Editing Calorimetry (section)

====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>