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**Enthalpy** /ˈɛnθəlpi/ (^{ [1] }^{ [2] } In a system enclosed so as to prevent matter transfer, for processes at constant pressure, the heat absorbed or released equals the change in enthalpy.

A **thermodynamic system** is a group of material and/or radiative contents. Its properties may be described by thermodynamic state variables such as temperature, entropy, internal energy, and pressure.

In thermodynamics, the **internal energy** of a system is the total energy contained within the system. It is the energy necessary to create or prepare the system in any given state, but does not include the kinetic energy of motion of the system as a whole, nor the potential energy of the system as a whole due to external force fields which includes the energy of displacement of the system's surroundings. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state.

- History
- Formal definition
- Other expressions
- Cardinal functions
- Physical interpretation
- Relationship to heat
- Applications
- Heat of reaction
- Specific enthalpy
- Enthalpy changes
- Open systems
- Diagrams
- Some basic applications
- Throttling
- Compressors
- See also
- Notes
- References
- Bibliography
- External links

The unit of measurement for enthalpy in the International System of Units (SI) is the joule. Other historical conventional units still in use include the British thermal unit (BTU) and the calorie.

The **International System of Units** is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the ampere, kelvin, second, metre, kilogram, candela, mole, and a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system also specifies names for 22 derived units, such as lumen and watt, for other common physical quantities.

The **joule** is a derived unit of energy in the International System of Units. It is equal to the energy transferred to an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre. It is also the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second. It is named after the English physicist James Prescott Joule (1818–1889).

The **British thermal unit** is a traditional unit of heat; it is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. It is also part of the United States customary units. Its counterpart in the metric system is the calorie, which is defined as the amount of heat required to raise the temperature of one gram of water by one degree Celsius. Heat is now known to be equivalent to energy, for which the SI unit is the joule; one BTU is about 1055 joules. While units of heat are often supplanted by energy units in scientific work, they are still used in many fields. As examples, in the United States the price of natural gas is quoted in dollars per million BTUs.

Enthalpy comprises a system's internal energy, which is the energy required to create the system, plus the amount of work required to make room for it by displacing its environment and establishing its volume and pressure.^{ [3] }

In physics, a force is said to do **work** if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the work done on the ball as it falls is equal to the weight of the ball multiplied by the distance to the ground. When the force is constant and the angle between the force and the displacement is θ, then the work done is given by *W* = *Fs* cos θ.

In science and engineering, a system is the part of the universe that is being studied, while the **environment** is the remainder of the universe that lies outside the boundaries of the system. It is also known as the **surroundings** or **neighborhood**, and in thermodynamics, as the **reservoir**. Depending on the type of system, it may interact with the environment by exchanging mass, energy, linear momentum, angular momentum, electric charge, or other conserved properties. In some disciplines, such as information theory, information may also be exchanged. The environment is ignored in analysis of the system, except in regard to these interactions.

Enthalpy is defined as a state function that depends only on the prevailing equilibrium state identified by the system's internal energy, pressure, and volume. It is an extensive quantity.

In thermodynamics, a **state function** or **function of state** or **point function** is a function defined for a system relating several *state variables* or *state quantities* that depends only on the current equilibrium state of the system, for example a gas, a liquid, a solid, crystal, or emulsion. State functions do not depend on the path by which the system arrived at its present state. A state function describes the equilibrium state of a system and thus also describes the type of system. For example, a state function could describe an atom or molecule in a gaseous, liquid, or solid form; a heterogeneous or homogeneous mixture; and the amounts of energy required to create such systems or change them into a different equilibrium state.

Physical properties of materials and systems can often be categorized as being either **intensive** or **extensive**, according to how the property changes when the size of the system changes. According to IUPAC, an intensive quantity is one whose magnitude is independent of the size of the system whereas an extensive quantity is one whose magnitude is additive for subsystems. This reflects the corresponding mathematical ideas of mean and measure, respectively.

Enthalpy is the preferred expression of system energy changes in many chemical, biological, and physical measurements at constant pressure, because it simplifies the description of energy transfer. In a system enclosed so as to prevent matter transfer, at constant pressure, the enthalpy change equals the energy transferred from the environment through heat transfer or work other than expansion work.

In thermodynamics, **heat** is energy in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work or transfer of matter. The mechanisms include conduction, through direct contact of immobile bodies, or through a wall or barrier that is impermeable to matter; or radiation between separated bodies; or isochoric mechanical work done by the surroundings on the system of interest; or Joule heating by an electric current driven through the system of interest by an external system; or a combination of these. When there is a suitable path between two systems with different temperatures, heat transfer occurs necessarily, immediately, and spontaneously from the hotter to the colder system. Thermal conduction occurs by the stochastic (random) motion of microscopic particles. In contrast, thermodynamic work is defined by mechanisms that act macroscopically and directly on the system's whole-body state variables; for example, change of the system's volume through a piston's motion with externally measurable force; or change of the system's internal electric polarization through an externally measurable change in electric field. The definition of heat transfer does not require that the process be in any sense smooth. For example, a bolt of lightning may transfer heat to a body.

The total enthalpy, *H*, of a system cannot be measured directly. The same situation exists in classical mechanics: only a change or difference in energy carries physical meaning. Enthalpy itself is a thermodynamic potential, so in order to measure the enthalpy of a system, we must refer to a defined reference point; therefore what we measure is the change in enthalpy, Δ*H*. The Δ*H* is a positive change in endothermic reactions, and negative in heat-releasing exothermic processes.

**For processes under constant pressure**, Δ*H* is equal to the change in the internal energy of the system, plus the pressure-volume work p ΔV done by the system on its surroundings (which is > 0 for an expansion and < 0 for a contraction).^{ [4] } This means that the change in enthalpy under such conditions is the heat absorbed or released by the system through a chemical reaction or by external heat transfer. Enthalpies for chemical substances at constant pressure usually refer to standard state: most commonly 1 bar pressure. Standard state does not, strictly speaking, specify a temperature (see standard state), but expressions for enthalpy generally reference the standard heat of formation at 25 °C.

Enthalpy of ideal gases and incompressible solids and liquids does not depend on pressure, unlike entropy and Gibbs energy. Real materials at common temperatures and pressures usually closely approximate this behavior, which greatly simplifies enthalpy calculation and use in practical designs and analyses.

The word *enthalpy* was coined relatively late, in the early 20th century, in analogy with the 19th-century terms * energy * (introduced in its modern sense by Thomas Young in 1802) and * entropy * (coined in analogy to *energy* by Rudolf Clausius in 1865). Where *energy* uses the root of the Greek word ἔργον (ergon) "work" to express the idea of "work-content" and where *entropy* uses the Greek word τροπή (tropi) "transformation" to express the idea of "transformation-content", so by analogy, *enthalpy* uses the root of the Greek word θάλπος (thalpos) "warmth, heat"^{ [5] } to express the idea of "heat-content". The term does in fact stand in for the older term "heat content",^{ [6] } a term which is now mostly deprecated as misleading, as dH refers to the amount of heat absorbed in a process at constant pressure only,^{ [7] } but not in the general case (when pressure is variable).^{ [8] } Josiah Willard Gibbs used the term "a heat function for constant pressure" for clarity.^{ [note 1] }

Introduction of the concept of "heat content" H is associated with Benoît Paul Émile Clapeyron and Rudolf Clausius (Clausius–Clapeyron relation, 1850).

The term *enthalpy* first appeared in print in 1909.^{ [9] } It is attributed to Heike Kamerlingh Onnes, who most likely introduced it orally the year before, at the first meeting of the Institute of Refrigeration in Paris.^{ [10] } It gained currency only in the 1920s, notably with the * Mollier Steam Tables and Diagrams *, published in 1927.

Until the 1920s, the symbol H was used, somewhat inconsistently, for "heat" in general. The definition of H as strictly limited to enthalpy or "heat content at constant pressure" was formally proposed by Alfred W. Porter in 1922.^{ [11] }^{ [12] }

The enthalpy of a thermodynamic system is defined as^{ [13] }^{ [14] }

- where
- is enthalpy
- is the internal energy of the system
- is pressure
- is the volume of the system

Enthalpy is an extensive property. This means that, for homogeneous systems, the enthalpy is proportional to the size of the system. It is convenient to introduce the specific enthalpy , where is the mass of the system, or the molar enthalpy , where is the number of moles ( and are intensive properties). For inhomogeneous systems the enthalpy is the sum of the enthalpies of the composing subsystems:

- where
- is the total enthalpy of all the subsystems
- refers to the various subsystems
- refers to the enthalpy of each subsystem
- refers to the sum of the enthalpies of all subsystems

A closed system may lie in thermodynamic equilibrium in a static gravitational field, so that its pressure, , varies continuously with altitude, while, because of the equilibrium requirement, its temperature is invariant with altitude. (Correspondingly, the system's gravitational potential energy density also varies with altitude.) Then the enthalpy summation becomes an integral:

- where
- ("rho") is density (mass per unit volume)
- is the specific enthalpy (enthalpy per unit mass)
- represents the enthalpy density (enthalpy per unit volume)
- denotes an infinitesimally small element of volume within the system, for example the volume of an infinitesimally thin horizontal layer
- represents the sum of the enthalpies of all the elements of the volume.

The enthalpy of a closed homogeneous system is its cardinal energy function, , with natural state variables its entropy and its pressure . A differential relation for it can be derived as follows. We start from the first law of thermodynamics for closed systems for an infinitesimal process:

- where
- is a small amount of heat added to the system
- a small amount of work performed by the system.

In a homogeneous system in which only reversible, or quasi-static, processes are considered, the second law of thermodynamics gives with the absolute temperature of the system. Furthermore, if only work is done, . As a result,

Adding to both sides of this expression gives

- or

So

The above expression of *dH* in terms of entropy and pressure may be unfamiliar to some readers. However, there are expressions in terms of more familiar variables such as temperature and pressure:^{ [13] }^{:88}^{ [15] }

Here *C*_{p} is the heat capacity at constant pressure and *α* is the coefficient of (cubic) thermal expansion:

With this expression one can, in principle, determine the enthalpy if *C _{p}* and

Note that for an ideal gas, *αT* = 1,^{ [note 2] } so that

In a more general form, the first law describes the internal energy with additional terms involving the chemical potential and the number of particles of various types. The differential statement for *dH* then becomes

where *μ*_{i} is the chemical potential per particle for an *i*-type particle, and *N*_{i} is the number of such particles. The last term can also be written as *μ _{i}*

The enthalpy, *H*(*S*[*p*],*p*,{*N _{i}*}), expresses the thermodynamics of a system in the

Alongside the enthalpy, with these arguments, the other cardinal function of state of a thermodynamic system is its entropy, as a function, *S*[*p*](*H*,*p*,{*N _{i}*}), of the same list of variables of state, except that the entropy,

The *U* term can be interpreted as the energy required to create the system, and the *pV* term as the work that would be required to "make room" for the system if the pressure of the environment remained constant. When a system, for example, *n* moles of a gas of volume *V* at pressure *p* and temperature *T*, is created or brought to its present state from absolute zero, energy must be supplied equal to its internal energy *U* plus *pV*, where *pV* is the work done in pushing against the ambient (atmospheric) pressure.

In basic physics and statistical mechanics it may be more interesting to study the internal properties of the system and therefore the internal energy is used.^{ [20] }^{ [21] } In basic chemistry, experiments are often conducted at constant atmospheric pressure, and the pressure-volume work represents an energy exchange with the atmosphere that cannot be accessed or controlled, so that Δ*H* is the expression chosen for the heat of reaction.

For a heat engine a change in its internal energy is the difference between the heat input and the pressure-volume work done by the working substance while a change in its enthalpy is the difference between the heat input and the work done by the engine:^{ [22] }

where the work W done by the engine is:

In order to discuss the relation between the enthalpy increase and heat supply, we return to the first law for closed systems, with the physics sign convention: *dU* = *δQ* − *δW*, where the heat *δQ* is supplied by conduction, radiation, and Joule heating. We apply it to the special case with a constant pressure at the surface. In this case the work term can be split into two contributions, the so-called *pV* work, given by *p dV* (where here *p* is the pressure at the surface, *dV* is the increase of the volume of the system), and the so-called isochoric mechanical work *δW′*, such as stirring by a shaft with paddles. Cases of long range electromagnetic interaction require further state variables in their formulation, and are not considered here. So we write *δW* = *p dV* + *δW′*. In this case the first law reads:

Now,

So

With sign convention of physics, *δW'* < 0, because isochoric shaft work done by an external device on the system adds energy to the system, and may be viewed as virtually adding heat. The only thermodynamic mechanical work done by the system is expansion work, *p dV*.^{ [23] }

The system is under constant pressure (*dp* = 0). Consequently, the increase in enthalpy of the system is equal to the added heat and virtual heat:

This is why the now obsolete term *heat content* was used in the nineteenth century.

In thermodynamics, one can calculate enthalpy by determining the requirements for creating a system from "nothingness"; the mechanical work required, *pV*, differs based upon the conditions that obtain during the creation of the thermodynamic system.

Energy must be supplied to remove particles from the surroundings to make space for the creation of the system, assuming that the pressure *p* remains constant; this is the *pV* term. The supplied energy must also provide the change in internal energy, *U*, which includes activation energies, ionization energies, mixing energies, vaporization energies, chemical bond energies, and so forth. Together, these constitute the change in the enthalpy *U* + *pV*. For systems at constant pressure, with no external work done other than the *pV* work, the change in enthalpy is the heat received by the system.

For a simple system, with a constant number of particles, the difference in enthalpy is the maximum amount of thermal energy derivable from a thermodynamic process in which the pressure is held constant.^{ [24] }

The total enthalpy of a system cannot be measured directly, the *enthalpy change* of a system is measured instead. Enthalpy change is defined by the following equation:

where

- Δ
*H*is the "enthalpy change", *H*_{f}is the final enthalpy of the system (in a chemical reaction, the enthalpy of the products),*H*_{i}is the initial enthalpy of the system (in a chemical reaction, the enthalpy of the reactants).

For an exothermic reaction at constant pressure, the system's change in enthalpy equals the energy released in the reaction, including the energy retained in the system and lost through expansion against its surroundings. In a similar manner, for an endothermic reaction, the system's change in enthalpy is equal to the energy *absorbed* in the reaction, including the energy *lost by* the system and *gained* from compression from its surroundings. If Δ*H* is positive, the reaction is endothermic, that is heat is absorbed by the system due to the products of the reaction having a greater enthalpy than the reactants. On the other hand, if Δ*H* is negative, the reaction is exothermic, that is the overall decrease in enthalpy is achieved by the generation of heat.^{ [25] }

From the definition of enthalpy as the enthalpy change at constant pressure However for most chemical reactions, the work term is much smaller than the internal energy change which is approximately equal to As an example, for the combustion of carbon monoxide 2 CO(g) + O_{2}(g) → 2 CO_{2}(g), = –566.0 kJ and = –563.5 kJ.^{ [26] } Since the differences are so small, reaction enthalpies are often loosely described as reaction energies and analyzed in terms of bond energies.

The specific enthalpy of a uniform system is defined as *h* = *H*/*m* where *m* is the mass of the system. The SI unit for specific enthalpy is joule per kilogram. It can be expressed in other specific quantities by *h* = *u* + *pv*, where *u* is the specific internal energy, *p* is the pressure, and *v* is specific volume, which is equal to 1/*ρ*, where *ρ* is the density.

An enthalpy change describes the change in enthalpy observed in the constituents of a thermodynamic system when undergoing a transformation or chemical reaction. It is the difference between the enthalpy after the process has completed, i.e. the enthalpy of the products, and the initial enthalpy of the system, i.e. the reactants. These processes are reversible^{[ why? ]} and the enthalpy for the reverse process is the negative value of the forward change.

A common standard enthalpy change is the enthalpy of formation, which has been determined for a large number of substances. Enthalpy changes are routinely measured and compiled in chemical and physical reference works, such as the CRC Handbook of Chemistry and Physics. The following is a selection of enthalpy changes commonly recognized in thermodynamics.

When used in these recognized terms the qualifier *change* is usually dropped and the property is simply termed *enthalpy of 'process'*. Since these properties are often used as reference values it is very common to quote them for a standardized set of environmental parameters, or standard conditions, including:

- A temperature of 25 °C or 298.15 K,
- A pressure of one atmosphere (1 atm or 101.325 kPa),
- A concentration of 1.0 M when the element or compound is present in solution,
- Elements or compounds in their normal physical states, i.e. standard state.

For such standardized values the name of the enthalpy is commonly prefixed with the term *standard*, e.g. *standard enthalpy of formation*.

Chemical properties:

- Enthalpy of reaction, defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of substance reacts completely.
- Enthalpy of formation, defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a compound is formed from its elementary antecedents.
- Enthalpy of combustion, defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a substance burns completely with oxygen.
- Enthalpy of hydrogenation, defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of an unsaturated compound reacts completely with an excess of hydrogen to form a saturated compound.
- Enthalpy of atomization, defined as the enthalpy change required to atomize one mole of compound completely.
- Enthalpy of neutralization, defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of water is formed when an acid and a base react.
- Standard Enthalpy of solution, defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a solute is dissolved completely in an excess of solvent, so that the solution is at infinite dilution.
- Standard enthalpy of Denaturation (biochemistry), defined as the enthalpy change required to denature one mole of compound.
- Enthalpy of hydration, defined as the enthalpy change observed when one mole of gaseous ions are completely dissolved in water forming one mole of aqueous ions.

Physical properties:

- Enthalpy of fusion, defined as the enthalpy change required to completely change the state of one mole of substance between solid and liquid states.
- Enthalpy of vaporization, defined as the enthalpy change required to completely change the state of one mole of substance between liquid and gaseous states.
- Enthalpy of sublimation, defined as the enthalpy change required to completely change the state of one mole of substance between solid and gaseous states.
- Lattice enthalpy, defined as the energy required to separate one mole of an ionic compound into separated gaseous ions to an infinite distance apart (meaning no force of attraction).
- Enthalpy of mixing, defined as the enthalpy change upon mixing of two (non-reacting) chemical substances.

In thermodynamic open systems, matter may flow in and out of the system boundaries. The first law of thermodynamics for open systems states: The increase in the internal energy of a system is equal to the amount of energy added to the system by matter flowing in and by heating, minus the amount lost by matter flowing out and in the form of work done by the system:

where *U*_{in} is the average internal energy entering the system, and *U*_{out} is the average internal energy leaving the system.

The region of space enclosed by the boundaries of the open system is usually called a control volume, and it may or may not correspond to physical walls. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of matter into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of matter out as if it were driving a piston of fluid. There are then two types of work performed: *flow work* described above, which is performed on the fluid (this is also often called *pV work*), and *shaft work*, which may be performed on some mechanical device.

These two types of work are expressed in the equation

Substitution into the equation above for the control volume (cv) yields:

The definition of enthalpy, *H*, permits us to use this thermodynamic potential to account for both internal energy and *pV* work in fluids for open systems:

If we allow also the system boundary to move (e.g. due to moving pistons), we get a rather general form of the first law for open systems.^{ [27] } In terms of time derivatives it reads:

with sums over the various places *k* where heat is supplied, matter flows into the system, and boundaries are moving. The *Ḣ _{k}* terms represent enthalpy flows, which can be written as

with *ṁ _{k}* the mass flow and

Note that the previous expression holds true only if the kinetic energy flow rate is conserved between system inlet and outlet.^{[ clarification needed ]} Otherwise, it has to be included in the enthalpy balance. During steady-state operation of a device (*see turbine, pump, and engine *), the average *dU*/*dt* may be set equal to zero. This yields a useful expression for the average power generation for these devices in the absence of chemical reactions:

where the angle brackets denote time averages. The technical importance of the enthalpy is directly related to its presence in the first law for open systems, as formulated above.

The enthalpy values of important substances can be obtained using commercial software. Practically all relevant material properties can be obtained either in tabular or in graphical form. There are many types of diagrams, such as *h*–*T* diagrams, which give the specific enthalpy as function of temperature for various pressures, and *h*–*p* diagrams, which give *h* as function of *p* for various *T*. One of the most common diagrams is the temperature–specific entropy diagram (*T*–*s*-diagram). It gives the melting curve and saturated liquid and vapor values together with isobars and isenthalps. These diagrams are powerful tools in the hands of the thermal engineer.

The points **a** through **h** in the figure play a role in the discussion in this section.

**a**:*T*= 300 K,*p*= 1 bar,*s*= 6.85 kJ/(kg K),*h*= 461 kJ/kg;**b**:*T*= 380 K,*p*= 2 bar,*s*= 6.85 kJ/(kg K),*h*= 530 kJ/kg;**c**:*T*= 300 K,*p*= 200 bar,*s*= 5.16 kJ/(kg K),*h*= 430 kJ/kg;**d**:*T*= 270 K,*p*= 1 bar,*s*= 6.79 kJ/(kg K),*h*= 430 kJ/kg;**e**:*T*= 108 K,*p*= 13 bar,*s*= 3.55 kJ/(kg K),*h*= 100 kJ/kg (saturated liquid at 13 bar);**f**:*T*= 77.2 K,*p*= 1 bar,*s*= 3.75 kJ/(kg K),*h*= 100 kJ/kg;**g**:*T*= 77.2 K,*p*= 1 bar,*s*= 2.83 kJ/(kg K),*h*= 28 kJ/kg (saturated liquid at 1 bar);**h**:*T*= 77.2 K,*p*= 1 bar,*s*= 5.41 kJ/(kg K),*h*= 230 kJ/kg (saturated gas at 1 bar);

One of the simple applications of the concept of enthalpy is the so-called throttling process, also known as Joule-Thomson expansion. It concerns a steady adiabatic flow of a fluid through a flow resistance (valve, porous plug, or any other type of flow resistance) as shown in the figure. This process is very important, since it is at the heart of domestic refrigerators, where it is responsible for the temperature drop between ambient temperature and the interior of the refrigerator. It is also the final stage in many types of liquefiers.

For a steady state flow regime, the enthalpy of the system (dotted rectangle) has to be constant. Hence

Since the mass flow is constant, the specific enthalpies at the two sides of the flow resistance are the same:

that is, the enthalpy per unit mass does not change during the throttling. The consequences of this relation can be demonstrated using the *T*–*s* diagram above. Point **c** is at 200 bar and room temperature (300 K). A Joule–Thomson expansion from 200 bar to 1 bar follows a curve of constant enthalpy of roughly 425 kJ/kg (not shown in the diagram) lying between the 400 and 450 kJ/kg isenthalps and ends in point **d**, which is at a temperature of about 270 K. Hence the expansion from 200 bar to 1 bar cools nitrogen from 300 K to 270 K. In the valve, there is a lot of friction, and a lot of entropy is produced, but still the final temperature is below the starting value!

Point **e** is chosen so that it is on the saturated liquid line with *h* = 100 kJ/kg. It corresponds roughly with *p* = 13 bar and *T* = 108 K. Throttling from this point to a pressure of 1 bar ends in the two-phase region (point **f**). This means that a mixture of gas and liquid leaves the throttling valve. Since the enthalpy is an extensive parameter, the enthalpy in **f** (*h*_{f}) is equal to the enthalpy in **g** (*h*_{g}) multiplied by the liquid fraction in **f** (*x*_{f}) plus the enthalpy in **h** (*h*_{h}) multiplied by the gas fraction in **f**(1 − *x*_{f}). So

With numbers: 100 = *x*_{f} × 28 + (1 − *x*_{f}) × 230, so *x*_{f} = 0.64. This means that the mass fraction of the liquid in the liquid–gas mixture that leaves the throttling valve is 64%.

A power *P* is applied e.g. as electrical power. If the compression is adiabatic, the gas temperature goes up. In the reversible case it would be at constant entropy, which corresponds with a vertical line in the *T*–*s* diagram. For example, compressing nitrogen from 1 bar (point **a**) to 2 bar (point **b**) would result in a temperature increase from 300 K to 380 K. In order to let the compressed gas exit at ambient temperature *T*_{a}, heat exchange, e.g. by cooling water, is necessary. In the ideal case the compression is isothermal. The average heat flow to the surroundings is *Q̇*. Since the system is in the steady state the first law gives

The minimal power needed for the compression is realized if the compression is reversible. In that case the second law of thermodynamics for open systems gives

Eliminating *Q̇* gives for the minimal power

For example, compressing 1 kg of nitrogen from 1 bar to 200 bar costs at least (*h*_{c} − *h*_{a}) − *T*_{a}(*s*_{c} − *s*_{a}). With the data, obtained with the *T*–*s* diagram, we find a value of (430 − 461) − 300 × (5.16 − 6.85) = 476 kJ/kg.

The relation for the power can be further simplified by writing it as

With *dh* = *T* *ds* + *v* *dp*, this results in the final relation

**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. The word *calorimetry* is derived from the Latin word *calor*, meaning heat and the Greek word *μέτρον* (metron), meaning measure. 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 **enthalpy of vaporization**, also known as the **(latent) heat of vaporization** or **heat of evaporation**, is the amount of energy (enthalpy) that must be added to a liquid substance, to transform a quantity of that substance into a gas. The enthalpy of vaporization is a function of the pressure at which that transformation takes place.

The **thermodynamic free energy** is a concept useful in the thermodynamics of chemical or thermal processes in engineering and science. The change in the free energy is the maximum amount of work that a thermodynamic system can perform in a process at constant temperature, and its sign indicates whether a 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.

The **second law of thermodynamics** states that the total entropy of an isolated system can never decrease over time. The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.

The **Gibbs–Helmholtz equation** is a thermodynamic equation used for calculating changes in the Gibbs energy of a system as a function of temperature. It is named after Josiah Willard Gibbs and Hermann von Helmholtz.

In thermodynamics, the **Gibbs free energy** is a thermodynamic potential that can be used to calculate the maximum of reversible work that may be performed by a thermodynamic system at a constant temperature and pressure. The Gibbs free energy is the *maximum* amount of non-expansion work that can be extracted from a thermodynamically closed system ; this maximum can be attained only in a completely reversible process. When a system transforms reversibly from an initial state to a final state, the decrease in Gibbs free energy equals the work done by the system to its surroundings, minus the work of the pressure forces.

A **thermodynamic potential** is a scalar quantity used to represent the thermodynamic state of a system. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term *fundamental functions*. One main thermodynamic potential that has a physical interpretation is the internal energy U. It is the energy of configuration of a given system of conservative forces and only has meaning with respect to a defined set of references. Expressions for all other thermodynamic energy potentials are derivable via Legendre transforms from an expression for U. In thermodynamics, external forces, such as gravity, are typically disregarded when formulating expressions for potentials. For example, while all the working fluid in a steam engine may have higher energy due to gravity while sitting on top of Mount Everest than it would at the bottom of the Mariana Trench, the gravitational potential energy term in the formula for the internal energy would usually be ignored because *changes* in gravitational potential within the engine during operation would be negligible. In a large system under even homogeneous external force, like the earth atmosphere under gravity, the intensive parameters should be studied locally having even in equilibrium different values in different places far from each other (see thermodynamic models of troposphere].

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 and volume. The negative of 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 volume is held constant. If the volume were not held constant, part of this work would be performed as boundary work. This makes the Helmholtz energy useful for systems held at constant volume. Furthermore, at constant temperature, the Helmholtz energy is minimized at equilibrium.

The **standard enthalpy of reaction** is the enthalpy change that occurs in a system when matter is transformed by a given chemical reaction, when all reactants and products are in their standard states.

In thermodynamics, an **isentropic process** is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no transfer of heat or matter. Such an idealized process is useful in engineering as a model of and basis of comparison for real processes.

An *isobaric process* is a thermodynamic process in which the pressure stays constant: Δ*P* = 0. The heat transferred to the system does work, but also changes the internal energy of the system. This article uses the chemistry sign convention for work, where positive work is work done *on* 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.

A **thermodynamic cycle** consists of a linked sequence of thermodynamic processes that involve transfer of heat and work into and out of the system, while varying pressure, temperature, and other state variables within the system, and that eventually returns the system to its initial state. In the process of passing through a cycle, the working fluid (system) may convert heat from a warm source into useful work, and dispose of the remaining heat to a cold sink, thereby acting as a heat engine. Conversely, the cycle may be reversed and use work to move heat from a cold source and transfer it to a warm sink thereby acting as a heat pump. At every point in the cycle, the system is in thermodynamic equilibrium, so the cycle is reversible.

**Entropy** is a property of thermodynamical systems. The term entropy was introduced by Rudolf Clausius who named it from the Greek word τρoπή, "transformation". He considered transfers of energy as heat and work between bodies of matter, taking temperature into account. Bodies of radiation are also covered by the same kind of reasoning.

In thermodynamics, the **fundamental thermodynamic relation** is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.

**Entropy** is an important concept in the branch of physics known as thermodynamics. The idea of "irreversibility" is central to the understanding of entropy. Everyone has an intuitive understanding of irreversibility. If one watches a movie of everyday life running forward and in reverse, it is easy to distinguish between the two. The movie running in reverse shows impossible things happening – water jumping out of a glass into a pitcher above it, smoke going down a chimney, water in a glass freezing to form ice cubes, crashed cars reassembling themselves, and so on. The intuitive meaning of expressions such as "you can't unscramble an egg", or "you can't take the cream out of the coffee" is that these are irreversible processes. No matter how long you wait, the cream won't jump out of the coffee into the creamer.

**Thermodynamic databases** contain information about thermodynamic properties for substances, the most important being enthalpy, entropy, and Gibbs free energy. Numerical values of these thermodynamic properties are collected as tables or are calculated from thermodynamic datafiles. Data is expressed as temperature-dependent values for one mole of substance at the standard pressure of 101.325 kPa, or 100 kPa. Unfortunately, both of these definitions for the standard condition for pressure are in use.

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. For example, when melting 1 kg of ice, 333.55 kJ of energy is absorbed with no temperature change. The **heat of solidification** is equal and opposite.

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*KNAW Proceedings*.**11**: 863–873. Bibcode:1908KNAB...11..863D. - Haase, R. (1971). Jost, W. (ed.).
*Physical Chemistry: An Advanced Treatise*. New York, NY: Academic. p. 29. - Gibbs, J. W.
*The Collected Works of J. Willard Gibbs, Vol. I*(1948 ed.). New Haven, CT: Yale University Press. p. 88.. - Howard, I. K. (2002). "
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*The World of Physical Chemistry*. Oxford: Oxford University Press. p. 110. - Kittel, C.; Kroemer, H. (1980).
*Thermal Physics*. New York, NY: S. R. Furphy & Co. p. 246. - DeHoff, R. (2006). "Thermodynamics in Materials Science".
*Entropy*(2nd ed.).**20**(7): 532. Bibcode:2018Entrp..20..532G. doi:10.3390/e20070532.

- Enthalpy - Eric Weisstein's World of Physics
- Enthalpy - Georgia State University
- Enthalpy example calculations - Texas A&M University Chemistry Department

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