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. [2]
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.
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. [10] [11] [12] [13] The concept of latent heat with respect to volume was perhaps first recognized by Joseph Black in 1762. [14] The term 'latent heat of expansion' is also used. [15] 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.
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
From the above rule of calculation of heat with respect to volume, there follows one with respect to pressure. [3] [7] [16] [17]
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: [3] [7] [17] [18]
where
and
The latent heats and are always of opposite sign. [19]
It is common to refer to the ratio of specific heats as
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.
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'.
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 [22]
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'. [23] [24] [25] Some books indicate this by writing instead of . [26] [27] Also, the notation đQ is used in some books. [23] [28] Carelessness about this can lead to error. [29]
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.
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.
Empirically, it is convenient to measure properties of calorimetric materials under experimentally controlled conditions.
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 [30]
For measurements at experimentally controlled pressure, it is assumed that the volume of the body of calorimetric material can be expressed as a function of its temperature and pressure . 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 [30] [31] [32] [33] [34] [35] [36]
For measurements at experimentally controlled temperature, it is again assumed that the volume of the body of calorimetric material can be expressed as a function of its temperature and pressure , 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 [31] [32] [33] [34] [35] [36]
Assuming that the rule is known, one can derive the function of that is used above in the classical heat calculation with respect to pressure. This function can be found experimentally from the coefficients and through the mathematically deducible relation
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..." [38] According to Kondepudi (2008): "Calorimetry is widely used in present day laboratories." [39]
In terms of thermodynamics, the internal energy of the calorimetric material can be considered as the value of a function of , with partial derivatives and .
Then it can be shown that one can write a thermodynamic version of the above calorimetric rules:
with
and
Again, further in terms of thermodynamics, the internal energy of the calorimetric material can sometimes, depending on the calorimetric material, be considered as the value of a function of , with partial derivatives and , and with being expressible as the value of a function of , with partial derivatives and .
Then, according to Adkins (1975), [44] it can be shown that one can write a further thermodynamic version of the above calorimetric rules:
with
and
Beyond the calorimetric fact noted above that the latent heats and are always of opposite sign, it may be shown, using the thermodynamic concept of work, that also
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 to a volume at constant temperature so as to incur a flow of heat into the body (known as an isothermal change)
(2) a change in volume from to a volume 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 to a volume at constant temperature 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 back to just such as to return the body to its starting temperature .
In isothermal segment (1), the heat that flows into the body is given by
and in isothermal segment (3) the heat that flows out of the body is given by
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
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, , is equal to zero, because the material of the working body has the special properties noted above.
The quantity , 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 . Provided that the temperature is measured in the thermodynamic absolute scale, the relation is expressed in the formula
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
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.
where
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
where
Enthalpy is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is conveniently provided by the large ambient atmosphere. The pressure–volume term expresses the work that was done against constant external pressure to establish the system's physical dimensions from to some final volume , i.e. to make room for it by displacing its surroundings. The pressure-volume term is very small for solids and liquids at common conditions, and fairly small for gases. Therefore, enthalpy is a stand-in for energy in chemical systems; bond, lattice, solvation, and other chemical "energies" are actually enthalpy differences. As a state function, enthalpy depends only on the final configuration of internal energy, pressure, and volume, not on the path taken to achieve it.
In thermodynamics, the specific heat capacity of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat capacity or as the specific heat. More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. The SI unit of specific heat capacity is joule per kelvin per kilogram, J⋅kg−1⋅K−1. For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.
In thermodynamics, the thermodynamic free energy is one of the state functions of a thermodynamic system. The change in the free energy is the maximum amount of work that the system can perform in a process at constant temperature, and its sign indicates whether the process is thermodynamically favorable or forbidden. Since free energy usually contains potential energy, it is not absolute but depends on the choice of a zero point. Therefore, only relative free energy values, or changes in free energy, are physically meaningful.
A calorimeter is a device 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 calorimeters and accelerated rate calorimeters are among the most common types. A simple calorimeter just consists of a thermometer attached to a metal container full of water suspended above a combustion chamber. It is one of the measurement devices used in the study of thermodynamics, chemistry, and biochemistry.
The second law of thermodynamics is a physical law based on universal empirical observation concerning heat and energy interconversions. A simple statement of the law is that heat always flows spontaneously from hotter to colder regions of matter. Another statement is: "Not all heat can be converted into work in a cyclic process."
The first law of thermodynamics is a formulation of the law of conservation of energy in the context of thermodynamic processes. The law distinguishes two principal forms of energy transfer, heat and thermodynamic work, that modify a thermodynamic system containing a constant amount of matter. The law also defines the internal energy of a system, an extensive property for taking account of the balance of heat and work in the system. Energy cannot be created or destroyed, but it can be transformed from one form to another. In an isolated system the sum of all forms of energy is constant.
In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. These are often chosen to be the standard temperature and pressure.
Heat capacity or thermal capacity is a 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 SI unit of heat capacity is joule per kelvin (J/K).
In thermodynamics, the Gibbs free energy is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure-volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. It also provides a necessary condition for processes such as chemical reactions that may occur under these conditions. The Gibbs free energy is expressed asWhere:
A thermodynamic potential is a scalar quantity used to represent the thermodynamic state of a system. Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions. While thermodynamic potentials cannot be measured directly, they can be predicted using computational chemistry.
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.
The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics.
In thermodynamics, an isobaric process is a type of thermodynamic process in which the pressure of the system stays constant: ΔP = 0. The heat transferred to the system does work, but also changes the internal energy (U) of the system. This article uses the physics sign convention for work, where positive work is work done by the system. Using this convention, by the first law of thermodynamics,
Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental set of postulates, that became the laws of thermodynamics.
The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. It is named after Rudolf Clausius and Benoît Paul Émile Clapeyron. However, this relation was in fact originally derived by Sadi Carnot in his Reflections on the Motive Power of Fire, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and Lord Kelvin decades later. Kelvin said of Carnot's argument that "nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning."
A calorimeter constant is a constant that quantifies the heat capacity of a calorimeter. It may be calculated by applying a known amount of heat to the calorimeter and measuring the calorimeter's corresponding change in temperature. In SI units, the calorimeter constant is then calculated by dividing the change in enthalpy (ΔH) in joules by the change in temperature (ΔT) in kelvins or degrees Celsius:
Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. This exchange results in externally measurable macroscopic forces on the system's surroundings, which can cause mechanical work, to lift a weight, for example, or cause changes in electromagnetic, or gravitational variables. The surroundings also can perform work on a thermodynamic system, which is measured by an opposite sign convention.
In chemical thermodynamics, isothermal titration calorimetry (ITC) is a physical technique used to determine the thermodynamic parameters of interactions in solution. It is most often used to study the binding of small molecules to larger macromolecules in a label-free environment. It consists of two cells which are enclosed in an adiabatic jacket. The compounds to be studied are placed in the sample cell, while the other cell, the reference cell, is used as a control and contains the buffer in which the sample is dissolved.
In thermodynamics, heat is the thermal energy transferred between systems due to a temperature difference. In colloquial use, heat sometimes refers to thermal energy itself. Thermal energy is the kinetic energy of vibrating and colliding atoms in a substance.
In thermodynamics, the enthalpy of fusion of a substance, also known as (latent) heat of fusion, is the change in its enthalpy resulting from providing energy, typically heat, to a specific quantity of the substance to change its state from a solid to a liquid, at constant pressure.