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Enthalpy

From Wikipedia, the free encyclopedia
Measure of energy in a thermodynamic system
Not to be confused withEntropy.
Enthalpy
Common symbols
H
SI unitjoule
InSI base unitskg⋅m2⋅s−2
Thermodynamics
The classicalCarnot heat engine

Enthalpy (/ˈɛnθəlpi/ ⓘ) is the sum of athermodynamic system'sinternal energy and the product of itspressure andvolume.[1] It is astate function inthermodynamics used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is conveniently provided by Earth's ambient atmosphere. The pressure–volume term expresses theworkW{\displaystyle W} that was done against constant external pressurePext{\displaystyle P_{\text{ext}}} to establish the system's physical dimensions fromVsystem, initial=0{\displaystyle V_{\text{system, initial}}=0} to some final volumeVsystem, final{\displaystyle V_{\text{system, final}}} (asW=PextΔV{\displaystyle W=P_{\text{ext}}\Delta V}), i.e. to make room for it by displacing its surroundings.[2][3]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 forenergy 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 theInternational System of Units (SI), the unit of measurement for enthalpy is thejoule. Other historical conventional units still in use include thecalorie and theBritish thermal unit (BTU).

The total enthalpy of a system cannot be measured directly because the internal energy contains components that are unknown, not easily accessible, or are not of interest for the thermodynamic problem at hand. In practice, a change in enthalpy is the preferred expression for measurements at constant pressure, because it simplifies the description ofenergy transfer. When transfer of matter into or out of the system is also prevented and no electrical or mechanical (stirring shaft or lift pumping) work is done, at constant pressure the enthalpy change equals the energy exchanged with the environment byheat.

In chemistry, the standardenthalpy of reaction is the enthalpy change when reactants in theirstandard states (p = 1bar; usuallyT = 298K) change to products in their standard states.[4]This quantity is thestandard heat of reaction at constant pressure and temperature, but it can be measured bycalorimetric methods even if the temperature does vary during the measurement, provided that the initial and final pressure and temperature correspond to the standard state. The value does not depend on the path from initial to final state because enthalpy is astate function.

Enthalpies of chemical substances are usually listed for 1 bar (100 kPa) pressure as a standard state. Enthalpies and enthalpy changes for reactions vary as a function of temperature,[5]but tables generally list the standard heats of formation of substances at 25 °C (298 K). Forendothermic (heat-absorbing) processes, the changeΔH is a positive value; forexothermic (heat-releasing) processes it is negative.

The enthalpy of anideal gas is independent of its pressure or volume, and depends only on its temperature, which correlates to its thermal energy. Real gases at common temperatures and pressures often closely approximate this behavior, which simplifies practical thermodynamic design and analysis.

The word "enthalpy" is derived from the Greek wordenthalpein, which means "to heat".[6][7]

Definition

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The enthalpyH of a thermodynamic system is defined as the sum of its internal energy and the product of its pressure and volume:[1]H=U+pV,{\displaystyle H=U+pV,}whereU is theinternal energy,p ispressure, andV is thevolume of the system;p V is sometimes referred to as the pressure energyƐp.[8]

Enthalpy is anextensive property; it is proportional to the size of the system (for homogeneous systems). Asintensive properties, thespecific enthalpyh =H/m is referenced to a unit ofmassm of the system, and themolar enthalpyHm =H/n, wheren is the number ofmoles. For inhomogeneous systems the enthalpy is the sum of the enthalpies of the component subsystems:H=∑kHk,{\displaystyle H=\sum _{k}H_{k},}where

H is the total enthalpy of all the subsystems,
k refers to the various subsystems,
Hk refers to the enthalpy of each subsystem.

A closed system may lie in thermodynamic equilibrium in a staticgravitational field, so that its pressurep varies continuously withaltitude, while, because of the equilibrium requirement, its temperatureT is invariant with altitude. (Correspondingly, the system'sgravitational potential energy density also varies with altitude.) Then the enthalpy summation becomes anintegral:H=∫ρhdV,{\displaystyle H=\int \rho h\,\mathrm {d} V,}where

ρ ("rho") isdensity (mass per unit volume),
h is the specific enthalpy (enthalpy per unit mass),
ρh represents theenthalpy density (enthalpy per unit volume),
dV denotes aninfinitesimally small element of volume within the system, for example, the volume of an infinitesimally thin horizontal layer.

The integral therefore represents the sum of the enthalpies of all the elements of the volume.

The enthalpy of a closed homogeneous system is its energy functionH(S,p), with its entropyS[p] and its pressurep asnatural state variables which provide a differential relation fordH of the simplest form, derived as follows. We start from thefirst law of thermodynamics for closed systems for an infinitesimal process:dU=δQ−δW,{\displaystyle \mathrm {d} U=\delta Q-\delta W,}where

δQ is a small amount of heat added to the system,
δW is a small amount of work performed by the system.

In a homogeneous system in which onlyreversible processes or pure heat transfer are considered, thesecond law of thermodynamics givesδQ =T dS, withT theabsolute temperature anddS the infinitesimal change inentropyS of the system. Furthermore, if onlypV work is done,δW =p dV. As a result,dU=TdS−pdV.{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V.}

Addingd(pV) to both sides of this expression givesdU+d(pV)=TdS−pdV+d(pV),{\displaystyle \mathrm {d} U+\mathrm {d} (pV)=T\,\mathrm {d} S-p\,\mathrm {d} V+\mathrm {d} (p\,V),}ord(U+pV)=TdS+Vdp.{\displaystyle \mathrm {d} (U+pV)=T\,\mathrm {d} S+V\,\mathrm {d} p.}SodH(S,p)=TdS+Vdp,{\displaystyle \mathrm {d} H(S,p)=T\,\mathrm {d} S+V\,\mathrm {d} p,}and the coefficients of the natural variable differentialsdS anddp are just the single variablesT andV.

Other expressions

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The above expression ofdH in terms of entropy and pressure may be unfamiliar to some readers. There are also expressions in terms of more directly measurable variables such as temperature and pressure:[9](p 88)[10]dH=CpdT+V(1−αT)dp,{\displaystyle \mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T+V(1-\alpha T)\,\mathrm {d} p,}whereCp is theheat capacity atconstantpressure, andα is thecoefficient of (cubic) thermal expansion:α=1V(∂V∂T)p.{\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}.}

With this expression one can, in principle, determine the enthalpy ifCp andV are known as functions ofp andT. However the expression is more complicated thandH=TdS+Vdp{\displaystyle \mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p} becauseT is not a natural variable for the enthalpyH.

At constant pressure,dp=0{\displaystyle \mathrm {d} p=0} so thatdH=CpdT.{\displaystyle \mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T.} For anideal gas,dH{\displaystyle \mathrm {d} H} reduces to this form even if the process involves a pressure change, becauseαT = 1.[note 1]

In a more general form, the first law describes the internal energy with additional terms involving thechemical potential and the number of particles of various types. The differential statement fordH then becomesdH=TdS+Vdp+∑iμidNi,{\displaystyle \mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p+\sum _{i}\mu _{i}\,\mathrm {d} N_{i},}whereμi is the chemical potential per particle for a type i particle, andNi is the number of such particles. The last term can also be written asμi dni (withdni 0 the number of moles of componenti added to the system and, in this case,μi the molar chemical potential) or asμi dmi (withdmi the mass of componenti added to the system and, in this case,μi the specific chemical potential).

Characteristic functions and natural state variables

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The enthalpyH(S[p],p, {Ni}) expresses the thermodynamics of a system in theenergy representation. As afunction of state, its arguments include one intensive and several extensivestate variables. The state variablesS[p],p, and{Ni} are said to be thenatural state variables in this representation. They are suitable for describing processes in which they are determined by factors in the surroundings. For example, when a virtual parcel of atmospheric air moves to a different altitude, the pressure surrounding it changes, and the process is often so rapid that there is too little time for heat transfer. This is the basis of the so-calledadiabatic approximation that is used inmeteorology.[11]

Conjugate with the enthalpy, with these arguments, the other characteristic function of state of a thermodynamic system is its entropy, as a functionS[p](H,p, {Ni}) of the same list of variables of state, except that the entropyS[p] is replaced in the list by the enthalpyH. It expresses theentropy representation. The state variablesH,p, and{Ni} are said to be thenatural state variables in this representation. They are suitable for describing processes in which they are experimentally controlled. For example,H andp can be controlled by allowing heat transfer, and by varying only the external pressure on the piston that sets the volume of the system.[12][13][14]

Physical interpretation

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TheU term is the energy of the system, and thepV term can be interpreted as thework that would be required to "make room" for the system if the pressure of the environment remained constant. When a system, for example,nmoles of a gas ofvolumeV atpressurep andtemperatureT, is created or brought to its present state fromabsolute zero, energy must be supplied equal to its internal energyU pluspV, wherepV is thework done in pushing against the ambient (atmospheric) pressure.

Inphysics andstatistical mechanics it may be more interesting to study the internal properties of a constant-volume system and therefore the internal energy is used.[15][16]Inchemistry, experiments are often conducted at constantatmospheric pressure, and the pressure–volume work represents a small, well-defined energy exchange with the atmosphere, so thatΔH is the appropriate expression for theheat of reaction. For aheat engine, the change in its enthalpy after a full cycle is equal to zero, since the final and initial state are equal.

Relationship to heat

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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 heatδQ is supplied by conduction, radiation,Joule heating. We apply it to the special case with a constant pressure at the surface. In this case the work is given byp dV (wherep is the pressure at the surface,dV is the increase of the volume of the system). Cases of long-range electromagnetic interaction require further state variables in their formulation and are not considered here. In this case the first law reads:dU=δQ−pdV.{\displaystyle \mathrm {d} U=\delta Q-p\,\mathrm {d} V.}Now,dH=dU+d(pV),{\displaystyle \mathrm {d} H=\mathrm {d} U+\mathrm {d} (pV),}sodH=δQ+Vdp+pdV−pdV=δQ+Vdp.{\displaystyle {\begin{aligned}\mathrm {d} H&=\delta Q+V\,\mathrm {d} p+p\,\mathrm {d} V-p\,\mathrm {d} V\\&=\delta Q+V\,\mathrm {d} p.\end{aligned}}}

If the system is underconstant pressure,dp = 0 and consequently, the increase in enthalpy of the system is equal to theheat added:dH=δQ.{\displaystyle \mathrm {d} H=\delta Q.}This is why the now-obsolete termheat content was used for enthalpy in the 19th century.

Applications

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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 thethermodynamic system.

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

For a simple system with a constant number of particles at constant pressure, the difference in enthalpy is the maximum amount of thermal energy derivable from an isobaric thermodynamic process.[17]

Heat of reaction

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Main article:Standard enthalpy of reaction

The total enthalpy of a system cannot be measured directly; theenthalpy change of asystem is measured instead. Enthalpy change is defined by the following equation:ΔH=Hf−Hi,{\displaystyle \Delta H=H_{\text{f}}-H_{\text{i}},}where

ΔH is the "enthalpy change",
Hf is the final enthalpy of the system (in a chemical reaction, the enthalpy of the products or the system at equilibrium),
Hi is the initial enthalpy of the system (in a chemical reaction, the enthalpy of the reactants).

For anexothermic reaction at constantpressure, the system's change in enthalpy,ΔH, is negative due to the products of the reaction having a smaller enthalpy than the reactants, and equals the heat released in the reaction if no electrical or mechanical work is done. In other words, the overall decrease in enthalpy is achieved by the generation of heat.[18]Conversely, for a constant-pressureendothermic reaction,ΔH is positive and equal to the heatabsorbed in the reaction.

From the definition of enthalpy asH = U + pV, the enthalpy change at constant pressure isΔH = ΔU +p ΔV. However, for most chemical reactions, the work termp ΔV is much smaller than the internal energy changeΔU, which is approximately equal toΔH. As an example, for the combustion of carbon monoxide2 CO(g) + O2(g) → 2 CO2(g),ΔH = −566.0 kJ andΔU = −563.5 kJ.[19]Since the differences are so small, reaction enthalpies are often described as reaction energies and analyzed in terms ofbond energies.

Specific enthalpy

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Thespecific enthalpy of a uniform system is defined ash =H/m, wherem is the mass of the system. ItsSI unit is joule per kilogram. It can be expressed in other specific quantities byh =u +pv, whereu is the specificinternal energy,p is the pressure, andv isspecific volume, which is equal to1/ρ, whereρ is thedensity.

Enthalpy changes

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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 theproducts assuming that the reaction goes to completion, and the initial enthalpy of the system, namely the reactants. These processes are specified solely by their initial and final states, so that the enthalpy change for the reverse is the negative of that for the forward process.

A common standard enthalpy change is theenthalpy 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 theCRC Handbook of Chemistry and Physics. The following is a selection of enthalpy changes commonly recognized in thermodynamics.

When used in these recognized terms the qualifierchange is usually dropped and the property is simply termedenthalpy 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, orstandard conditions, including:

  • Apressure of one atmosphere (1 atm = 1013.25 hPa) or 1 bar
  • Atemperature of 25 °C = 298.15 K
  • Aconcentration 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 termstandard, e.g.standard enthalpy of formation.

Chemical properties

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Enthalpy of reaction is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of substance reacts completely.

Enthalpy of formation is 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 is 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 is 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 asaturated compound.

Enthalpy of atomization is defined as the enthalpy change required to separate one mole of a substance into its constituentatoms completely.

Enthalpy of neutralization is 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.

Standardenthalpy of solution is 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 ofdenaturation is defined as the enthalpy change required to denature one mole of compound.

Enthalpy of hydration is defined as the enthalpy change observed when one mole of gaseous ions is completely dissolved in water forming one mole of aqueous ions.

Physical properties

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Enthalpy of fusion is defined as the enthalpy change required to completely change the state of one mole of substance from solid to liquid.

Enthalpy of vaporization is defined as the enthalpy change required to completely change the state of one mole of substance from liquid to gas.

Enthalpy of sublimation is defined as the enthalpy change required to completely change the state of one mole of substance from solid to gas.

Lattice enthalpy is 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 is defined as the enthalpy change upon mixing of two (non-reacting) chemical substances.

Open systems

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Inthermodynamicopen systems, mass (of substances) 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 mass flowing in and by heating, minus the amount lost by mass flowing out and in the form of work done by the system:dU=δQ+dUin−dUout−δW,{\displaystyle \mathrm {d} U=\delta Q+\mathrm {d} U_{\text{in}}-\mathrm {d} U_{\text{out}}-\delta W,}whereUin is the average internal energy entering the system, andUout is the average internal energy leaving the system.

Duringsteady, continuous operation, an energy balance applied to an open system equates shaft work performed by the system to heat added plus net enthalpy added

The region of space enclosed by the boundaries of the open system is usually called acontrol 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 mass 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 mass 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 calledpV work), andmechanical work (shaft work), which may be performed on some mechanical device such as a turbine or pump.

These two types of work are expressed in the equationδW=d(poutVout)−d(pinVin)+δWshaft.{\displaystyle \delta W=\mathrm {d} (p_{\text{out}}V_{\text{out}})-\mathrm {d} (p_{\text{in}}V_{\text{in}})+\delta W_{\text{shaft}}.}Substitution into the equation above for the control volume (cv) yieldsdUcv=δQ+dUin+d(pinVin)−dUout−d(poutVout)−δWshaft.{\displaystyle \mathrm {d} U_{\text{cv}}=\delta Q+\mathrm {d} U_{\text{in}}+\mathrm {d} (p_{\text{in}}V_{\text{in}})-\mathrm {d} U_{\text{out}}-\mathrm {d} (p_{\text{out}}V_{\text{out}})-\delta W_{\text{shaft}}.}

The definition of enthalpyH permits us to use thisthermodynamic potential to account for both internal energy andpV work in fluids for open systems:dUcv=δQ+dHin−dHout−δWshaft.{\displaystyle \mathrm {d} U_{\text{cv}}=\delta Q+\mathrm {d} H_{\text{in}}-\mathrm {d} H_{\text{out}}-\delta W_{\text{shaft}}.}

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.[20]In terms of time derivatives, usingNewton's dot notation for time derivatives, it reads:dUdt=∑kQ˙k+∑kH˙k−∑kpkdVkdt−P,{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\mathrm {d} V_{k}}{\mathrm {d} t}}-P,}with sums over the various placesk where heat is supplied, mass flows into the system, and boundaries are moving. The.Hk terms represent enthalpy flows, which can be written asH˙k=hkm˙k=Hmn˙k,{\displaystyle {\dot {H}}_{k}=h_{k}{\dot {m}}_{k}=H_{\text{m}}{\dot {n}}_{k},}withm˙k{\displaystyle {\dot {m}}_{k}} the mass flow andn˙k{\displaystyle {\dot {n}}_{k}} the molar flow at positionk respectively. The termdVk/dt represents the rate of change of the system volume at positionk that results inpV power done by the system. The parameterP represents all other forms of power done by the system such as shaft power, but it can also be, say, electric power produced by an electrical power plant.

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. Duringsteady-state operation of a device (seeTurbine,Pump, andEngine), the averagedU/dt may be set equal to zero. This yields a useful expression for the averagepower generation for these devices in the absence of chemical reactions:P=∑k⟨Q˙k⟩+∑k⟨H˙k⟩−∑k⟨pkdVkdt⟩,{\displaystyle P=\sum _{k}{\big \langle }{\dot {Q}}_{k}{\big \rangle }+\sum _{k}{\big \langle }{\dot {H}}_{k}{\big \rangle }-\sum _{k}\left\langle p_{k}{\frac {\mathrm {d} V_{k}}{\mathrm {d} t}}\right\rangle ,}where theangle 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.

Diagrams

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T −s diagram of nitrogen.[21] The red curve at the left is the melting curve. The red dome represents the two-phase region with the low-entropy side the saturated liquid and the high-entropy side the saturated gas. The black curves give theT −s relation along isobars. The pressures are indicated in bar. The blue curves are isenthalps (curves of constant enthalpy). The values are indicated in blue in⁠ kJ /kg⁠. The specific points a,b, etc., are treated in the main text.

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 ash −T diagrams, which give the specific enthalpy as function of temperature for various pressures, andh −p diagrams, which giveh as function ofp for variousT. 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.

Some basic applications

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The points a throughh in the figure play a role in the discussion in this section.

PointTpsh
UnitKbar⁠ kJ / kg K ⁠⁠ kJ / kg ⁠
a30016.85461
b38026.85530
c3002005.16430
d27016.79430
e108133.55100
f77.213.75100
g77.212.8328
h77.215.41230

Points e andg are saturated liquids, and point h is a saturated gas.

Throttling

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Main article:Joule–Thomson effect
Schematic diagram of a throttling in the steady state. Fluid enters the system (dotted rectangle) at point 1 and leaves it at point 2. The mass flow isṁ .

One of the simple applications of the concept of enthalpy is the so-called throttling process, also known asJoule–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 domesticrefrigerators, 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 ofliquefiers.

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

0=m˙h1−m˙h2 .{\displaystyle 0={\dot {m}}h_{1}-{\dot {m}}h_{2}~.}

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

h1=h2,{\displaystyle h_{1}=h_{2}\;,}

that is, the enthalpy per unit mass does not change during the throttling. The consequences of this relation can be demonstrated using theT − s diagram above.

Example 1

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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.

Example 2

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Point e is chosen so that it is on the saturated liquid line withh = 100⁠ kJ /kg⁠ . It corresponds roughly withp = 13 bar andT = 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 inf(hf ) is equal to the enthalpy ing(hg ) multiplied by the liquid fraction inf(xf ) plus the enthalpy inh(hh ) multiplied by the gas fraction inf (1 −xf ) . So

hf=xfhg+(1−xf)hh .{\displaystyle h_{\mathbf {\mathsf {f}} }=x_{\mathbf {\mathsf {f}} }h_{\mathbf {\mathsf {g}} }+(1-x_{\mathbf {\mathsf {f}} })h_{\mathsf {\mathbf {h} }}~.}

With numbers:

100 =xf × 28 +(1 −xf) × 230 , soxf = 0.64 .

This means that the mass fraction of the liquid in the liquid–gas mixture that leaves the throttling valve is 64%.

Compressors

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Main article:Compressor
Schematic diagram of a compressor in the steady state. Fluid enters the system (dotted rectangle) at point 1 and leaves it at point 2. The mass flow isṁ. A powerP is applied and a heat flowQ̇ is released to the surroundings at ambient temperatureTa .

A powerP is applied e.g. as electrical power. If the compression isadiabatic, the gas temperature goes up. In the reversible case it would be at constant entropy, which corresponds with a vertical line in theT −s diagram. For example, compressing nitrogen from 1 bar (pointa) to 2 bar (pointb) would result in a temperature increase from 300 K to 380 K. In order to let the compressed gas exit at ambient temperatureTa, heat exchange, e.g. by cooling water, is necessary. In the ideal case the compression is isothermal. The average heat flow to the surroundings isQ̇. Since the system is in the steady state the first law gives

0=−Q˙+m˙h1−m˙h2+P .{\displaystyle 0=-{\dot {Q}}+{\dot {m}}\,h_{1}-{\dot {m}}\,h_{2}+P~.}

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

0=−Q˙Ta+m˙s1−m˙s2 .{\displaystyle 0=-{\frac {\,{\dot {Q}}\,}{T_{\mathsf {a}}}}+{\dot {m}}\,s_{1}-{\dot {m}}\,s_{2}~.}

EliminatingQ̇ gives for the minimal power

Pminm˙=h2−h1−Ta(s2−s1) .{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=h_{2}-h_{1}-T_{\mathsf {a}}\left(s_{2}-s_{1}\right)~.}

For example, compressing 1 kg of nitrogen from 1 bar to 200 bar costs at least :(hc −ha) −Ta(sc −sa ) .With the data, obtained with theT −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

Pminm˙=∫12(dh−Tads) .{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=\int _{1}^{2}\left(\mathrm {d} h-T_{\mathsf {a}}\,\mathrm {d} s\right)~.}

With

dh =T ds +v dp ,

this results in the final relation

Pminm˙=∫12vdp .{\displaystyle {\frac {\,P_{\mathsf {min}}}{\dot {m}}}=\int _{1}^{2}v\,\mathrm {d} p~.}

History and etymology

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The termenthalpy was coined relatively late in the history of thermodynamics, in the early 20th century.Energy was introduced in a modern sense byThomas Young in 1802, whileentropy byRudolf Clausius in 1865.Energy uses the root of theGreek wordἔργον (ergon), meaning "work",[22] to express the idea of capacity to perform work.Entropy uses the Greek wordτροπή (tropē) meaningtransformation orturning.[23]Enthalpy uses the root of the Greek wordθάλπος (thalpos) "warmth, heat".[24]

The term expresses the obsolete concept ofheat content,[note 2] asdH refers to the amount of heat gained in a process at constant pressure only,[25] but not in the general case when pressure is variable.[18]J. W. Gibbs used the term "a heat function for constant pressure" for clarity.[note 3]

Introduction of the concept of "heat content"H is associated withBenoît Paul Émile Clapeyron andRudolf Clausius (Clausius–Clapeyron relation, 1850).

The termenthalpy first appeared in print in 1909.[28] It is attributed toHeike Kamerlingh Onnes, who most likely introduced it orally the year before, at the first meeting of the Institute of Refrigeration in Paris.[29] It gained currency only in the 1920s, notably with theMollier Steam Tables and Diagrams, published in 1927.

Until the 1920s, the symbolH was used, somewhat inconsistently, for "heat" in general. The definition ofH as strictly limited to enthalpy or "heat content at constant pressure" was formally proposed by A. W. Porter in 1922.[30][31]

Notes

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  1. ^αT=TV(∂(nRTP)∂T)p=nRTPV=1.{\displaystyle \alpha T={\frac {T}{V}}\left({\frac {\partial ({\frac {nRT}{P}})}{\partial T}}\right)_{p}={\frac {nRT}{PV}}=1.}
  2. ^Howard (2002) quotesJ. R. Partington inAn Advanced Treatise on Physical Chemistry (1949) as saying that the functionH was "usually called the heat content."
  3. ^Volume I of Gibbs'Collected Works[26] does not contain the wordenthalpy, but uses the phrase"heat function for constant pressure" instead, for the same quantity.[27]

See also

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References

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  1. ^abIUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "enthalpy".doi:10.1351/goldbook.E02141
  2. ^Zemansky, Mark W. (1968).Heat and Thermodynamics (5th ed.). New York, NY: McGraw-Hill. chapter 11p 275.
  3. ^van Wylen, G.J.; Sonntag, R.E. (1985).Fundamentals of Classical Thermodynamics (3rd ed.). New York, NY: John Wiley & Sons. section 5.5.ISBN 978-0-471-82933-1.
  4. ^Atkins, Peter; de Paula, Julio (2006).Atkins' Physical Chemistry (8th ed.). W.H.Freeman. p. 51.ISBN 0-7167-8759-8.
  5. ^Laidler, Keith J.; Meiser, John H. (1999).Physical Chemistry (3rd ed.). Boston, MA: Houghton Mifflin. p. 66.ISBN 0-395-91848-0.
  6. ^Çengel, Yunus A.; Boles, Michael A.; Kanoglu, Mehmet (2019).Thermodynamics: an engineering approach (Ninth ed.). New York, NY: McGraw-Hill Education. p. 123.ISBN 978-1-259-82267-4.
  7. ^ἐνθάλπειν,ἐν,θάλπειν.Liddell, Henry George;Scott, Robert;A Greek–English Lexicon at thePerseus Project.
  8. ^"1st Law of Thermodynamics".Chemistry LibreTexts. 2013-10-02. Retrieved2023-10-17.
  9. ^Guggenheim, E. A. (1959).Thermodynamics. Amsterdam, NL: North-Holland Publishing Company.
  10. ^Moran, M. J.; Shapiro, H. N. (2006).Fundamentals of Engineering Thermodynamics (5th ed.). John Wiley & Sons. p. 511.ISBN 9780470030370.
  11. ^Iribarne, J. V.; Godson, W. L. (1981).Atmospheric Thermodynamics (2nd ed.). Dordrecht, NL: Kluwer Academic Publishers. pp. 235–236.ISBN 90-277-1297-2.
  12. ^Tschoegl, N. W. (2000).Fundamentals of Equilibrium and Steady-State Thermodynamics. Amsterdam, NL: Elsevier. p. 17.ISBN 0-444-50426-5.
  13. ^Callen, H. B. (1985) [1960].Thermodynamics and an Introduction to Thermostatistics (1st (1960), 2nd (1985) ed.). New York, NY: John Wiley & Sons. Chapter 5.ISBN 0-471-86256-8.
  14. ^Münster, A. (1970).Classical Thermodynamics. Translated by Halberstadt, E. S. London, UK: Wiley–Interscience. p. 6.ISBN 0-471-62430-6.
  15. ^Reif, F. (1967).Statistical Physics. London, UK: McGraw-Hill.
  16. ^Kittel, C.; Kroemer, H. (1980).Thermal Physics. London, UK: Freeman.
  17. ^Rathakrishnan (2015).High Enthalpy Gas Dynamics. John Wiley and Sons Singapore Pte. Ltd.ISBN 978-1118821893.
  18. ^abLaidler, K. J.; Meiser, John H. (1982).Physical Chemistry. Benjamin / Cummings. p. 53.ISBN 978-0-8053-5682-3.
  19. ^Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002).General Chemistry (8th ed.). Prentice Hall. pp. 237–238.ISBN 978-0-13-014329-7.
  20. ^Moran, M. J.; Shapiro, H. N. (2006).Fundamentals of Engineering Thermodynamics (5th ed.). John Wiley & Sons. p. 129.ISBN 9780470030370.
  21. ^Figure composed with data obtained withREFPROP, NIST Standard Reference Database 23.
  22. ^ἔργον inLiddell andScott.
  23. ^τροπή inLiddell andScott.
  24. ^θάλπος inLiddell andScott.
  25. ^Tinoco, Ignacio Jr.; Sauer, Kenneth; Wang, James C. (1995).Physical Chemistry (3rd ed.). Prentice-Hall. p. 41.ISBN 978-0-13-186545-7.
  26. ^Gibbs (1948)
  27. ^Henderson, Douglas; Eyring, Henry; Jost, Wilhelm (1967).Physical Chemistry: An advanced treatise. Academic Press. p. 29.
  28. ^Dalton (1909), p. 864, footnote (1).
  29. ^Laidler (1995), p. 110;van Ness (2003), p. 486.
  30. ^Porter, Alfred W. (1922)."The Generation and Utilisation of Cold. A General Discussion".Transactions of the Faraday Society.18: 139–143 esp. p. 140.doi:10.1039/tf9221800139.
  31. ^Howard (2002), p. 697.

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