Editing Calorimetry (section)

==Classical calorimetric calculation of heat==

===Cases with differentiable equation of state for a one-component body===

====Basic classical calculation with respect to volume====

Calorimetry requires that a reference material that changes temperature have known definite thermal constitutive properties. The classical rule, recognized by [[Rudolf Clausius|Clausius]] and [[William Thomson, 1st Baron Kelvin|Kelvin]], is that the pressure exerted by the calorimetric material is fully and rapidly determined solely by its temperature and volume; this rule is for changes that do not involve phase change, such as melting of ice. There are many materials that do not comply with this rule, and for them, the present formula of classical calorimetry does not provide an adequate account. Here the classical rule is assumed to hold for the calorimetric material being used, and the propositions are mathematically written:

The thermal response of the calorimetric material is fully described by its pressure <math>p\ </math> as the value of its constitutive function <math>p(V,T)\ </math> of just the volume <math>V\ </math> and the temperature <math>T\ </math>. All increments are here required to be very small. This calculation refers to a domain of volume and temperature of the body in which no phase change occurs, and there is only one phase present. An important assumption here is continuity of property relations. A different analysis is needed for phase change

When a small increment of heat is gained by a calorimetric body, with small increments, <math>\delta V\ </math> of its volume, and <math>\delta T\ </math> of its temperature, the increment of heat, <math>\delta Q\ </math>, gained by the body of calorimetric material, is given by

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

where

:<math>C^{(V)}_T(V,T)\ </math> denotes the latent heat with respect to volume, of the calorimetric material at constant controlled temperature <math>T</math>. The surroundings' pressure on the material is instrumentally adjusted to impose a chosen volume change, with initial volume <math>V\ </math>. To determine this latent heat, the volume change is effectively the independently instrumentally varied quantity. This latent heat is not one of the widely used ones, but is of theoretical or conceptual interest.
:<math>C^{(T)}_V(V,T)\ </math> denotes the heat capacity, of the calorimetric material at fixed constant volume <math>V\ </math>, while the pressure of the material is allowed to vary freely, with initial temperature <math>T\ </math>. The temperature is forced to change by exposure to a suitable heat bath. It is customary to write <math>C^{(T)}_V(V,T)\ </math> simply as <math>C_V(V,T)\ </math>, or even more briefly as <math>C_V\ </math>. This latent heat is one of the two widely used ones.<ref name="Bryan 1907 21–22">{{harvnb|Bryan|1907|pp=21–22}}</ref><ref>{{harvnb|Partington|1949|pp=155–7}}</ref><ref>{{cite book |last1=Prigogine |first1=I. |last2=Defay |first2=R. |title=Chemical Thermodynamics |publisher=Longmans, Green & Co. |location=London |date=1954 |oclc=8502081 |pages=22–23 |url=}}</ref><ref>{{harvnb|Crawford|1963|loc=§ 5.9, pp. 120–121}}</ref><ref name="Adkins 3.6">{{harvnb|Adkins|1975|loc=§ 3.6, pp. 43–46}}</ref><ref>{{harvnb|Truesdell|Bharatha|1977|pp=20–21}}</ref><ref>{{harvnb|Landsberg|1978|p=11}}</ref>

The latent heat with respect to volume is the heat required for unit increment in volume at constant temperature. It can be said to be 'measured along an isotherm', and the pressure the material exerts is allowed to vary freely, according to its constitutive law <math>p=p(V,T)\ </math>. For a given material, it can have a positive or negative sign or exceptionally it can be zero, and this can depend on the temperature, as it does for water about 4 C.<ref>{{harvnb|Maxwell|1871|pp=232–3}}</ref><ref>{{harvnb|Lewis|Randall|1961|pp=378–9}}</ref><ref>{{harvnb|Truesdell|Bharatha|1977|pp=9–10, 15–18, 36–37}}</ref><ref>{{cite book |first=C.A. |last=Truesdell |title=The Tragicomical History of Thermodynamics, 1822–1854 |publisher=Springer |date=1980 |isbn=0-387-90403-4 |pages= |url=}}</ref> The concept of latent heat with respect to volume was perhaps first recognized by [[Joseph Black]] in 1762.<ref>{{harvnb|Lewis|Randall|1961|p=29}}</ref> The term 'latent heat of expansion' is also used.<ref>{{harvnb|Callen|1985|p=73}}</ref> The latent heat with respect to volume can also be called the 'latent energy with respect to volume'. For all of these usages of 'latent heat', a more systematic terminology uses 'latent heat capacity'.

The heat capacity at constant volume is the heat required for unit increment in temperature at constant volume. It can be said to be 'measured along an isochor', and again, the pressure the material exerts is allowed to vary freely. It always has a positive sign. This means that for an increase in the temperature of a body without change of its volume, heat must be supplied to it. This is consistent with common experience.

Quantities like <math>\delta Q\ </math> are sometimes called 'curve differentials', because they are measured along curves in the <math>(V,T)\ </math> surface.

====Classical theory for constant-volume (isochoric) calorimetry====
Constant-volume calorimetry is calorimetry performed at a constant [[volume]]. This involves the use of a [[constant-volume calorimeter]]. Heat is still measured by the above-stated principle of calorimetry.

This means that in a suitably constructed calorimeter, called a bomb calorimeter, the increment of volume <math>\delta V\ </math> can be made to vanish, <math>\delta V=0\ </math>. For constant-volume calorimetry:

:<math>\delta Q = C_V \delta T\ </math>

where

:<math>\delta T\ </math> denotes the increment in [[temperature]] and
:<math>C_V\ </math> denotes the [[heat capacity]] at constant volume.

====Classical heat calculation with respect to pressure====

From the above rule of calculation of heat with respect to volume, there follows one with respect to pressure.<ref name="Bryan 1907 21–22"/><ref name="Adkins 3.6"/><ref>{{harvnb|Crawford|1963|loc=§ 5.10, pp. 121–122}}</ref><ref name="TB 1977 23">{{harvnb|Truesdell|Bharatha|1977|p=23}}</ref>

In a process of small increments, <math>\delta p\ </math> of its pressure, and <math>\delta T\ </math> of its temperature, the increment of heat, <math>\delta Q\ </math>, gained by the body of calorimetric material, is given by

:<math>\delta Q\ =C^{(p)}_T(p,T)\, \delta p\,+\,C^{(T)}_p(p,T)\,\delta T</math>

where

:<math>C^{(p)}_T(p,T)\ </math> denotes the latent heat with respect to pressure, of the calorimetric material at constant temperature, while the volume and pressure of the body are allowed to vary freely, at pressure <math>p\ </math> and temperature <math>T\ </math>;
:<math>C^{(T)}_p(p,T)\ </math> denotes the heat capacity, of the calorimetric material at constant pressure, while the temperature and volume of the body are allowed to vary freely, at pressure <math>p\ </math> and temperature <math>T\ </math>. It is customary to write <math>C^{(T)}_p(p,T)\ </math> simply as <math>C_p(p,T)\ </math>, or even more briefly as <math>C_p\ </math>.

The new quantities here are related to the previous ones:<ref name="Bryan 1907 21–22"/><ref name="Adkins 3.6"/><ref name="TB 1977 23"/><ref>{{harvnb|Crawford|1963|loc=§ 5.11, pp. 123–124}}</ref>

:<math>C^{(p)}_T(p,T)=\frac{C^{(V)}_T(V,T)}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}} </math>

:<math>C^{(T)}_p(p,T)=C^{(T)}_V(V,T)-C^{(V)}_T(V,T) \frac{\left.\cfrac{\partial p}{\partial T}\right|_{(V,T)}}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}} </math>
where
:<math>\left.\frac{\partial p}{\partial V}\right|_{(V,T)}</math> denotes the [[partial derivative]] of <math>p(V,T)\ </math> with respect to <math>V\ </math> evaluated for <math>(V,T)\ </math>

and

:<math>\left.\frac{\partial p}{\partial T}\right|_{(V,T)}</math> denotes the partial derivative of <math>p(V,T)\ </math> with respect to <math>T\ </math> evaluated for <math>(V,T)\ </math>.

The latent heats <math>C^{(V)}_T(V,T)\ </math> and <math>C^{(p)}_T(p,T)\ </math> are always of opposite sign.<ref>{{harvnb|Truesdell|Bharatha|1977|p=24}}</ref>

It is common to refer to the ratio of specific heats as

:<math>\gamma(V,T)=\frac{C^{(T)}_p(p,T)}{C^{(T)}_V(V,T)}</math> often just written as <math>\gamma=\frac{C_p}{C_V}</math>.<ref>{{harvnb|Truesdell|Bharatha|1977|pp=25}}</ref><ref>{{harvnb|Kondepudi|2008|pp=66–67}}</ref>

===Calorimetry through phase change, equation of state shows one jump discontinuity===

An early calorimeter was that used by [[Pierre-Simon Laplace|Laplace]] and [[Antoine Lavoisier|Lavoisier]], as shown in the figure above. It worked at constant temperature, and at atmospheric pressure. The latent heat involved was then not a latent heat with respect to volume or with respect to pressure, as in the above account for calorimetry without phase change. The latent heat involved in this calorimeter was with respect to phase change, naturally occurring at constant temperature. This kind of calorimeter worked by measurement of mass of water produced by the melting of ice, which is a [[phase transition|phase change]].

===Cumulation of heating===

For a time-dependent process of heating of the calorimetric material, defined by a continuous joint progression <math>P(t_1,t_2)\ </math> of <math>V(t)\ </math> and <math>T(t)\ </math>, starting at time <math>t_1\ </math> and ending at time <math>t_2\ </math>, there can be calculated an accumulated quantity of heat delivered, <math>\Delta Q(P(t_1,t_2))\, </math> . This calculation is done by [[line integral|mathematical integration along the progression]] with respect to time. This is because increments of heat are 'additive'; but this does not mean that heat is a conservative quantity. The idea that heat was a conservative quantity was invented by [[Antoine Lavoisier|Lavoisier]], and is called the '[[caloric theory]]'; by the middle of the nineteenth century it was recognized as mistaken. Written with the symbol <math>\Delta\ </math>, the quantity <math>\Delta Q(P(t_1,t_2))\, </math> is not at all restricted to be an increment with very small values; this is in contrast with <math>\delta Q\ </math>.

One can write

:<math>\Delta Q(P(t_1,t_2))\ </math>

::<math>=\int_{P(t_1,t_2)} \dot Q(t)dt</math>

::<math>=\int_{P(t_1,t_2)} C^{(V)}_T(V,T)\, \dot V(t)\, dt\,+\,\int_{P(t_1,t_2)}C^{(T)}_V(V,T)\,\dot T(t)\,dt </math>.

This expression uses quantities such as <math>\dot Q(t)\ </math> which are defined in the section below headed 'Mathematical aspects of the above rules'.

===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.

===Physical scope of the above rules of calorimetry===

The above rules refer only to suitable calorimetric materials. The terms 'rapidly' and 'very small' call for empirical physical checking of the domain of validity of the above rules.

The above rules for the calculation of heat belong to pure calorimetry. They make no reference to [[thermodynamics]], and were mostly understood before the advent of thermodynamics. They are the basis of the 'thermo' contribution to thermodynamics. The 'dynamics' contribution is based on the idea of [[thermodynamic work|work]], which is not used in the above rules of calculation.