calorimetry calorimetry in chemistry and thermodynamics, calorimetry (from latin calor 'heat', and greek 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 state due, for example, to chemical reactions, physical changes, or phase transitions 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. 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 changes; calculations which were based on joseph black’s prior discovery of latent heat. these experiments mark the foundation of thermochemistry. snellen direct calorimetry chamber, university of ottawa. indirect calorimetry metabolic cart measuring oxygen uptake and co2 production of a spontaneously breathing subject (dilution method with canopy hood). 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. 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 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 and 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 p as the value of its constitutive function p ( v , t ) 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, δ v of its volume, and δ t of its temperature, the increment of heat, δ q , gained by the body of calorimetric material, is given by δ q = c t ( v ) ( v , t ) δ v + c v ( t ) ( v , t ) δ t ^(v,t)\,\delta v\,+\,c_^(v,t)\,\delta t} where c t ( v ) ( v , t ) ^(v,t)\ } 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 ( t ) ( v , t ) ^(v,t)\ } 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 ( t ) ( v , t ) ^(v,t)\ } simply as c v ( v , t ) (v,t)\ } , or even more briefly as c v \ } . this latent heat is one of the two widely used ones. 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 ( v , t ) . 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 δ q are sometimes called 'curve differentials', because they are measured along curves in the ( v , t ) 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 δ v can be made to vanish, δ v = 0 . for constant-volume calorimetry: δ q = c v δ t \delta t\ } where δ t denotes the increment in temperature and c v \ } 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. in a process of small increments, δ p of its pressure, and δ t of its temperature, the increment of heat, δ q , gained by the body of calorimetric material, is given by δ q = c t ( p ) ( p , t ) δ p + c p ( t ) ( p , t ) δ t ^(p,t)\,\delta p\,+\,c_^(p,t)\,\delta t} where c t ( p ) ( p , t ) ^(p,t)\ } 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 ( t ) ( p , t ) ^(p,t)\ } 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 ( t ) ( p , t ) ^(p,t)\ } simply as c p ( p , t ) (p,t)\ } , or even more briefly as c p \ } . the new quantities here are related to the previous ones: c t ( p ) ( p , t ) = c t ( v ) ( v , t ) ∂ p ∂ v | ( v , t ) ^(p,t)=^(v,t)}\right|_ c p ( t ) ( p , t ) = c v ( t ) ( v , t ) − c t ( v ) ( v , t ) ∂ p ∂ t | ( v , t ) ∂ p ∂ v | ( v , t ) ^(p,t)=c_^(v,t)-c_^(v,t)\right|_\right|_ where ∂ p ∂ v | ( v , t ) \right|_ denotes the partial derivative of p ( v , t ) with respect to v evaluated for ( v , t ) and ∂ p ∂ t | ( v , t ) \right|_ denotes the partial derivative of p ( v , t ) with respect to t evaluated for ( v , t ) . the latent heats c t ( v ) ( v , t ) ^(v,t)\ } and c t ( p ) ( p , t ) ^(p,t)\ } are always of opposite sign. it is common to refer to the ratio of specific heats as γ ( v , t ) = c p ( t ) ( p , t ) c v ( t ) ( v , t ) ^(p,t)}^(v,t)}}} often just written as γ = c p c v . calorimetry through phase change, equation of state shows one jump discontinuity an early calorimeter was that used by laplace 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 ( t 1 , t 2 ) ,t_)\ } of v ( t ) and t ( t ) , starting at time t 1 \ } and ending at time t 2 \ } , there can be calculated an accumulated quantity of heat delivered, δ q ( p ( t 1 , t 2 ) ) ,t_))\,} . 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'; by the middle of the nineteenth century it was recognized as mistaken. written with the symbol δ , the quantity δ q ( p ( t 1 , t 2 ) ) ,t_))\,} is not at all restricted to be an increment with very small values; this is in contrast with δ q . one can write δ q ( p ( t 1 , t 2 ) ) ,t_))\ } = ∫ p ( t 1 , t 2 ) q ˙ ( t ) d t ,t_)}(t)dt} = ∫ p ( t 1 , t 2 ) c t ( v ) ( v , t ) v ˙ ( t ) d t + ∫ p ( t 1 , t 2 ) c v ( t ) ( v , t ) t ˙ ( t ) d t ,t_)}c_^(v,t)\,(t)\,dt\,+\,\int _,t_)}c_^(v,t)\,(t)\,dt} . this expression uses quantities such as q ˙ ( t ) (t)\ } 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 δ q is related to the physical requirement for the quantity p ( v , t ) 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 approach to the infinitesimal calculus. the newton