Editing Calorimetry

{{Short description|Determining heat transfer in a system by measuring its other properties}}
{{Distinguish|Colorimetry|Colometry}}
[[File:Ice-calorimeter.jpg|upright=1.2|thumb|The world's first '''ice-calorimeter''', used in the winter of 1782–83, by [[Antoine Lavoisier]] and [[Pierre-Simon Laplace]], to determine the [[heat]] involved in various [[chemical change]]s; calculations which were based on [[Joseph Black]]’s prior discovery of [[latent heat]].  These experiments mark the foundation of [[thermochemistry]].]]
[[File:Snellen human calorimeter, uOttawa.jpg|thumb|upright|Snellen direct calorimetry chamber, University of Ottawa<ref>{{cite journal |vauthors=Reardon FD, Leppik KE, Wegmann R, Webb P, Ducharme MB, Kenny GP |title=The Snellen human calorimeter revisited, re-engineered and upgraded: design and performance characteristics |journal=Med Biol Eng Comput |volume=44 |issue=8 |pages=721–8 |date=August 2006 |pmid=16937214 |doi=10.1007/s11517-006-0086-5 }}</ref>]]
[[File:Indirect calorimetry laboratory with canopy hood.jpg|thumb|Indirect calorimetry metabolic cart measuring oxygen uptake and {{CO2}} production of a spontaneously breathing subject (dilution method with canopy hood).]]

In [[chemistry]] and [[thermodynamics]], '''calorimetry''' ({{ety|la|calor|heat|el|''μέτρον'' (metron)|measure}}) is the science or act of measuring changes in ''[[state variables]]'' of a body for the purpose of deriving the [[heat transfer]] associated with changes of its [[Thermodynamic state|state]] due, for example, to [[chemical reaction]]s, [[physical change]]s, or [[phase transition]]s under specified constraints. Calorimetry is performed with a [[calorimeter]]. Scottish physician and scientist [[Joseph Black]], who was the first to recognize the distinction between [[heat]] and [[temperature]], is said to be the founder of the science of calorimetry.<ref name="Laider" >{{cite book |last=Laidler |first=K.J. |author-link=Keith J. Laidler |title=The World of Physical Chemistry|url=https://archive.org/details/worldofphysicalc0000laid |url-access=registration |publisher=Oxford University Press|year=1993|isbn=0-19-855919-4 |oclc=27034547}}</ref>

[[Indirect calorimetry]] calculates heat that living organisms produce by measuring either their production of [[carbon dioxide]] and nitrogen waste (frequently [[ammonia]] in aquatic organisms, or [[urea]] in terrestrial ones), or from their consumption of [[oxygen]]. [[Antoine Lavoisier|Lavoisier]] noted in 1780 that heat production can be predicted from oxygen consumption this way, using [[multiple regression]]. The [[dynamic energy budget]] theory explains why this procedure is correct. Heat generated by living organisms may also be measured by ''direct calorimetry'', in which the entire organism is placed inside the calorimeter for the measurement.

A widely used modern instrument is the [[differential scanning calorimeter]], a device which allows thermal data to be obtained on small amounts of material. It involves heating the sample at a controlled rate and recording the heat flow either into or from the specimen.

==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.

==Experimentally conveniently measured coefficients==

Empirically, it is convenient to measure properties of calorimetric materials under experimentally controlled conditions.

===Pressure increase at constant volume===

For measurements at experimentally controlled volume, one can use the assumption, stated above, that the pressure of the body of calorimetric material is can be expressed as a function of its volume and temperature.

For measurement at constant experimentally controlled volume, the isochoric coefficient of pressure rise with temperature, is defined by <ref name="IG 46">{{harvnb|Iribarne|Godson|1981|p=46}}</ref>

:<math>\alpha _V(V,T)\ = \frac{1}{p(V,T)}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}}  </math>

===Expansion at constant pressure===

For measurements at experimentally controlled pressure, it is assumed that the volume <math>V\ </math> of the body of calorimetric material can be expressed as a function <math>V(T,p)\ </math> of its temperature <math>T\ </math> and pressure <math>p\ </math>. This assumption is related to, but is not the same as, the above used assumption that the pressure of the body of calorimetric material is known as a function of its volume and temperature; anomalous behaviour of materials can affect this relation.

The quantity that is conveniently measured at constant experimentally controlled pressure, the isobar volume expansion coefficient, is defined by <ref name="IG 46"/><ref name="LR 54">{{harvnb|Lewis|Randall|1961|p=54}}</ref><ref name="Guggenheim 38">{{harvnb|Guggenheim|1967|p=38}}</ref><ref name="Callen 84">{{harvnb|Callen|1985|p=84}}</ref><ref name="Adkins 38">{{harvnb|Adkins|1975|p=38}}</ref><ref name="Bailyn 49">{{harvnb|Bailyn|1994|p=49}}</ref><ref name="Kondepudi 180">{{harvnb|Kondepudi|2008|p=180}}</ref>

:<math>\beta _p(T,p)\ = \frac{1}{V(T,p)}{\left.\cfrac{\partial V}{\partial T}\right|_{(T,p)}}   </math>

===Compressibility at constant temperature===

For measurements at experimentally controlled temperature, it is again assumed that the volume <math>V\ </math> of the body of calorimetric material can be expressed as a function <math>V(T,p)\ </math> of its temperature <math>T\ </math> and pressure <math>p\ </math>, with the same provisos as mentioned just above.

The quantity that is conveniently measured at constant experimentally controlled temperature, the isothermal compressibility, is defined by <ref name="LR 54"/><ref name="Guggenheim 38"/><ref name="Callen 84"/><ref name="Adkins 38"/><ref name="Bailyn 49"/><ref name="Kondepudi 180"/>

:<math>\kappa _T(T,p)\ = -\frac{1}{V(T,p)}{\left.\cfrac{\partial V}{\partial p}\right|_{(T,p)}} </math>

==Relation between classical calorimetric quantities==

Assuming that the rule <math>p=p(V,T)\ </math> is known, one can derive the function of <math>\frac{\partial p}{\partial T}\ </math> that is used above in the classical heat calculation with respect to pressure. This function can be found experimentally from the coefficients <math>\beta _p(T,p)\ </math> and <math>\kappa _T(T,p)\ </math> through the mathematically deducible relation

:<math>\frac{\partial p}{\partial T}=\frac{\beta _p(T,p)}{\kappa _T(T,p)}</math>.<ref name="Kondepudi 181">{{harvnb|Kondepudi|2008|p=181}}</ref>

==Connection between calorimetry and thermodynamics==

[[Thermodynamics]] developed gradually over the first half of the nineteenth century, building on the above theory of calorimetry which had been worked out before it, and on other discoveries. According to Gislason and Craig (2005): "Most thermodynamic data come from calorimetry..."<ref>{{cite journal |last1=Gislason |first1=E.A. |last2=Craig |first2=N.C. |title=Cementing the foundations of thermodynamics:comparison of system-based and surroundings-based definitions of work and heat |journal=J. Chem. Thermodynamics |volume=37 |issue=9 |pages=954–966 |date=2005 |doi=10.1016/j.jct.2004.12.012 |bibcode=2005JChTh..37..954G }}</ref> According to Kondepudi (2008): "Calorimetry is widely used in present day laboratories."<ref>{{harvnb|Kondepudi|2008|p=63}}</ref>

In terms of thermodynamics, the [[internal energy]] <math>U\ </math> of the calorimetric material can be considered as the value of a function <math>U(V,T)\ </math> of <math>(V,T)\ </math>, with partial derivatives <math>\frac{\partial U}{\partial V}\ </math> and <math>\frac{\partial U}{\partial T}\ </math>.

Then it can be shown that one can write a thermodynamic version of the above calorimetric rules:

:<math>\delta Q\ =\left [p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}\right ]\, \delta V\,+\,\left.\frac{\partial U}{\partial T}\right|_{(V,T)}\,\delta T</math>

with

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

and

:<math>C^{(T)}_V(V,T)=\left.\frac{\partial U}{\partial T}\right|_{(V,T)}\ </math> .<ref name="Planck 1923/1926 57"/><ref>{{cite book |last=Preston |first=T. |editor-first=J.R. |editor-last=Cotter |title=The Theory of Heat |publisher=Macmillan |location=London |date=1904 |oclc=681492070 |pages=700–701 |edition=2nd}}</ref><ref>{{harvnb|Adkins|1975|p=45}}</ref><ref>{{harvnb|Truesdell|Bharatha|1977|pp=134}}</ref><ref>{{harvnb|Kondepudi|2008|p=64}}</ref>

Again, further in terms of thermodynamics, the [[internal energy]] <math>U\ </math> of the calorimetric material can sometimes, depending on the calorimetric material, be considered as the value of a function <math>U(p,T)\ </math> of <math>(p,T)\ </math>, with partial derivatives <math>\frac{\partial U}{\partial p}\ </math> and <math>\frac{\partial U}{\partial T}\ </math>, and with <math>V\ </math> being expressible as the value of a function <math>V(p,T)\ </math> of <math>(p,T)\ </math>, with partial derivatives <math>\frac{\partial V}{\partial p}\ </math> and <math>\frac{\partial V}{\partial T}\ </math> .

Then, according to Adkins (1975),<ref name="Adkins 46">{{harvnb|Adkins|1975|p=46}}</ref> it can be shown that one can write a further thermodynamic version of the above calorimetric rules:

:<math>\delta Q\ =\left [\left. \frac{\partial U}{\partial p}\right |_{(p,T)}\,+\,p \left.\frac{\partial V}{\partial p}\right |_{(p,T)}\right ]\delta p\,+\,\left [ \left.\frac{\partial U}{\partial T}\right|_{(p,T)}\,+\,p \left.\frac{\partial V}{\partial T}\right |_{(p,T)}\right ]\delta T</math>

with

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

and

:<math>C^{(T)}_p(p,T)=\left.\frac{\partial U}{\partial T}\right|_{(p,T)}\,+\,p\left.\frac{\partial V}{\partial T}\right|_{(p,T)}\ </math> .<ref name="Adkins 46"/>

Beyond the calorimetric fact noted above that the latent heats <math>C^{(V)}_T(V,T)\ </math> and <math>C^{(p)}_T(p,T)\ </math> are always of opposite sign, it may be shown, using the thermodynamic concept of work, that also

:<math>C^{(V)}_T(V,T)\,\left.\frac{\partial p}{\partial T}\right|_{(V,T)} \geq 0\,.</math><ref>{{harvnb|Truesdell|Bharatha|1977|p=59}}</ref>

==Special interest of thermodynamics in calorimetry: the isothermal segments of a Carnot cycle==

Calorimetry has a special benefit for thermodynamics. It tells about the heat absorbed or emitted in the isothermal segment of a [[Carnot cycle]].

A Carnot cycle is a special kind of cyclic process affecting a body composed of material suitable for use in a heat engine. Such a material is of the kind considered in calorimetry, as noted above, that exerts a pressure that is very rapidly determined just by temperature and volume. Such a body is said to change reversibly. A Carnot cycle consists of four successive stages or segments:

(1) a change in volume from a volume <math>V_a\ </math> to a volume <math>V_b\ </math> at constant temperature <math>T^+\ </math> so as to incur a flow of heat into the body (known as an isothermal change)

(2) a change in volume from <math>V_b\ </math> to a volume <math>V_c\ </math> at a variable temperature just such as to incur no flow of heat (known as an adiabatic change)

(3) another isothermal change in volume from <math>V_c\ </math> to a volume <math>V_d\ </math> at constant temperature <math>T^-\ </math> such as to incur a flow or heat out of the body and just such as to precisely prepare for the following change

(4) another adiabatic change of volume from <math>V_d\ </math> back to <math>V_a\ </math> just such as to return the body to its starting temperature <math>T^+\ </math>.

In isothermal segment (1), the heat that flows into the body is given by

:&nbsp;&nbsp;&nbsp;<math>\Delta Q(V_a,V_b;T^+)\,=\,\,\,\,\,\,\,\,\int_{V_a}^{V_b} C^{(V)}_T(V,T^+)\, dV\ </math>

and in isothermal segment (3) the heat that flows out of the body is given by

:<math>-\Delta Q(V_c,V_d;T^-)\,=\,-\int_{V_c}^{V_d} C^{(V)}_T(V,T^-)\, dV\ </math>.<ref>{{harvnb|Truesdell|Bharatha|1977|pp=52–53}}</ref>

Because the segments (2) and (4) are adiabats, no heat flows into or out of the body during them, and consequently the net heat supplied to the body during the cycle is given by

:<math>\Delta Q(V_a,V_b;T^+;V_c,V_d;T^-)\,=\,\Delta Q(V_a,V_b;T^+)\,+\,\Delta Q(V_c,V_d;T^-)\,=\,\int_{V_a}^{V_b} C^{(V)}_T(V,T^+)\, dV\,+\,\int_{V_c}^{V_d} C^{(V)}_T(V,T^-)\, dV\ </math>.

This quantity is used by thermodynamics and is related in a special way to the net [[Work (thermodynamics)|work]] done by the body during the Carnot cycle. The net change of the body's internal energy during the Carnot cycle, <math>\Delta U(V_a,V_b;T^+;V_c,V_d;T^-)\ </math>, is equal to zero, because the material of the working body has the special properties noted above.

==Special interest of calorimetry in thermodynamics: relations between classical calorimetric quantities==

===Relation of latent heat with respect to volume, and the equation of state===

The quantity <math>C^{(V)}_T(V,T)\ </math>, the latent heat with respect to volume, belongs to classical calorimetry. It accounts for the occurrence of energy transfer by work in a process in which heat is also transferred; the quantity, however, was considered before the relation between heat and work transfers was clarified by the invention of thermodynamics. In the light of thermodynamics, the classical calorimetric quantity is revealed as being tightly linked to the calorimetric material's equation of state <math>p=p(V,T)\ </math>. Provided that the temperature <math>T\, </math> is measured in the thermodynamic absolute scale, the relation is expressed in the formula

:<math>C^{(V)}_T(V,T)=T \left.\frac{\partial p}{\partial T}\right|_{(V,T)}\ </math>.<ref>{{harvnb|Truesdell|Bharatha|1977|p=150}}</ref>

===Difference of specific heats===

Advanced thermodynamics provides the relation

:<math>C_p(p,T)-C_V(V,T)=\left [p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}\right ]\, \left.\frac{\partial V}{\partial T}\right|_{(p,T)}</math>.

From this, further mathematical and thermodynamic reasoning leads to another relation between classical calorimetric quantities. The difference of specific heats is given by

:<math>C_p(p,T)-C_V(V,T)=\frac{TV\,\beta _p^2(T,p)}{\kappa _T(T,p)}</math>.<ref name="LR 54"/><ref name="Kondepudi 181"/><ref>{{harvnb|Callen|1985|p=86}}</ref>

==Practical constant-volume calorimetry (bomb calorimetry) for thermodynamic studies==
Constant-volume calorimetry is calorimetry performed at a constant [[volume]]. This involves the use of a [[constant-volume calorimeter]].

No work is performed in constant-volume calorimetry, so the heat measured equals the change in internal energy of the system. The heat capacity at constant volume is assumed to be independent of temperature.

Heat is measured by the principle of calorimetry.

:<math>q = C_V \Delta T = \Delta U \,,</math>

where

:Δ''U'' is change in [[internal energy]],
:Δ''T'' is change in [[temperature]] and
:''C<sub>V</sub>'' is the [[heat capacity]] at constant volume.
In ''constant-volume calorimetry'' the [[pressure]] is not held constant. If there is a pressure difference between initial and final states, the heat measured needs adjustment to provide the ''[[enthalpy]] change''. One then has
:<math>\Delta H = \Delta U + \Delta (PV) = \Delta U + V \Delta P \,,</math>

where

:Δ''H'' is change in [[enthalpy]] and
:''V'' is the unchanging volume of the sample chamber.

==See also==

* [[Isothermal microcalorimetry (IMC)]]
* [[Isothermal titration calorimetry]]
* [[Sorption calorimetry]]
* [[Reaction calorimeter]]

==References==
{{Reflist}}

==Books==
{{refbegin}}
*{{cite book |first=C.J. |last=Adkins |title=Equilibrium Thermodynamics |publisher=McGraw-Hill |edition=2nd |date=1975 |isbn=0-07-084057-1 |oclc=1174680 }}
*{{cite book |first=M. |last=Bailyn |title=A Survey of Thermodynamics |publisher=American Institute of Physics |date=1994 |isbn=0-88318-797-3 |oclc=28587332 |url=http://catalog.hathitrust.org/api/volumes/oclc/28587332.html }}
*{{cite book |first=G.H. |last=Bryan |title=Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications |publisher=B.G. Tuebner |location=Leipzig |date=1907 |oclc=1843575 |pages= |url=https://archive.org/details/thermodynamicsin00bryarich}}		
*{{cite book |first=H.B. |last=Callen |title=Thermodynamics and an Introduction to Thermostatistics |publisher=Wiley |edition=2nd |date=1985 |isbn=0-471-86256-8 |oclc=11916089 |url=https://archive.org/details/thermodynamicsan0000call1960/}}
*{{cite book |first=F.H. |last=Crawford |title=Heat, Thermodynamics, and Statistical Physics |publisher=Harcourt, Brace, & World |location=London |date=1963 |oclc=1170987551 |pages= |url=http://catalog.hathitrust.org/api/volumes/oclc/535026.html}}	
*{{cite book |first=E.A. |last=Guggenheim |title=Thermodynamics. An Advanced Treatment for Chemists and Physicists |publisher=North-Holland |location=Amsterdam |date=1967 |oclc=324553 |pages= |url=}}.	
*{{cite book |last1=Iribarne |first1=J.V. |last2=Godson |first2=W.L. |title=Atmospheric Thermodynamics |publisher=Kluwer Academic |date=1981 |isbn=90-277-1296-4 |oclc=7573676 |doi=10.1007/978-94-017-0815-9 |edition=2nd}}
*{{cite book |first=D. |last=Kondepudi |title=Introduction to Modern Thermodynamics |publisher=Wiley |date=2008 |isbn=978-0-470-01598-8 |oclc=180204969}}
*{{cite book |first=P.T. |last=Landsberg |title=Thermodynamics and Statistical Mechanics |publisher=Oxford University Press |date=1978 |isbn=0-19-851142-6 |oclc=5257434}}
*{{cite book |last1=Lewis |first1=G.N. |last2=Randall |first2=M. |title=Thermodynamics |publisher=McGraw-Hill |series=McGraw-Hill series in advanced chemistry |date=1961 |oclc=271687278 }}
*{{cite book |first=J.C. |last=Maxwell |title=Theory of Heat |publisher=Cambridge University Press  |orig-date=1871 |date=2011 |isbn=978-1-139-05794-3 |doi=10.1017/CBO9781139057943 |ref={{harvid|Maxwell|1871}}}}
*{{cite book |first=J.R. |last=Partington |title=Fundamental Principles. The Properties of Gases |series=An Advanced Treatise on Physical Chemistry |volume=1 |publisher=Longmans, Green, & Co. |location=London |date=1949 |oclc=544118 |pages= |url=}}
*{{cite book |first=M. |last=Planck |title=Treatise on Thermodynamics |publisher=Dover |edition=3rd English |date=1990 |orig-date=1926 |isbn=0-486-31928-8 |oclc=841526209 }}
*{{cite book |last1=Truesdell |first1=C. |last2=Bharatha |first2=S. |title=The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech |publisher=Springer |series=Texts and monographs in physics |date=1977 |isbn=0-387-07971-8 |oclc=2542716 |doi=10.1007/978-3-642-81077-0}}
{{refend}}

==External links==
{{EB1911 poster|Calorimetry}}
*{{cite web |title= Differential scanning calorimetry protocol: MOST |date= |publisher=Appropedia |url=https://www.appropedia.org/Differential_scanning_calorimetry_protocol:_MOST}}

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