approach uses instead 'fluxions' such as v ˙ ( t ) = d v d t | t (t)=\left.\right|_ , which makes it more obvious that p ( v , t ) must be 'rapidly determined'. in terms of fluxions, the above first rule of calculation can be written q ˙ ( t ) = c t ( v ) ( v , t ) v ˙ ( t ) + c v ( t ) ( v , t ) t ˙ ( t ) (t)\ =c_^(v,t)\,(t)\,+\,c_^(v,t)\,(t)} where t denotes the time q ˙ ( t ) (t)\ } denotes the time rate of heating of the calorimetric material at time t v ˙ ( t ) (t)\ } denotes the time rate of change of volume of the calorimetric material at time t t ˙ ( t ) (t)\ } denotes the time rate of change of temperature of the calorimetric material. the increment δ q and the fluxion q ˙ ( t ) (t)\ } 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 q ( v , t ) . for this reason, the increment δ q is said to be an 'imperfect differential' or an 'inexact differential'. some books indicate this by writing q instead of δ q . also, the notation đq is used in some books. carelessness about this can lead to error. the quantity δ q ( p ( t 1 , t 2 ) ) ,t_))\ } is properly said to be a functional of the continuous joint progression p ( t 1 , t 2 ) ,t_)\ } of v ( t ) and t ( t ) , but, in the mathematical definition of a function, δ q ( p ( t 1 , t 2 ) ) ,t_))\ } is not a function of ( v , t ) . although the fluxion q ˙ ( t ) (t)\ } is defined here as a function of time t , the symbols q and q ( v , t ) 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 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 α v ( v , t ) = 1 p ( v , t ) ∂ p ∂ v | ( v , t ) (v,t)\ =\right|_ 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 ( t , p ) 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 β p ( t , p ) = 1 v ( t , p ) ∂ v ∂ t | ( t , p ) (t,p)\ =\right|_ 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 ( t , p ) 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 κ t ( t , p ) = − 1 v ( t , p ) ∂ v ∂ p | ( t , p ) (t,p)\ =-\right|_ relation between classical calorimetric quantities assuming that the rule p = p ( v , t ) is known, one can derive the function of ∂ p ∂ t \ } that is used above in the classical heat calculation with respect to pressure. this function can be found experimentally from the coefficients β p ( t , p ) (t,p)\ } and κ t ( t , p ) (t,p)\ } through the mathematically deducible relation ∂ p ∂ t = β p ( t , p ) κ t ( t , p ) =(t,p)}(t,p)}}} . 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..." according to kondepudi (2008): "calorimetry is widely used in present day laboratories." in terms of thermodynamics, the internal energy u of the calorimetric material can be considered as the value of a function u ( v , t ) of ( v , t ) , with partial derivatives ∂ u ∂ v \ } and ∂ u ∂ t \ } . then it can be shown that one can write a thermodynamic version of the above calorimetric rules: δ q = [ p ( v , t ) + ∂ u ∂ v | ( v , t ) ] δ v + ∂ u ∂ t | ( v , t ) δ t \right|_\,\delta t} with c t ( v ) ( v , t ) = p ( v , t ) + ∂ u ∂ v | ( v , t ) ^(v,t)=p(v,t)\,+\,\left.\right|_\ } and c v ( t ) ( v , t ) = ∂ u ∂ t | ( v , t ) ^(v,t)=\left.\right|_\ } . again, further in terms of thermodynamics, the internal energy u of the calorimetric material can sometimes, depending on the calorimetric material, be considered as the value of a function u ( p , t ) of ( p , t ) , with partial derivatives ∂ u ∂ p \ } and ∂ u ∂ t \ } , and with v being expressible as the value of a function v ( p , t ) of ( p , t ) , with partial derivatives ∂ v ∂ p \ } and ∂ v ∂ t \ } . then, according to adkins (1975), it can be shown that one can write a further thermodynamic version of the above calorimetric rules: δ q = [ ∂ u ∂ p | ( p , t ) + p ∂ v ∂ p | ( p , t ) ] δ p + [ ∂ u ∂ t | ( p , t ) + p ∂ v ∂ t | ( p , t ) ] δ t with c t ( p ) ( p , t ) = ∂ u ∂ p | ( p , t ) + p ∂ v ∂ p | ( p , t ) ^(p,t)=\left.\right|_\,+\,p\left.\right|_\ } and c p ( t ) ( p , t ) = ∂ u ∂ t | ( p , t ) + p ∂ v ∂ t | ( p , t ) ^(p,t)=\left.\right|_\,+\,p\left.\right|_\ } . beyond the calorimetric fact noted above that the latent heats c t ( v ) ( v , t ) ^(v,t)\ } and c t ( p ) ( p , t ) ^(p,t)\ } are always of opposite sign, it may be shown, using the thermodynamic concept of work, that also c t ( v ) ( v , t ) ∂ p ∂ t | ( v , t ) ≥ 0 . ^(v,t)\,\left.\right|_\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 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 δ q ( v a , v b ; t + ) = ∫ v a v b c t ( v ) ( v , t + ) d v ,v_;t^)\,=\,\,\,\,\,\,\,\,\int _^c_^(v,t^)\,dv\ } and in isothermal segment (3) the heat that flows out of the body is given by − δ q ( v c , v d ; t − ) = − ∫ v c v d c t ( v ) ( v , t − ) d v ,v_;t^)\,=\,-\int _^c_^(v,t^)\,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 δ q ( v a , v b ; t + ; v c , v d ; t − ) = δ q ( v a , v b ; t + ) + δ q ( v c , v d ; t − ) = ∫ v a v b c t ( v ) ( v , t + ) d v + ∫ v c v d c t ( v ) ( v , t − ) d v ,v_;t^;v_,v_;t^)\,=\,\delta q(v_,v_;t^)\,+\,\delta q(v_,v_;t^)\,=\,\int _^c_^(v,t^)\,dv\,+\,\int _^c_^(v,t^)\,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, δ u ( v a , v b ; t + ; v c , v d ; t − ) ,v_;t^;v_,v_;t^)\ } , 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 ( v ) ( v