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In
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties ...
and
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, calorimetry () is the science or act of measuring changes in '' state variables'' of a body for the purpose of deriving the
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic s ...
associated with changes of its state due, for example, to
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
s,
physical change Physical changes are changes affecting the form of a chemical substance, but not its chemical composition. Physical changes are used to separate mixtures into their component Chemical compound, compounds, but can not usually be used to separate c ...
s, or
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of ...
s under specified constraints. Calorimetry is performed with a
calorimeter A calorimeter is an object used for calorimetry, or the process of measuring the heat of chemical reactions or physical changes as well as heat capacity. Differential scanning calorimeters, isothermal micro calorimeters, titration calorimeter ...
. Scottish physician and scientist Joseph Black, who was the first to recognize the distinction between
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
, is said to be the founder of the science of calorimetry.
Indirect calorimetry Indirect calorimetry calculates heat that Organism, living organisms produce by measuring either their production of carbon dioxide and Metabolic waste, nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from ...
calculates heat that living organisms produce by measuring either their production of
carbon dioxide Carbon dioxide ( chemical formula ) is a chemical compound made up of molecules that each have one carbon Carbon () is a chemical element with the chemical symbol, symbol C and atomic number 6. It is nonmetallic and tetravalence, tetraval ...
and nitrogen waste (frequently
ammonia Ammonia is an inorganic chemical compound, compound of nitrogen and hydrogen with the chemical formula, formula . A Binary compounds of hydrogen, stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a dis ...
in aquatic organisms, or
urea Urea, also known as carbamide, is an organic compound with chemical formula . This amide has two Amine, amino groups (–) joined by a carbonyl functional group (–C(=O)–). It is thus the simplest amide of carbamic acid. Urea serves an impor ...
in terrestrial ones), or from their consumption of
oxygen Oxygen is the chemical element with the chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen Group (periodic table), group in the periodic table, a highly Chemical reaction, reactive nonmetal, and an oxidizing a ...
. 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
Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Nicolas Léonard Sadi Ca ...
and
Kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its metric prefix, prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based eng ...
, 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 $p\$ as the value of its constitutive function $p\left(V,T\right)\$ of just the volume $V\$ and the temperature $T\$. 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, $\delta V\$ of its volume, and $\delta T\$ of its temperature, the increment of heat, $\delta Q\$, gained by the body of calorimetric material, is given by :$\delta Q\ =C^_T\left(V,T\right)\, \delta V\,+\,C^_V\left(V,T\right)\,\delta T$ where :$C^_T\left(V,T\right)\$ denotes the latent heat with respect to volume, of the calorimetric material at constant controlled temperature $T$. The surroundings' pressure on the material is instrumentally adjusted to impose a chosen volume change, with initial volume $V\$. To determine this latent heat, the volume change is effectively the independently instrumentally varied quantity. This latent heat is not one of the widely used ones, but is of theoretical or conceptual interest. :$C^_V\left(V,T\right)\$ denotes the heat capacity, of the calorimetric material at fixed constant volume $V\$, while the pressure of the material is allowed to vary freely, with initial temperature $T\$. The temperature is forced to change by exposure to a suitable heat bath. It is customary to write $C^_V\left(V,T\right)\$ simply as $C_V\left(V,T\right)\$, or even more briefly as $C_V\$. This latent heat is one of the two widely used ones.Bryan, G.H. (1907), pages 21–22.Adkins, C.J. (1975), Section 3.6, pages 43-46. 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 $p=p\left(V,T\right)\$. 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. The concept of latent heat with respect to volume was perhaps first recognized by Joseph Black in 1762. The term 'latent heat of expansion' is also used. 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 $\delta Q\$ are sometimes called 'curve differentials', because they are measured along curves in the $\left(V,T\right)\$ surface.

### Classical theory for constant-volume (isochoric) calorimetry

Constant-volume calorimetry is calorimetry performed at a constant
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
. 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 $\delta V\$ can be made to vanish, $\delta V=0\$. For constant-volume calorimetry: :$\delta Q = C_V \delta T\$ where :$\delta T\$ denotes the increment in
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
and :$C_V\$ denotes the
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
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.Truesdell, C., Bharatha, S. (1977), page 23. In a process of small increments, $\delta p\$ of its pressure, and $\delta T\$ of its temperature, the increment of heat, $\delta Q\$, gained by the body of calorimetric material, is given by :$\delta Q\ =C^_T\left(p,T\right)\, \delta p\,+\,C^_p\left(p,T\right)\,\delta T$ where :$C^_T\left(p,T\right)\$ 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 $p\$ and temperature $T\$; :$C^_p\left(p,T\right)\$ 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 $p\$ and temperature $T\$. It is customary to write $C^_p\left(p,T\right)\$ simply as $C_p\left(p,T\right)\$, or even more briefly as $C_p\$. The new quantities here are related to the previous ones: :$C^_T\left(p,T\right)=\frac$ :$C^_p\left(p,T\right)=C^_V\left(V,T\right)-C^_T\left(V,T\right) \frac$ where :$\left.\frac\_$ denotes the
partial derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
of $p\left(V,T\right)\$ with respect to $V\$ evaluated for $\left(V,T\right)\$ and :$\left.\frac\_$ denotes the partial derivative of $p\left(V,T\right)\$ with respect to $T\$ evaluated for $\left(V,T\right)\$. The latent heats $C^_T\left(V,T\right)\$ and $C^_T\left(p,T\right)\$ are always of opposite sign. It is common to refer to the ratio of specific heats as :$\gamma\left(V,T\right)=\frac$ often just written as $\gamma=\frac$.

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

An early calorimeter was that used by
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
and 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 change.

## Cumulation of heating

For a time-dependent process of heating of the calorimetric material, defined by a continuous joint progression $P\left(t_1,t_2\right)\$ of $V\left(t\right)\$ and $T\left(t\right)\$, starting at time $t_1\$ and ending at time $t_2\$, there can be calculated an accumulated quantity of heat delivered, $\Delta Q\left(P\left(t_1,t_2\right)\right)\,$ . This calculation is done by 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 Lavoisier, and is called the '
caloric theory The caloric theory is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in ...
'; by the middle of the nineteenth century it was recognized as mistaken. Written with the symbol $\Delta\$, the quantity $\Delta Q\left(P\left(t_1,t_2\right)\right)\,$ is not at all restricted to be an increment with very small values; this is in contrast with $\delta Q\$. One can write :$\Delta Q\left(P\left(t_1,t_2\right)\right)\$ ::$=\int_ \dot Q\left(t\right)dt$ ::$=\int_ C^_T\left(V,T\right)\, \dot V\left(t\right)\, dt\,+\,\int_C^_V\left(V,T\right)\,\dot T\left(t\right)\,dt$. This expression uses quantities such as $\dot Q\left(t\right)\$ 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 $\delta Q\$ is related to the physical requirement for the quantity $p\left(V,T\right)\$ to be 'rapidly determined' by $V\$ and $T\$; such 'rapid determination' refers to a physical process. These 'very small' quantities are used in the
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
approach to the
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. The Newton approach uses instead ' fluxions' such as $\dot V\left(t\right) = \left.\frac\_t$, which makes it more obvious that $p\left(V,T\right)\$ must be 'rapidly determined'. In terms of fluxions, the above first rule of calculation can be written :$\dot Q\left(t\right)\ =C^_T\left(V,T\right)\, \dot V\left(t\right)\,+\,C^_V\left(V,T\right)\,\dot T\left(t\right)$ where :$t\$ denotes the time :$\dot Q\left(t\right)\$ denotes the time rate of heating of the calorimetric material at time $t\$ :$\dot V\left(t\right)\$ denotes the time rate of change of volume of the calorimetric material at time $t\$ :$\dot T\left(t\right)\$ denotes the time rate of change of temperature of the calorimetric material. The increment $\delta Q\$ and the fluxion $\dot Q\left(t\right)\$ are obtained for a particular time $t\$ 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
mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the D ...
$Q\left(V,T\right)\$. For this reason, the increment $\delta Q\$ is said to be an 'imperfect differential' or an '
inexact differential An inexact differential or imperfect differential is a Differential (infinitesimal), differential whose integral is path dependent. It is most often used in thermodynamics to express changes in path dependent quantities such as heat and work, but i ...
'.Adkins, C.J. (1975), Section 1.9.3, page 16. Some books indicate this by writing $q\$ instead of $\delta Q\$. Also, the notation ''đQ'' is used in some books.Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, , page 7. Carelessness about this can lead to error.Planck, M. (1923/1926), page 57. The quantity $\Delta Q\left(P\left(t_1,t_2\right)\right)\$ is properly said to be a functional of the continuous joint progression $P\left(t_1,t_2\right)\$ of $V\left(t\right)\$ and $T\left(t\right)\$, but, in the mathematical definition of a function, $\Delta Q\left(P\left(t_1,t_2\right)\right)\$ is not a function of $\left(V,T\right)\$. Although the fluxion $\dot Q\left(t\right)\$ is defined here as a function of time $t\$, the symbols $Q\$ and $Q\left(V,T\right)\$ 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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, 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 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 Iribarne, J.V., Godson, W.L. (1973/1981), page 46. :$\alpha _V\left(V,T\right)\ = \frac$

## Expansion at constant pressure

For measurements at experimentally controlled pressure, it is assumed that the volume $V\$ of the body of calorimetric material can be expressed as a function $V\left(T,p\right)\$ of its temperature $T\$ and pressure $p\$. This assumption is related to, but is not the same as, the above used assumption that the pressure of the body of calorimetric material is known as a function of its volume and temperature; anomalous behaviour of materials can affect this relation. The quantity that is conveniently measured at constant experimentally controlled pressure, the isobar volume expansion coefficient, is defined by Lewis, G.N., Randall, M. (1923/1961), page 54.Guggenheim, E.A. (1949/1967), page 38.Callen, H.B. (1960/1985), page 84.Adkins, C.J. (1975), page 38.Bailyn, M. (1994), page 49.Kondepudi, D. (2008), page 180. :$\beta _p\left(T,p\right)\ = \frac$

## Compressibility at constant temperature

For measurements at experimentally controlled temperature, it is again assumed that the volume $V\$ of the body of calorimetric material can be expressed as a function $V\left(T,p\right)\$ of its temperature $T\$ and pressure $p\$, with the same provisos as mentioned just above. The quantity that is conveniently measured at constant experimentally controlled temperature, the isothermal compressibility, is defined by :$\kappa _T\left(T,p\right)\ = -\frac$

# Relation between classical calorimetric quantities

Assuming that the rule $p=p\left(V,T\right)\$ is known, one can derive the function of $\frac\$ that is used above in the classical heat calculation with respect to pressure. This function can be found experimentally from the coefficients $\beta _p\left(T,p\right)\$ and $\kappa _T\left(T,p\right)\$ through the mathematically deducible relation :$\frac=\frac$.Kondepudi, D. (2008), page 181.

# Connection between calorimetry and thermodynamics

Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
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..." According to Kondepudi (2008): "Calorimetry is widely used in present day laboratories." In terms of thermodynamics, the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
$U\$ of the calorimetric material can be considered as the value of a function $U\left(V,T\right)\$ of $\left(V,T\right)\$, with partial derivatives $\frac\$ and $\frac\$. Then it can be shown that one can write a thermodynamic version of the above calorimetric rules: : with :$C^_T\left(V,T\right)=p\left(V,T\right)\,+\,\left.\frac\_\$ and :$C^_V\left(V,T\right)=\left.\frac\_\$ . Again, further in terms of thermodynamics, the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
$U\$ of the calorimetric material can sometimes, depending on the calorimetric material, be considered as the value of a function $U\left(p,T\right)\$ of $\left(p,T\right)\$, with partial derivatives $\frac\$ and $\frac\$, and with $V\$ being expressible as the value of a function $V\left(p,T\right)\$ of $\left(p,T\right)\$, with partial derivatives $\frac\$ and $\frac\$ . Then, according to Adkins (1975),Adkins, C.J. (1975), page 46. it can be shown that one can write a further thermodynamic version of the above calorimetric rules: : with :$C^_T\left(p,T\right)=\left.\frac\_\,+\,p\left.\frac\_\$ and :$C^_p\left(p,T\right)=\left.\frac\_\,+\,p\left.\frac\_\$ . Beyond the calorimetric fact noted above that the latent heats $C^_T\left(V,T\right)\$ and $C^_T\left(p,T\right)\$ are always of opposite sign, it may be shown, using the thermodynamic concept of work, that also :$C^_T\left(V,T\right)\,\left.\frac\_ \geq 0\,.$

# 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 an ideal thermodynamic cycle proposed by French physicist Nicolas Léonard Sadi Carnot, Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem (thermodynamics), Carnot's theorem, it provides ...
. 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 $V_a\$ to a volume $V_b\$ at constant temperature $T^+\$ so as to incur a flow of heat into the body (known as an isothermal change) (2) a change in volume from $V_b\$ to a volume $V_c\$ 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 $V_c\$ to a volume $V_d\$ at constant temperature $T^-\$ 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 $V_d\$ back to $V_a\$ just such as to return the body to its starting temperature $T^+\$. In isothermal segment (1), the heat that flows into the body is given by :   $\Delta Q\left(V_a,V_b;T^+\right)\,=\,\,\,\,\,\,\,\,\int_^ C^_T\left(V,T^+\right)\, dV\$ and in isothermal segment (3) the heat that flows out of the body is given by :$-\Delta Q\left(V_c,V_d;T^-\right)\,=\,-\int_^ C^_T\left(V,T^-\right)\, dV\$. 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 :$\Delta Q\left(V_a,V_b;T^+;V_c,V_d;T^-\right)\,=\,\Delta Q\left(V_a,V_b;T^+\right)\,+\,\Delta Q\left(V_c,V_d;T^-\right)\,=\,\int_^ C^_T\left(V,T^+\right)\, dV\,+\,\int_^ C^_T\left(V,T^-\right)\, dV\$. This quantity is used by thermodynamics and is related in a special way to the net work done by the body during the Carnot cycle. The net change of the body's internal energy during the Carnot cycle, $\Delta U\left(V_a,V_b;T^+;V_c,V_d;T^-\right)\$, 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 $C^_T\left(V,T\right)\$, 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 $p=p\left(V,T\right)\$. Provided that the temperature $T\,$ is measured in the thermodynamic absolute scale, the relation is expressed in the formula :$C^_T\left(V,T\right)=T \left.\frac\_\$.

## Difference of specific heats

Advanced thermodynamics provides the relation :. From this, further mathematical and thermodynamic reasoning leads to another relation between classical calorimetric quantities. The difference of specific heats is given by :$C_p\left(p,T\right)-C_V\left(V,T\right)=\frac$.Callen, H.B. (1960/1985), page 86.

# Practical constant-volume calorimetry (bomb calorimetry) for thermodynamic studies

Constant-volume calorimetry is calorimetry performed at a constant
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
. 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. :$q = C_V \Delta T = \Delta U \,,$ where :Δ''U'' is change in
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
, :Δ''T'' is change in
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
and :''CV'' is the
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
at constant volume. In ''constant-volume calorimetry'' the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
is not held constant. If there is a pressure difference between initial and final states, the heat measured needs adjustment to provide the ''
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...
change''. One then has :$\Delta H = \Delta U + \Delta \left(PV\right) = \Delta U + V \Delta P \,,$ where :Δ''H'' is change in
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...
and :''V'' is the unchanging volume of the sample chamber.