Editing Exergy (section)

==Mathematical description==

===An application of the second law of thermodynamics===
{{See also|Second law of thermodynamics}}

Exergy uses [[System (thermodynamics)|system boundaries]] in a way that is unfamiliar to many. We imagine the presence of a [[Carnot heat engine|Carnot engine]] between the system and its reference environment even though this engine does not exist in the real world.  Its only purpose is to measure the results of a "what-if" scenario to represent the most efficient work interaction possible between the system and its surroundings.

If a real-world reference environment is chosen that behaves like an unlimited reservoir that remains unaltered by the system, then Carnot's speculation about the consequences of a system heading towards equilibrium with time is addressed by two equivalent mathematical statements. Let ''B'', the exergy or available work, decrease with time, and ''S''<sub>total</sub>, the entropy of the system and its reference environment enclosed together in a larger [[isolated system]], increase with time:

{{NumBlk|:|<math> \frac{\mathrm{d}B}{\mathrm{d}t} \le 0 \mbox{ is equivalent to } \frac {\mathrm{d}S_\text{total}}{\mathrm{d}t} \ge 0 </math>|{{EquationRef|1}}}}

For macroscopic systems (above the [[thermodynamic limit]]), these statements are both expressions of the [[second law of thermodynamics]] if the following expression is used for exergy:

{{NumBlk|:|<math> B = U + P_R V - T_R S - \sum_i \mu_{i,R} N_i </math>|{{EquationRef|2}}}}
where the [[Extensive quantity|extensive quantities]] for the system are: ''U'' = [[Internal energy]], ''V'' = [[Volume]], and ''N''<sub>i</sub> = [[Mole (unit)|Moles]] of component ''i''. The [[Intensive quantity|intensive quantities]] for the surroundings are: ''P''<sub>R</sub> = [[Pressure]], ''T''<sub>R</sub> = [[temperature]], ''μ''<sub>i, R
</sub> = [[Chemical potential]] of component ''i''. Indeed the total entropy of the universe reads:
{{NumBlk|:|<math> S_{\mathrm{total}} = -B/T_R = S - U/T_R - P_R V /T_R + \sum_i \mu_{i,R} N_i /T_R </math>|{{EquationRef|3}}}}
the second term <math>  - U/T_R - P_R V /T_R + \sum_i \mu_{i,R} N_i /T_R </math> being the entropy of the surroundings to within a constant. 

Individual terms also often have names attached to them: <math>P_R V</math> is called "available PV work", <math>T_R S</math> is called "entropic loss" or "heat loss" and the final term is called "available chemical energy."

Other [[thermodynamic potential]]s may be used to replace internal energy so long as proper care is taken in recognizing which natural variables correspond to which potential. For the recommended nomenclature of these potentials, see (Alberty, 2001){{ref|alberty01}}. Equation ({{EquationNote|2}}) is useful for processes where system volume, entropy, and the number of moles of various components change because internal energy is also a function of these variables and no others.

An alternative definition of internal energy does not separate available chemical potential from ''U''. This expression is useful (when substituted into equation ({{EquationNote|1}})) for processes where system volume and entropy change, but no chemical reaction occurs:

{{NumBlk|:|<math> B=U[\mu_1, \mu_2, \ldots, \mu_n] +P_RV -T_RS=U[\boldsymbol{\mu}] +P_RV -T_RS </math>|{{EquationRef|4}}}}

In this case, a given set of chemicals at a given entropy and volume will have a single numerical value for this thermodynamic potential. A [[phase (matter)|multi-state]] system may complicate or simplify the problem because the [[Gibbs phase rule]] predicts that intensive quantities will no longer be completely independent from each other.

===A historical and cultural tangent===
In 1848, [[William Thomson, 1st Baron Kelvin]], asked (and immediately answered) the question

:Is there any principle on which an absolute thermometric scale can be founded? It appears to me that Carnot's theory of the motive power of heat enables us to give an affirmative answer.{{ref|kelvin1848}}

With the benefit of the hindsight contained in equation ({{EquationNote|5}}), we are able to understand the historical impact of Kelvin's idea on physics.  Kelvin suggested that the best temperature scale would describe a constant ability for a unit of temperature in the surroundings to alter the available work from Carnot's engine. From equation ({{EquationNote|3}}):

{{NumBlk|:|<math> \frac{\mathrm{d}B}{\mathrm{d}T_R}=-S </math>|{{EquationRef|5}}}}

[[Rudolf Clausius]] recognized the presence of a [[Proportionality (mathematics)|proportionality]] constant in Kelvin's analysis and gave it the name [[entropy]] in 1865 from the Greek for "transformation" because it quantifies the amount of energy lost during the conversion from heat to work. The available work from a Carnot engine is at its maximum when the surroundings are at a temperature of [[absolute zero]].

Physicists then, as now, often look at a property with the word "available" or "utilizable" in its name with a certain unease.  The idea of what is available raises the question of "available to what?" and raises a concern about whether such a property is [[Anthropocentrism|anthropocentric]]. Laws derived using such a property may not describe the universe but instead, describe what people wish to see.

The field of [[statistical mechanics]] (beginning with the work of [[Ludwig Boltzmann]] in developing the [[Boltzmann equation]]) relieved many physicists of this concern.  From this discipline, we now know that macroscopic properties may all be determined from properties on a microscopic scale where entropy is more "real" than temperature itself (''see [[Thermodynamic temperature]]''). Microscopic kinetic fluctuations among particles cause entropic loss, and this energy is unavailable for work because these fluctuations occur randomly in all directions.  The anthropocentric act is taken, in the eyes of some physicists and engineers today, when someone draws a hypothetical boundary, in fact, he says: "This is my system. What occurs beyond it is surroundings." In this context, exergy is sometimes described as an anthropocentric property, both by some who use it and by some who don't. However, exergy is based on the dis-equilibrium between a system and its environment, so its very real and necessary to define the system distinctly from its environment. It can be agreed that entropy is generally viewed as a more fundamental property of matter than exergy.

===A potential for every thermodynamic situation===

In addition to <math>U </math> and {{nowrap|<math> U[\boldsymbol{\mu}]</math>,}} the other [[thermodynamic potential]]s are frequently used to determine exergy.  For a given set of chemicals at a given entropy and pressure, [[enthalpy]] ''H'' is used in the expression:

{{NumBlk|:|<math> B =  H - T_R S </math>|{{EquationRef|6}}}}

For a given set of chemicals at a given temperature and volume, [[Helmholtz free energy]] ''A'' is used in the expression:

{{NumBlk|:|<math> B = A + P_R V </math>|{{EquationRef|7}}}}

For a given set of chemicals at a given temperature and pressure, [[Gibbs free energy]] ''G'' is used in the expression:

{{NumBlk|:|<math> G = H - TS = B - (T - T_R)S  </math>|{{EquationRef|8}}}}

where <math>G</math> is evaluated at the isothermal system temperature (<math>T</math>), and <math>B</math> is defined with respect to the isothermal temperature of the system's environment (<math>T_R</math>). The exergy <math>B</math> is the energy <math>H</math> reduced by the product of the entropy times the environment temperature <math>T_R</math>, which is the slope or partial derivative of the internal energy with respect to entropy in the environment.<ref>S.E. Wright, Comparison of the theoretical performance potential of fuel cells and heat engines, Renewable Energy 29 (2004) 179–195</ref> That is, higher entropy reduces the exergy or free energy available relative to the energy level <math>H</math>.

Work can be produced from this energy, such as in an isothermal process, but any entropy generation during the process will cause the destruction of exergy ([[irreversibility]]) and the reduction of these thermodynamic potentials. Further, exergy losses can occur if mass and energy is transferred out of the system at non-ambient or elevated temperature, pressure or chemical potential. Exergy losses are potentially recoverable though because the exergy has not been destroyed, such as what occurs in waste heat recovery systems (although the energy quality or exergy content is typically low). As a special case, an isothermal process operating at ambient temperature will have no thermally related exergy losses.

=== Exergy Analysis involving Radiative Heat Transfer ===

All matter emits radiation continuously as a result of its non-zero (absolute) temperature. This emitted energy flow is proportional to the material’s temperature raised to the fourth power. As a result, any radiation conversion device that seeks to absorb and convert radiation (while reflecting a fraction of the incoming source radiation) inherently emits its own radiation. Also, given that reflected and emitted radiation can occupy the same direction or solid angle, the entropy flows, and as a result, the exergy flows, are generally not independent. The entropy and exergy balance equations for a control volume (CV), re-stated to correctly apply to situations involving radiative transfer,<ref name="dx.doi.org"/><ref name="ReferenceA"/><ref name="Wright 2001 1691–1706">{{Cite journal |last=Wright |date=2001 |title=On the entropy of radiative heat transfer in engineering thermodynamics |journal=Int. J. Eng. Sci. |volume=39 |issue=15 |pages=1691–1706|doi=10.1016/S0020-7225(01)00024-6 }}</ref><ref>{{Cite journal |last=Wright |first=S.E. |date=December 2007 |title=The Clausius inequality corrected for heat transfer involving radiation |url=http://dx.doi.org/10.1016/j.ijengsci.2007.08.005 |journal=International Journal of Engineering Science |volume=45 |issue=12 |pages=1007–1016 |doi=10.1016/j.ijengsci.2007.08.005 |issn=0020-7225|url-access=subscription }}</ref> are expressed as,

<math display="block"> \frac {dS_{CV}}{dt} = \int_{CV boundary} (\frac{q_{cc}}{T_b} + J_{NetRad})dA+\sum_{i}^m({\dot{m}_i}s_i)-\sum_{o}^n({\dot{m}_o}s_o)+\dot{S}_{gen}</math>
where <math>{S}_{gen}</math> or {{math|''Π''}} denotes entropy production within the control volume, and,

<math display="block"> \frac {{dX}_{CV}}{dt} = \int_{CV boundary}[{q_{cc}}(1-\frac{T_o}{T_b}) + M_{NetRad}]dA-({\dot{W}_{CV}}-P_o(\frac{dV_{CV}}{dt}))+\sum_{i}^r({\dot{m}_i}(h_i-{T_o}{s_i}))-\dot{I}_{CV}</math>

This rate equation for the exergy within an open system X ({{math|''Ξ or B''}}) takes into account the exergy transfer rates across the system boundary by heat transfer ({{math|q}} for conduction & convection, and {{math|''M''}} by radiative fluxes), by mechanical or electrical work transfer ({{math|''W''}}), and by mass transfer ({{math|''m''}}), as well as taking into account the exergy destruction ({{math|''I''}}) that occurs within the system due to irreversibility’s or non-ideal processes. Note that chemical exergy, kinetic energy, and gravitational potential energy have been excluded for simplicity.

The exergy irradiance or flux M, and the exergy radiance N (where M = πN for isotropic radiation), depend on the spectral and directional distribution of the radiation (for example, see the next section on ‘Exergy Flux of Radiation with an Arbitrary Spectrum’). Sunlight can be crudely approximated as blackbody, or more accurately, as graybody radiation. Noting that, although a graybody spectrum looks similar to a blackbody spectrum, the entropy and exergy are very different.
Petela<ref name="R. Petela 1964, pp. 187-192">R. Petela, 1964, “Exergy of Heat Radiation”, ASME Journal of Heat Transfer, Vol. 86, pp. 187-192</ref> determined that the exergy of isotropic blackbody radiation was given by the expression,

<math display="block">M_{BR} = \frac{cX}{4V} = \sigma T^4 (1-\frac{4}{3}x+\frac{1}{3}x^4)</math>

where the exergy within the enclosed system is X ({{math|''Ξ or B''}}), c is the speed of light, V is the volume occupied by the enclosed radiation system or void, T is the material emission temperature, To is the environmental temperature, and x is the dimensionless temperature ratio To/T.

However, for decades this result was contested in terms of its relevance to the conversion of radiation fluxes, and in particular, solar radiation. For example, Bejan<ref>A. Bejan, 1997, Advanced Engineering Thermodynamics, 2nd edition, John Wiley and Sons, New York</ref> stated that “Petela’s efficiency is no more than a convenient, albeit artificial way, of non-dimensionalizing the calculated work output” and that Petela’s efficiency “is not a ‘conversion efficiency.’ ” However, it has been shown that Petela’s result represents the exergy of blackbody radiation.<ref name="dx.doi.org"/> This was done by resolving a number of issues, including that of inherent irreversibility, defining the environment in terms of radiation, the effect of inherent emission by the conversion device and the effect of concentrating source radiation.

==== Exergy Flux of Radiation with an Arbitrary Spectrum (including Sunlight) ====
In general, terrestrial solar radiation has an arbitrary non-blackbody spectrum. Ground level spectrums can vary greatly due to reflection, scattering and absorption in the atmosphere. While the emission spectrums of thermal radiation in engineering systems can vary widely as well.

In determining the exergy of radiation with an arbitrary spectrum, it must be considered whether reversible or ideal conversion (zero entropy production) is possible. It has been shown that reversible conversion of blackbody radiation fluxes across an infinitesimal temperature difference is theoretically possible <ref name="dx.doi.org"/><ref name="ReferenceA"/>]. However, this reversible conversion can only be theoretically achieved because equilibrium can exist between blackbody radiation and matter.<ref name="ReferenceA"/> However, non-blackbody radiation cannot even exist in equilibrium with itself, nor with its own emitting material.

Unlike blackbody radiation, non-blackbody radiation cannot exist in equilibrium with matter, so it appears likely that the interaction of non-blackbody radiation with matter is always an inherently irreversible process. For example, an enclosed non-blackbody radiation system (such as a void inside a solid mass) is unstable and will spontaneously equilibriate to blackbody radiation unless the enclosure is perfectly reflecting (i.e., unless there is no thermal interaction of the radiation with its enclosure – which is not possible in actual, or real, non-ideal systems). Consequently, a cavity initially devoid of thermal radiation inside a non-blackbody material will spontaneously and rapidly (due to the high velocity of the radiation), through a series of absorption and emission interactions, become filled with blackbody radiation rather than non-blackbody radiation.

The approaches by Petela<ref name="R. Petela 1964, pp. 187-192"/> and Karlsson<ref>S. Karlsson, 1982, “Exergy of Incoherent Electromagnetic Radiation,” Physica Scripta, Vol. 26, pp. 329-332</ref> both assume that reversible conversion of non-blackbody radiation is theoretically possible, that is, without addressing or considering the issue. Exergy is not a property of the system alone, it’s a property of both the system and its environment. Thus, it is of key importance non-blackbody radiation cannot exist in equilibrium with matter, indicating that the interaction of non-blackbody radiation with matter is an inherently irreversible process.

The flux (irradiance) of radiation with an arbitrary spectrum, based on the inherent irreversibility of non-blackbody radiation conversion, is given by the expression,

<math display="block">M=H-T_o(\frac{4}{3}\sigma^{0.25} H^{0.75})+\frac{\sigma}{3}T_o^4)</math>

The exergy flux <math>M</math> is expressed as a function of only the energy flux or irradiance <math>H</math> and the environment temperature <math>T_o</math>. For graybody radiation, the exergy flux is given by the expression,

<math display="block">M_{GR}=\sigma T^4 (\epsilon-\frac{4}{3}x\epsilon^{0.75}+\frac{1}{3}x^4)</math>

As one would expect, the exergy flux of non-blackbody radiation reduces to the result for blackbody radiation when emissivity is equal to one.

Note that the exergy flux of graybody radiation can be a small fraction of the energy flux. For example, the ratio of exergy flux to energy flux <math>(M/H)</math> for graybody radiation  with emissivity <math>\epsilon = 0.50</math> is equal to 40.0%, for <math>T = 500^oC</math> and <math> T_o = 27^oC  (x = 0.388)</math>. That is, a maximum of only 40% of the graybody energy flux can be converted to work in this case (already only 50% of that of the blackbody energy flux with the same emission temperature). Graybody radiation has a spectrum that looks similar to the blackbody spectrum, but the entropy and exergy flux cannot be accurately approximated as that of blackbody radiation with the same emission temperature. However, it can be reasonably approximated by the entropy flux of blackbody radiation with the same energy flux (lower emission temperature).

Blackbody radiation has the highest entropy-to-energy ratio of all radiation with the same energy flux, but the lowest entropy-to-energy ratio, and the highest exergy content, of all radiation with the same emission temperature.<ref name="Wright 2001 1691–1706"/><ref name="ReferenceA"/> For example, the exergy content of graybody radiation is lower than that of blackbody radiation with the same emission temperature and decreases as emissivity decreases. For the example above with <math>x = 0.388</math> the exergy flux of the blackbody radiation source flux is 52.5% of the energy flux compared to 40.0% for graybody radiation with <math>\epsilon = 0.50</math>, or compared to 15.5% for graybody radiation with <math>\epsilon = 0.10</math>.

==== The Exergy Flux of Sunlight ====
In addition to the production of power directly from sunlight, solar radiation provides most of the exergy for processes on Earth, including processes that sustain living systems directly, as well as all fuels and energy sources that are used for transportation and electric power production (directly or indirectly). This is primarily with the exception of nuclear fission power plants and geothermal energy (due to natural fission decay). Solar energy is, for the most part, thermal radiation from the Sun with an emission temperature near 5762 Kelvin, but it also includes small amounts of higher energy radiation from the fusion reaction or higher thermal emission temperatures within the Sun. The source of most energy on Earth is nuclear in origin.

The figure below depicts typical solar radiation spectrums under clear sky conditions for AM0 (extraterrestrial solar radiation), AM1 (terrestrial solar radiation with solar zenith angle of 0 degrees) and AM4 (terrestrial solar radiation with solar zenith angle of 75.5 degrees). The solar spectrum at sea level (terrestrial solar spectrum) depends on a number of factors including the position of the Sun in the sky, atmospheric turbidity, the level of local atmospheric pollution, and the amount and type of cloud cover. These spectrums are for relatively clear air (α = 1.3, β = 0.04) assuming a U.S. standard atmosphere with 20&nbsp;mm of precipitable water vapor and 3.4&nbsp;mm of ozone. The Figure shows the spectral energy irradiance (W/m2μm) which does not provide information regarding the directional distribution of the solar radiation. The exergy content of the solar radiation, assuming that it is subtended by the solid angle of the ball of the Sun (no circumsolar), is 93.1%, 92.3% and 90.8%, respectively, for the AM0, AM1 and the AM4 spectrums.<ref name="ReferenceB"/>

[[File:Spectrums AM0 AM1 AM4.jpg|thumb|Typical clear sky solar spectrums for AM0, AM1 and AM4]]

The exergy content of terrestrial solar radiation<ref name="ReferenceB"/> is also reduced because of the diffuse component caused by the complex interaction of solar radiation, originally in a very small solid angle beam, with material in the Earth’s atmosphere. The characteristics and magnitude of diffuse terrestrial solar radiation depends on a number of factors, as mentioned, including the position of the Sun in the sky, atmospheric turbidity, the level of local atmospheric pollution, and the amount and type of cloud cover. Solar radiation under clear sky conditions exhibits a maximum intensity towards the Sun (circumsolar radiation) but also exhibits an increase in intensity towards the horizon (horizon brightening). In contrast for opaque overcast skies the solar radiation can be completely diffuse with a maximum intensity in the direction of the zenith and monotonically decreasing towards the horizon. The magnitude of the diffuse component generally varies with frequency, being highest in the ultraviolet region.

The dependence of the exergy content on directional distribution can be illustrated by considering, for example, the AM1 and AM4 terrestrial spectrums depicted in the figure, with the following simplified cases of directional distribution:

•	For AM1: 80% of the solar radiation is contained in the solid angle subtended by the Sun, 10% is contained and isotropic in a solid angle 0.008 sr (this field of view includes circumsolar radiation), while the remaining 10% of the solar radiation is diffuse and isotropic in the solid angle 2π sr.

•	For AM4: 65% of the solar radiation is contained in the solid angle subtended by the Sun, 20% of the solar radiation is contained and isotropic in a solid angle 0.008 sr, while the remaining 15% of the solar radiation is diffuse and isotropic in the solid angle 2π sr. Note that when the Sun is low in the sky the diffuse component can be the dominant part of the incident solar radiation.

For these cases of directional distribution, the exergy content of the terrestrial solar radiation for the AM1 and AM4 spectrum<ref name="ReferenceB"/> depicted are 80.8% and 74.0%, respectively. From these sample calculations it is evideνnt that the exergy content of terrestrial solar radiation is strongly dependent on the directional distribution of the radiation. This result is interesting because one might expect that the performance of a conversion device would depend on the incoming rate of photons and their spectral distribution but not on the directional distribution of the incoming photons. However, for a given incoming flux of photons with a certain spectral distribution, the entropy (level of disorder) is higher the more diffuse the directional distribution. From the second law of thermodynamics, the incoming entropy of the solar radiation cannot be destroyed and consequently reduces the maximum work output that can be obtained by a conversion device.

===Chemical exergy===
Similar to thermomechanical exergy, chemical exergy depends on the temperature and pressure of a system as well as on the composition. The key difference in evaluating chemical exergy versus thermomechanical exergy is that thermomechanical exergy does not take into account the difference in a system and the environment's chemical composition. If the temperature, pressure or composition of a system differs from the environment's state, then the overall system will have exergy.<ref name="moran">
{{cite book|last=Moran|first=Michael
|title=Fundamentals of Engineering Thermodynamics.|year=2010
|publisher=John Wiley & Sons Canada, Limited|location=Hoboken, N.J.
|isbn=978-0-470-49590-2|pages=816–817|edition=7th}}
</ref>

The definition of chemical exergy resembles the standard definition of thermomechanical exergy, but with a few differences. Chemical exergy is defined as the maximum work that can be obtained when the considered system is brought into reaction with reference substances present in the environment.<ref>{{cite web|last=Szargut|first=Jan|title=Towards an International Reference Environment of Chemical Exergy|url=http://www.exergoecology.com/papers/towards_int_re.pdf|access-date=15 April 2012|archive-date=25 November 2011|archive-url=https://web.archive.org/web/20111125032943/http://www.exergoecology.com/papers/towards_int_re.pdf|url-status=dead}}</ref> Defining the exergy reference environment is one of the most vital parts of analyzing chemical exergy. In general, the environment is defined as the composition of air at 25&nbsp;°C and 1 atm of pressure. At these properties air consists of N<sub>2</sub>=75.67%, O<sub>2</sub>=20.35%, H<sub>2</sub>O(g)=3.12%, CO<sub>2</sub>=0.03% and other gases=0.83%.<ref name="moran" /> These molar fractions will become of use when applying Equation 8 below.

C<sub>a</sub>H<sub>b</sub>O<sub>c</sub> is the substance that is entering a system that one wants to find the maximum theoretical work of. By using the following equations, one can calculate the chemical exergy of the substance in a given system. Below, Equation 9 uses the Gibbs function of the applicable element or compound to calculate the chemical exergy. Equation 10 is similar but uses standard molar chemical exergy, which scientists have determined based on several criteria, including the ambient temperature and pressure that a system is being analyzed and the concentration of the most common components.<ref>
{{cite journal|last=Rivero|first=R.|author2=Garfias, M.
|title=Standard chemical exergy of elements updated|journal=Energy|date=1 December 2006
|volume=31|issue=15|pages=3310–3326
|doi=10.1016/j.energy.2006.03.020}}</ref> These values can be found in thermodynamic books or in online tables.<ref>
{{cite journal|last=Zanchini|first=Enzo|author2=Terlizzese, Tiziano
|title=Molar exergy and flow exergy of pure chemical fuels|journal=Energy
|date=1 September 2009|volume=34|issue=9|pages=1246–1259
|doi=10.1016/j.energy.2009.05.007}}</ref>

====Important equations====
{{NumBlk|:|<math>\bar{e}^{ch}=\left[ \bar{g}_{\mathrm{F}}+\left( a+\frac{b}{4}-\frac{c}{2}
\right)\bar{g}_{\mathrm{O_{2}}}-a\bar{g}_{\mathrm{CO_{2}}}-\, \frac{b}{2}\bar{g}_{\mathrm{H_{2}O}(g)}
\right]\, \left( T_{0,}p_{0} \right)+\bar{R}T_{0}\, ln\left[
\frac{{{(y}_{\mathrm{O_{2}}}^{e})}^{a+\frac{b}{4}-\, \frac{c}{2}}}{\left(
y_{\mathrm{CO_{2}}}^{e} \right)^{a}\left( y_{\mathrm{H_{2}O}}^{e} \right)^{\frac{b}{2}}}
\right] </math>|{{EquationRef|9}}}}
where:
*<math>\bar{g}_{x}</math> is the Gibbs function of the specific substance in the system at <math>\left( T_{0}, p_{0} \right)</math>. (<math>\bar{g}_{F}</math> refers to the substance that is entering the system)

*<math>\bar{R}</math> is the Universal gas constant (8.314462 J/mol•K)<ref>{{cite web | title=The Individual and Universal Gas Constant | url=http://www.engineeringtoolbox.com/individual-universal-gas-constant-d_588.html |access-date=15 April 2012}}</ref>
*<math>T_{0}</math> is the temperature that the system is being evaluated at in absolute temperature
*<math>y_{x}^{e}</math> is the molar fraction of the given substance in the environment, i.e. air

{{NumBlk|:|<math>\bar{e}^{ch}=\left[ \bar{g}_{\mathrm{F}}+\left( a+\frac{b}{4} - \frac{c}{2} \right) \bar{g}_{\mathrm{O_2}}-a\bar{g}_{\mathrm{CO_2}}-\, \frac{b}{2} \bar{g}_{\mathrm{H_2 O}(g)}
\right]\, \left( T_{0,}p_{0} \right)+a\bar{e}_{\mathrm{CO_2}}^{ch}+\, \left(
\frac{b}{2} \right)\bar{e}_{\mathrm{H_{2}O}(l)}^{ch}-\, \left( a+\, \frac{b}{4}
\right)\bar{e}_{\mathrm{O_{2}}}^{ch} </math>|{{EquationRef|10}}}}
where <math>\bar{e}_{x}^{ch}</math> is the standard molar chemical exergy taken from a table for the specific conditions that the system is being evaluated.

Equation 10 is more commonly used due to the simplicity of only having to look up the standard chemical exergy for given substances. Using a standard table works well for most cases, even if the environmental conditions vary slightly, the difference is most likely negligible.

====Total exergy====
After finding the chemical exergy in a given system, one can find the total exergy by adding it to the thermomechanical exergy. Depending on the situation, the amount of chemical exergy added can be very small. If the system being evaluated involves combustion, the amount of chemical exergy is very large and necessary to find the total exergy of the system.

===Irreversibility===
Irreversibility accounts for the amount of exergy destroyed in a closed system, or in other words, the wasted work potential. This is also called dissipated energy. For highly efficient systems, the value of {{math|''I''}}, is low, and vice versa. The equation to calculate the irreversibility of a closed system, as it relates to the exergy of that system, is as follows:<ref name="ohio">
{{cite web|title=Exergy (Availability) – Part a (updated 3/24/12)|url=http://www.ohio.edu/mechanical/thermo/Applied/Chapt.7_11/Chapter7a.html|access-date=1 April 2015}}</ref>

{{NumBlk|:|<math> I = T_0 S_\text{gen} </math>|{{EquationRef|11}}}}

where <math> S_\text{gen} </math>, also denoted by {{math|''Π''}}, is the entropy generated by processes within the system. If <math>I > 0</math> then there are irreversibilities present in the system. If <math>I = 0</math> then there are no irreversibilities present in the system. The value of {{math|''I''}}, the irreversibility, can not be negative, as this implies entropy destruction, a direct violation of the second law of thermodynamics.

Exergy analysis also relates the actual work of a work producing device to the maximal work, that could be obtained in the reversible or ideal process:

{{NumBlk|:|<math> W_\text{act} = W_\text{max} - I </math>|{{EquationRef|12}}}}

That is, the irreversibility is the ideal maximum work output minus the actual work production. Whereas, for a work consuming device such as refrigeration or heat pump, irreversibility is the actual work input minus the ideal minimum work input.

The first term at the right part is related to the difference in exergy at inlet and outlet of the system:<ref name="ohio" />

{{NumBlk|:|<math> W_\text{max} = \Delta B = B_\text{in} - B_\text{out} </math>|{{EquationRef|13}}}}

where {{math|''B''}} is also denoted by {{math|''Ξ or X''}}.

For an isolated system there are no heat or work interactions or transfers of exergy between the system and its surroundings. The exergy of an isolated system can therefore only decrease, by a magnitude equal to the irreversibility of that system or process,

{{NumBlk|:|<math> \Delta B = -I </math>|{{EquationRef|14}}}}