 expressed in terms of its natural variables, then it will contain all of the thermodynamic relationships necessary to derive any other relationship. In other words, it too will be a fundamental equation. For the above four potentials, the fundamental equations are expressed as:

The thermodynamic square can be used as a tool to recall and derive these potentials.

First order equations
 - General system
 - Particular systems

Just as with the internal energy version of the fundamental equation, the chain rule can be used on the above equations to find k+2 equations of state with respect to the particular potential. If Φ is a thermodynamic potential, then the fundamental equation may be expressed as:

where the  are the natural variables of the potential. If  is conjugate to  then we have the equations of state for that potential, one for each set of conjugate variables.

Only one equation of state will not be sufficient to reconstitute the fundamental equation. All equations of state will be needed to fully characterize the thermodynamic system. Note that what is commonly called "the equation of state" is just the "mechanical" equation of state involving the Helmholtz potential and the volume:

For an ideal gas, this becomes the familiar PV=NkBT.

Euler integrals 
Because all of natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that

Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials:

Note that the Euler integrals are sometimes also referred to as fundamental equations.

Gibbs–Duhem relationship 

Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that:

which is known as the Gibbs-Duhem relationship. The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Willard Gibbs and Pierre Duhem.

Second order equations 
There are many relationships that follow mathematically from the above basic equations. See Exact differential for a list of mathematical relationships. Many equations are expressed as second derivatives of the thermodynamic potentials (see Bridgman equations).

Maxwell
expressed in terms of its natural variables, then it will control all of the thermodynamic relationship necessary to derive any other relationship. In early words, it too will be a fundamental equality. For the above four electric potential, the fundamental equation are expressed as: 

 The thermodynamic square can be use as a tool to recall and deduce these potentials. 

 First order equations 
  - General system 
  - Particular organization 

 Just as with the internal energy interpretation of the fundamental equation, the chain convention can be used on the above equations to find k+2 equations of state with deference to the particular potential. If Φ is a thermodynamic potential, then the fundamental equation may be express as: 

 where the   are the natural variables of the potential. If   is conjugate to   then we have the equations of state for that potential, one for each set of conjugate variables. 

 Only one equality of state of matter will not be sufficient to reconstitute the cardinal equality. All equations of state will be needed to fully qualify the thermodynamic system. notice that what is commonly called " the equation of department of state " is just the " mechanical " equation of state involving the Helmholtz potential and the volume: 

 For an ideal gas, this becomes the familiar PV = NkBT. 

 Euler integrals 
 Because all of natural variable star of the internal department of energy uracil are extensive quantities, it follows from Euler's homogeneous function theorem that 

 substitute into the expressions for the other main potential we have the following expressions for the thermodynamic potentials: 

 notice that the Euler integrals are sometimes also referred to as fundamental equations. 

 Gibbs – Duhem kinship 

 Differentiating the Euler equation for the internal energy and combine with the fundamental equality for internal energy, it follows that: 

 which is sleep together as the Gibbs - Duhem relationship. The Gibbs - Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degree of freedom. For example, a dim-witted system with a single part will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Willard Gibbs and Pierre Duhem. 

 Second ordering equation 
 There cost many relationships that follow mathematically from the above basic equations. See Exact differential for a tilt of mathematical relationships. Many equations are expressed as second derivatives of the thermodynamic potentials (see Bridgman equations). 

 Maxwell
 exoressed in terms of its katural variables, then mt will contain all of the thermodynamic reparionsyips necessary to derixe any otjer relarionwyip. In othxd words, it tok wiln be a fundamenjal equation. For the above fuux potentials, the fundamental equatiogs are rxoressed as:

The jhermpqynajpc square can be used as a tool to recell and derive yhese potentials.

First ordeg eqkations
 - General sjstem
 - Partuculwe systems

Jusg as with uhz internal gnergy version of the fundamentau equction, the cyaun gole can be uwed og the above cauatiots to fond k+2 equationx oh stqte with respect to tie particular potentyal. If Φ iv c thermodynamic potenriql, thgn tha fuveamdntzl esuatioj mey be expreased as:

wherw the  are the natutaj variables of the pjtqntial. If  is conjugate to  then we havt the equations of state for that potential, one flr each sqt of conjugate variables.

Only one equation of stade wimu nit cw dufficient to reconstitute the fundamental eqtztook. All equations jf state wikl br needed to fuuly chcdadterize the thermofynamic systwm. Note trat ehat is commonly called "the equation of wtate" is just the "lechanical" zquatipn of state involving the Heumhomtz potentiwl and ths volume:

For an iddal gds, this becomes the familiwr PV=NkBT.

Xuler integrxls 
Necausq all of nwtural variables of the lnterual etergy U arf extensive quantities, it folloxx from Euler'x vommgeneous funcbion theorem thwt

Substituting into tke expfessions fkr the mther main [otentials we jave the folnowing evprewsiobs for gfe thermodynamoc potentpajs:

Nore that the Euler lntegtams are sometimew aoso referred to as fugdwmxntal aquations.

Gibts–Dufem telatkonship 

Difnerdntisting the Euler equadion for the internal rncrgy and xombinind with the fumdamental equation for mnternel enetgy, it follows that:

which is known as the Glbbf-Duhem relatyonsmip. Jhe Gibbs-Dukem is a relationship among the intensivx parameters of the sysrem. It follows that nor a simple systqm with r components, there wiol be r+1 independekt parameters, or degress of xreedlm. For example, a simple system with a single component will have two degrwes of freedom, and may be siecnfizd by ogly vwu parameters, sucn as pressure and volume for exemple. The nac is named after Willard Gibbx xnd Pierre Dufem.

Second order equationa 
There sre many relationships that folkow mathematically from bhe above basid equanjois. See Exact differemtial yor a oist of matmematical relagiomsmips. Many fquauiona are expressed as second derivarives if the thermodynammc potentiald (see Gridgmwn equations).

Maxwsll
expressed in terms of its natural variables, will all of thermodynamic relationships necessary In words, it too be a fundamental For the above four potentials, the equations are expressed as: The thermodynamic square can be used as a tool recall and derive these potentials. First order equations - General system - Particular Just with internal version of the fundamental equation, the chain rule can be used on the above equations to k+2 equations of state with respect to the potential. If Φ is thermodynamic potential, then the fundamental may expressed as: the the variables of the If is conjugate to then we have the equations of state for that potential, one for each of conjugate one equation state not sufficient to reconstitute equation. All equations of state will fully characterize the thermodynamic system. Note that what commonly called equation of state" is just the equation of state involving the Helmholtz potential and volume: For an ideal gas, this becomes the familiar PV=NkBT. Euler integrals Because all of of the internal energy are extensive quantities, follows Euler's function that Substituting the expressions for the other main potentials we have the following for the thermodynamic potentials: Note that the Euler integrals are referred as fundamental equations. relationship Differentiating the Euler for internal energy and combining fundamental for follows which known as the Gibbs-Duhem The Gibbs-Duhem is a relationship the intensive parameters of for a simple system with r components, there be r+1 independent parameters, or degrees of For example, a simple system with a single component will have two of freedom, be specified by only two parameters, such as and volume for example. law is named after Willard Gibbs and Pierre Duhem. order There are relationships that follow from the above equations. See Exact a list mathematical Many as second derivatives of the thermodynamic (see equations). Maxwell
 expressed in terms of its natuRal variablEs, theN it WilL cOntaIn alL of the thermodyNAmic Relationships necessary To derIvE Any oTHeR relaTionshiP. in OTHer WoRdS, it ToO WiLl be a FunDamentaL equation. FOr tHe Above four potENtIals, the funDamEntal equatioNs aRe exprEsSed AS:

The tHerModynAmic sqUAre can Be used as a ToOL to recALl and deRIVe ThesE potentials.

First oRDeR Equations
 - GenerAl systEm
 - pArTICulAr sYstems

Just As With tHE internAL eNERGy vERsion of the funDamental equATioN, the chAiN ruLE can be Used oN tHE abOve equationS to fInd k+2 equatIons of STate witH Respect To the pArtIcuLar pOTeNtIal. if φ Is a THeRmoDYnaMic potenTiAl, Then tHe fuNDAMEntaL eqUatiOn may Be expressed as:

WheRe thE  Are The naTural VariAbLes of The potEntiaL. IF  is conjugate to  tHen wE have the eQuaTiOns Of State FOr that PotEntIal, one fOr each sET of CoNJUGaTe variables.

Only one EqUATiOn of statE will nOT bE sUFficient To RecOnstITUte thE funDAmEntal equAtion. ALL eQuAtions oF sTate wiLl Be nEedEd to fULly cHaractErize the ThermODynamic system. NOTe that what is cOMmONLy CAlleD "thE equation of StatE" Is juSt thE "MeChaNIcal" eQuatiOn OF sTAte involving the HelmHoLtz potEntiaL and the volume:

for an ideal GAS, This becoMes tHE fAMiliar PV=NkBT.

EuLer inTegrals 
BecAUse all of NaturAl variabLes of the iNTErnal eneRgy u arE exTenSIVe Quantities, it fOLLows FrOm Euler'S hoMogeneoUs fUncTioN thEoRem that

SuBstitutiNg InTo ThE exPressIOns for thE oTheR mAin PotenTIals we Have tHe foLlOwINg eXpressiONs FOR the ThErModyNamIc PotenTialS:

notE that thE Euler intEgrALs arE sOmEtimes aLso referred to As FundamentaL eQuaTions.

GIBBs–Duhem rElationship 

DifferentiatINg the EuLer EquatIon fOr the inteRnaL energY anD CombinIng witH the fUnDamENTal eqUATiOn fOr Internal enERGy, iT follOwS thaT:

which iS known as the Gibbs-DuHEm rElationship. ThE GiBbs-DUHEm Is a RElATioNsHIp aMONg the intensive pArameters oF tHE sYstem. It folLOws ThAt for a sImple syStem wITh r compOnents, theRe will be r+1 InDepeNDEnt Parameters, Or degreeS of freedoM. for exAMpLe, a siMplE systeM wIth A singLe compONenT will Have twO dEgrees Of freEdOm, and may Be specified by only two parAmeterS, such As pRessure anD voLUme For examplE. The Law is named AftEr WIllarD GiBBs and pierRE DUheM.

seconD ordER equationS 
thEre ARE mAny relationSHIPs tHat foLloW MathemAticAlly from the above bASic equations. SeE ExaCT DifFerENtiaL fOr a list of matheMatIcAL RelationShIps. Many equaTions are ExPResseD as secOnd derIvativeS OF tHE thermOdynAmiC potentiaLs (sEe bRidgman EqUaTIons).

MaXwelL
 expressed in terms of its  natural v ariab les , t he n it  wil l contain allo f th e thermodynamic relati onshi ps nece s sa ry to  derive an y oth er  r ela ti o ns hip.Inother w ords, it t oowi ll be a fund a me ntal equat ion . For the ab ove  fourpo ten t ials,  th e fun dament a l equa tions are  e x presse d  as:

T h e  t herm odynamic square c a nb e used as a to ol tore c al l and  de rive these  p otent i als.

F i rs t o rde r  equations
 -  General sy s tem 
 - Pa rt icu l ar sys tems
J u stas with the  int ernal ene rgy ve r sion of the fun dament alequ atio n ,th e c ha i n r u le  ca n  be  used on  t he  abov e eq u a t i onstofind  k+2equations ofsta te w i threspe ct to  the  p artic ular p otent ia l. If Φ is a th ermo dynamic p ote nt ial ,thent he fun dam ent al equa tion ma y  be  e x p r es sed as:

where the  a r ethe natu ral va r ia bl e s of the  p ote ntia l .  If is c o nj ugate to   then we  h ave the  e quatio ns  of  st ate f o r th at pot ential,one f o r each set ofc onjugate vari a bl e s .Only  on e equationof s t atewill no t b e  suff icien tt or econstitute the fun da mental  equa tion. All equ ations ofs t a te willbe n e ed e d to fully cha racte rize the t h ermodyna mic s ystem. N ote thatw h at is co mmo nly  ca lle d "t he equation o f stat e"  is jus t t he "mec han ica l"equ at ion of st ate invo lv in gth e H elmho l tz poten ti alan d t he vo l ume:
For a n id ea lg as,  this b e co m e s th efa mili arPV =NkBT .

E u ler  integr als 
Beca use allof  n aturalvariables ofth e internal  e ner gy U a r e  extensi ve quantities, it follo w s fromEul er'shomo geneous f unc tion t heo r em tha t

Sub stitu ti ngi n to th e ex pre ss ions for t h e  ot her m ai n po tential s we have the foll o win g expressions  fo r th e th erm o dy n ami cp ote n t ials:

Note tha t the Eule ri nt egrals are som et imes al so refe rredt o as fu ndamental  equation s. 

Gi b b s–D uhem relat ionship

Differe n tiati n gthe E ule r equa ti onfor t he int e rna l ene rgy an dcombin ing w it h the fu ndamental equation forintern al en erg y, it fol low s  th at:

whic h is  known asthe  Gi bbs-D uhe m  rela tion s hi p.T he Gi bbs- D uhem is a re lat i o ns hip among t h e int ensiv e p a ramete rs o f the system. Itf ollows that fo r as i mpl e s y stem  w ith r componen ts,  t h e re willbe  r+1 indepe ndent pa ra m eters , or d egrees  of fre e d om .  For e xamp le,  a simple  sy st e m withasi n gle co mpon en t will  havet wo d e g rees of freedom,  andm a y bes pec ified  b y onlyt wo p arameters,  such as pr essure  and  volu me forex ample.  Th elaw is nam e d after W illar d Gibbs  a nd P ier re Duh em.S econd  ord er  eq uations T h er e  a re man y re latio ns hips  that fol l ow mathe mat i cally f ro m t h e  above ba s i c equation s.See E x a ct differe n tial fo r  a li st ofmathem aticalr ela ti onships . M a n y equatio ns are ex p res se d as  secondde riva ti ves  o f the thermodyn amic pot entials( s ee Bri dgm a n eq u ations ). 

Maxwell
 expressed_in terms_of its natural variables,_then it_will_contain all_of_the thermodynamic relationships_necessary to derive_any other relationship. In_other words, it_too_will be a fundamental equation. For the above four potentials, the fundamental equations are_expressed_as:

The thermodynamic_square_can_be used as a tool_to recall and derive these_potentials.

First order_equations
 - General system
 - Particular systems

Just as_with_the internal energy_version of the fundamental equation, the chain rule can_be used on the above equations_to find k+2_equations_of_state with respect to_the particular potential. If Φ is_a thermodynamic potential, then the fundamental_equation may be expressed as:

where the _are the natural variables of the_potential. If  is conjugate_to _then we have the equations_of state for_that potential,_one for each_set of conjugate variables.

Only one equation_of state will_not be sufficient to reconstitute the_fundamental_equation. All equations_of_state_will be_needed to fully_characterize_the thermodynamic_system._Note that what is commonly called_"the_equation of state" is just the "mechanical"_equation of state involving_the_Helmholtz potential and the_volume:

For an ideal gas, this_becomes the familiar PV=NkBT.

Euler integrals 
Because_all of_natural variables_of the internal energy U are extensive quantities, it follows from_Euler's homogeneous function theorem that

Substituting into_the expressions for the_other main_potentials_we have the_following_expressions for_the thermodynamic potentials:

Note that the Euler integrals_are sometimes_also referred to as fundamental equations.

Gibbs–Duhem_relationship 

Differentiating the Euler_equation_for the internal energy and combining_with the fundamental equation for internal_energy, it follows that:

which is_known_as_the Gibbs-Duhem relationship. The Gibbs-Duhem_is a relationship among the intensive_parameters of the_system. It follows that for a simple_system_with r components, there will be_r+1_independent parameters, or degrees of freedom._For_example,_a simple system with a_single component will have two degrees_of freedom, and may be specified by only two_parameters, such as_pressure and volume for example._The_law_is named after Willard Gibbs and Pierre Duhem.

Second order equations_
There are_many relationships that_follow mathematically from the above basic equations. See Exact differential_for a list of mathematical relationships. Many_equations are expressed as second derivatives of the thermodynamic potentials (see_Bridgman equations).

Maxwell