Calorimetry
In chemistry and thermodynamics, calorimetry 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.
Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste, 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.
History
The concept of heat has intrigued scientists for more than 2500 years. In the Graeco-Roman era, Plato and Aristotle regarded heat as a manifestation of fire. Newton proposed that it was transmitted by vibrations of the particles of aether, while Descartes described it as an accelerated motion of air particles induced by light. Robert Hooke viewed heat as a property of matter arising from the motion of its parts. For centuries, the prevailing theory imagined heat as a self-repelling, weightless fluid called "caloric".The measurement of heat began to take shape about three centuries ago. In 1750, Georg Wilhelm Richmann formulated the first general calorimetric equation, later known as Richmann's law, allowing calculation of the equilibrium temperature of mixed substances of the same kind. A decade later, Joseph Black made a decisive contribution: in 1761, he discovered that adding heat to ice at its melting point or to boiling water did not change their temperature. His identification of latent and specific heat marked the birth of thermodynamics and introduced the distinction between heat and temperature.
In 1789, Antoine Lavoisier and Pierre-Simon Laplace constructed the first calorimeter, launching quantitative calorimetry. The next major advance came from Sir Benjamin Thompson in the 1790s. Observing heat generated during cannon boring under water, he concluded that heat must be a form of energy, not a material substance.
This idea was quantified by James Prescott Joule, who in the 1840s determined the mechanical equivalent of heat. Using falling weights to drive a paddle in an insulated liquid, Joule demonstrated that 4.184 joules of work produce the same heating effect as one calorie, establishing a numerical link between mechanical work and thermal energy.
Around the same time, in 1840, the chemist Germain Henri Hess formulated Hess's law, showing that the total enthalpy change of a chemical reaction is independent of the reaction path. This principle remains fundamental to modern thermochemistry and calorimetry.
Finally, in the 1870s, Pierre Eugène Berthelot developed the first modern bomb calorimeter and introduced the concepts of endothermic and exothermic reactions.
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 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 as the value of its constitutive function of just the volume and the temperature. 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, of its volume, and of its temperature, the increment of heat,, gained by the body of calorimetric material, is given by
where
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. 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 are sometimes called 'curve differentials', because they are measured along curves in the 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 can be made to vanish,. For constant-volume calorimetry:
where
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, of its pressure, and of its temperature, the increment of heat,, gained by the body of calorimetric material, is given by
where
The new quantities here are related to the previous ones:
where
and
The latent heats and are always of opposite sign.
It is common to refer to the ratio of specific heats as
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 of and, starting at time and ending at time, there can be calculated an accumulated quantity of heat delivered, . 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, the quantity is not at all restricted to be an increment with very small values; this is in contrast with.One can write
This expression uses quantities such as 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 is related to the physical requirement for the quantity to be 'rapidly determined' by and ; such 'rapid determination' refers to a physical process. These 'very small' quantities are used in the Leibniz approach to the infinitesimal calculus. The Newton approach uses instead 'fluxions' such as, which makes it more obvious that must be 'rapidly determined'.In terms of fluxions, the above first rule of calculation can be written
where
The increment and the fluxion are obtained for a particular time 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. For this reason, the increment is said to be an 'imperfect differential' or an 'inexact differential'. Some books indicate this by writing instead of. Also, the notation đQ is used in some books. Carelessness about this can lead to error.Planck, M., page 57.
The quantity is properly said to be a functional of the continuous joint progression of and, but, in the mathematical definition of a function, is not a function of. Although the fluxion is defined here as a function of time, the symbols and respectively standing alone are not defined here.