by redefining the term candela, as a fixed fraction of the ‘weight’, by adopting a conventional, radiant intensity received from a fixed, standard value of the monochromatic light source of a fixed ‘acceleration due to gravity’ unrelated frequency and power. That leaves to place, so that the mass has the unanswered the question of the same ‘weight’ everywhere. It would be luminous intensity of light received called its ‘standard weight’. In the from a source of a different frequency; same way luminous intensity, a it has to be obtained by applying an property that is personal to the owner empirically determined ‘standard of a human eye and that differs from luminosity function’ (quoted in neither person to person, is made instead a ISO 31 nor the SI Brochure) that property of the light source, averages widely differing results independent of that eye by replacing it obtained from real human beings by by an invariable, highly empiracle, variously differing means. The SI unit mathematical function, unrelated to the is of a quantity that differs from all human eye that perceives the light. It other base quantities in that it includes might at least be renamed ‘standard arbitrarily in its definition an luminous intensity’. I find it hard to impersonal, empirically derived imagine ‘standard weight’ ever being function that purports to represent a adopted as a base quantity of the ISQ, subjective quantity. with its own SI unit. Nor should The standard luminosity function is a standard luminosity and the candela. ratio, so luminous intensity at a particular wavelength is defined as a Acoustic engineers also have a set of fixed fraction of radiant intensity at that units that almost parallel those of 3 lighting engineers, including units associated with perceived loudness; but loudness, another physiological quantity, is not a base quantity of the ISQ. References VIM3 2008 International Vocabulary of Metrology—Basic and General Concepts and General Terms 3rd edition JCGM 200:2008 (ISQ) 2009 ISO 80000-1:2009 Quantities and Units Geneva; International Standards Organisation (SI8) 2006 The International System of Units (SI) Sêvres, BIPM New SI Resolution 1 of the CGPM (2011): On the possible future revision of the International System of Units, the SI. BIPM Website. April 7, 2014 4 .