Editing Calorimetry (section)

===Mathematical aspects of the above rules===

The use of 'very small' quantities such as <math>\delta Q\ </math> is related to the physical requirement for the quantity <math>p(V,T)\ </math> to be 'rapidly determined' by <math>V\ </math> and <math>T\ </math>; such 'rapid determination' refers to a physical process. These 'very small' quantities are used in the [[Gottfried Leibniz|Leibniz]] approach to the [[Calculus|infinitesimal calculus]]. The [[Isaac Newton|Newton]] approach uses instead '[[Method of Fluxions|fluxions]]' such as <math>\dot V(t) = \left.\frac{dV}{dt}\right|_t</math>, which makes it more obvious that <math>p(V,T)\ </math> must be 'rapidly determined'.

In terms of fluxions, the above first rule of calculation can be written<ref>{{harvnb|Truesdell|Bharatha|1977|p=20}}</ref>

:<math>\dot Q(t)\ =C^{(V)}_T(V,T)\, \dot V(t)\,+\,C^{(T)}_V(V,T)\,\dot T(t)</math>

where

:<math>t\ </math> denotes the time

:<math>\dot Q(t)\ </math> denotes the time rate of heating of the calorimetric material at time <math>t\ </math>

:<math>\dot V(t)\ </math> denotes the time rate of change of volume of the calorimetric material at time <math>t\ </math>

:<math>\dot T(t)\ </math> denotes the time rate of change of temperature of the calorimetric material.

The increment <math>\delta Q\ </math> and the fluxion <math>\dot Q(t)\ </math> are obtained for a particular time <math>t\ </math> that determines the values of the quantities on the righthand sides of the above rules. But this is not a reason to expect that there should exist a [[Function (mathematics)|mathematical function]] <math>Q(V,T)\ </math>. For this reason, the increment <math>\delta Q\ </math> is said to be an 'imperfect differential' or an '[[inexact differential]]'.<ref name="Adkins 1975 16">{{harvnb|Adkins|1975|loc=§ 1.9.3, p. 16}}</ref><ref>{{harvnb|Landsberg|1978|pp=8–9}}</ref><ref>An account of this is given by {{harvnb|Landsberg|1978|loc=Ch. 4, pp 26–33}}</ref> Some books indicate this by writing <math>q\ </math> instead of <math>\delta Q\ </math>.<ref>{{cite book |last1=Fowler |first1=R. |last2=Guggenheim |first2=E.A. |title=Statistical Thermodynamics. A version of Statistical Mechanics for Students of Physics and Chemistry |publisher=Cambridge University Press |date=1965 |oclc=123179003 |page=57 }}</ref><ref>{{harvnb|Guggenheim|1967|loc=§ 1.10, pp. 9–11}}</ref> Also, the notation ''đQ'' is used in some books.<ref name="Adkins 1975 16"/><ref name="Lebon Jou Casas-Vázquez 2008">{{cite book |last1=Lebon |first1=G. |last2=Jou |first2=D. |last3=Casas-Vázquez |first3=J. |title=Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers |publisher=Springer |location= |date=2008 |isbn=978-3-540-74252-4 |pages=7 |doi=10.1007/978-3-540-74251-7|doi-broken-date=14 January 2025 }}</ref> Carelessness about this can lead to error.<ref name="Planck 1923/1926 57">Planck, M. (1923/1926), page 57.</ref>

The quantity <math>\Delta Q(P(t_1,t_2))\ </math> is properly said to be a [[Functional (mathematics)|functional]] of the continuous joint progression <math>P(t_1,t_2)\ </math> of <math>V(t)\ </math> and <math>T(t)\ </math>, but, in the mathematical definition of a [[Function (mathematics)|function]], <math>\Delta Q(P(t_1,t_2))\ </math> is not a function of <math>(V,T)\ </math>. Although the fluxion <math>\dot Q(t)\ </math> is defined here as a function of time <math>t\ </math>, the symbols <math>Q\ </math> and <math>Q(V,T)\ </math> respectively standing alone are not defined here.