Editing Calorimetry (section)

===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>