Maxwell Relations

8 Key excerpts on "Maxwell Relations"

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  Statistical and Thermal Physics

  An Introduction

  • Michael J.R. Hoch(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  (7.50) (e) Summary of the Maxwell Relations. ThefourMaxwellrelationsaresum-marizedasfollows: fromEquation7.41, M1, ( / ) ( / ) ∂ ∂ = -∂ ∂ T V P S S V ; fromEquation7.44, M2, ( / ) ( / ) ∂ ∂ = ∂ ∂ T P V S S P ; fromEquation7.47, M3, ( / ) ( / ) ∂ ∂ = ∂ ∂ S V P T T V ; and fromEquation7.50, M4, ( / ) ( / ) ∂ ∂ = -∂ ∂ S P V T T P . As mentioned above, the Maxwell Relations are a consequence of the factthat T , S , P ,and V arenotindependent.TheMaxwellrelationsmay be written down immediately with the aid of the fundamental relation (Equation3.18)intheform d d d E T S P V = -,withthefollowingrulethat involvesthevariables T , S , P ,and V ,whichoccurontheright-handside oftherelations.IfinM1toM4 thevariables,withrespecttowhichone differentiates,occurasdifferentialsinthefundamentalrelation,aminus signmustbeusedinthecorrespondingMaxwellrelationwhileanysingle Application of Thermodynamics to Gases: The Maxwell Relations ◾ 145 permutationawayfromthisresultsinachangeinsign.Thephysicalsig-nificanceoftherelationsmaybeunderstoodfromanexaminationofthe partialderivatives.InM1(Equation7.41), ( / ) ∂ ∂ T V S givestheinfinitesimal change in temperature with volume in an isentropic (quasistatic, adia-batic)process,whereas ( / ) ∂ ∂ P S V givestheinfinitesimalchangeinpressure whenagasisheatedatconstantvolume.Thenegativesignsignifiesthat thetwoinfinitesimalchangeshaveoppositesigns.Similarconsideration may be used in interpreting the other relations. The great usefulness of theMaxwellrelationsisthatdifferentpropertiescanbeconnected,and thispermitsacompletedescriptionofasystemonthebasisofafewofits experimentallymeasuredproperties.WemakeuseoftheMaxwellrela-tionsinSections7.8and7.9.

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  Thermodynamics in Materials Science

  • Robert DeHoff(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  Recall Equation 4.24: d Z ¼ M d X þ N d Y þ P d U þ R d V þ · · · ð 4 : 24 Þ Maxwell Relations must hold for each pair of terms in this equation, since each Thermodynamic Variables and Relations 67 coefficient represents a differential of the state function, Z . Thus, for example, › R › X . / Y ; U ; V ; … ¼ › M › V . / X ; Y ; U ; … ; › R › Y . / X ; U ; V ; … ¼ › N › V . / X ; Y ; U ; … : etc : ð 4 : 28 Þ E XAMPLE 4.1 It will be shown in Section 4.2 that the form of the equation relating entropy to temperature and pressure is S ¼ S ð T ; P Þ d S ¼ C P T d T 2 V a d P Write the coefficient and Maxwell Relations for this equation. The coefficients of d T and d P are, respectively, › S › T . / P ¼ C P T › S › P . / T ¼ 2 V a The Maxwell relation is › C P T . / › P 0 B B @ 1 C C A T ¼ › ð 2 V a Þ › T . / P Evidently how the heat capacity varies with pressure is related to the coefficient of thermal expansion for a system. Two other relationships among partial derivatives that may be defined between the variables X , Y , and Z may be derived with the aid of results presented in this section. The reciprocal relation : › Z › X . / Y ¼ 1 › X › Z . / Y ð 4 : 29 Þ and the ratio relation (see Problem 4.2): › Z › X . / Y › X › Y . / Z › Y › Z . / X ¼ 2 1 ð 4 : 30 Þ These equations are useful in manipulating partial derivatives in the process of developing relations between thermodynamic state functions. As a further step in the development of the structure of thermodynamics, four classes of relationships between state functions have been presented. The laws of thermodynamics provide the physical basis for all of these relations. Definitions introduce energy functions that are convenient in specific applications and a set of Thermodynamics in Materials Science 68 experimental variables that form the core of information needed to solve practical problems. Coefficient and Maxwell Relations are consequences of the mathematical properties of differentials of state functions.

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  Atmospheric and Oceanic Fluid Dynamics

  Fundamentals and Large-Scale Circulation

  • Geoffrey K. Vallis(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  Much of classical thermodynamics follows from these two expressions. 1.5 Thermodynamic Relations 17 1.5.2 Thermodynamic Potentials and Maxwell Relations Given the fundamental thermodynamic relation, various other ‘thermodynamic potentials’ and relations between variables can be derived that prove extremely useful. The thermodynamic po- tentials are, like internal energy and entropy, functions of the state but they have different natural variables by which they are expressed. If we begin with the internal energy itself, then from (1.60) and (1.65) it follows that  = (   ) , ,  = − (   ) , ,  = (   ) , . (1.66a,b,c) These may be regarded as the defining relations for these variables; because of the connection between (1.61) and (1.65) these are not just formal definitions, and the pressure and temperature so defined are indeed related to our intuitive concepts of these variables and to the motion of the fluid molecules. If we write d = 1  d +   d −   d, (1.67) it is also clear that  =  (   ) , ,  −1 = (   ) , ,  = − (   ) , . (1.68a,b,c) We also see that  and  (and ) are the natural variables for entropy. Because the right-hand side of (1.65) is equal to an exact differential, the second derivatives are independent of the order of differentiation. That is,  2    =  2    , (1.69) and therefore, using (1.66) (   )  = − (   )  . (1.70) This is one of the Maxwell Relations, which are a collection of four similar relations that follow directly from the fundamental thermodynamic relation (1.65) and simple relations between second derivatives. (Additional Maxwell-like relations exist if we consider chemical effects.) To derive the other Maxwell Relations we will introduce thermodynamic potentials enthalpy, ℎ, the Gibbs function, g , and the free energy, .

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  Engineering and Chemical Thermodynamics

  • Milo D. Koretsky(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  We will learn more about such things in Chapter 6. Maxwell Relations Additional relations between thermodynamic properties and their derivatives can be derived from the second derivatives of the fundamental property relationships. These relations are called Maxwell Relations and can be obtained by noting that the order of partial differentiation of an exact differential does not matter. For example, we can equate the following two sets of partial derivatives of the exact differential du from the fundamental grouping 5 u, s, v 6 : B ' 'v a 'u 's b v R s 5 B ' 's a 'u 'v b s R v (5.15) Substitution of Equation (5.11) into Equation (5.15) gives: a 'T 'v b s 5 2a 'P 's b v (5.16) Similarly, from the other three fundamental property relationships we get: a 'T 'P b s 5 a 'v 's b P (5.17) a 's 'v b T 5 a 'P 'T b v (5.18) 2a 's 'P b T 5 a 'v 'T b P (5.19) Can you verify these last three relations? In the Maxwell Relations given by Equations (5.18) and (5.19), all the properties on the right-hand side are measured properties! These permit the calculation of change in entropy from the measured PvT data. The derivative relations of Equations (5.6), (5.7), and (5.9) then enable us to calculate changes in u, h, and g. 272 ► Chapter 5. The Thermodynamic Web Other Useful Mathematical Relations In this section, we present three other mathematical relations that will be of use in helping us surf the thermodynamic web. The first relationship is the chain rule, which can be written in general as follows: a 'z 'x b a 5 a 'z 'y b a a 'y 'x b a (5.20) Derivative inversion allows us to flip partial derivatives as follows: a 'x 'z b y 5 1 a 'z 'x b y (5.21) We are not through yet. We can derive an additional relation based on the mathematical behavior of state functions.

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  Statistical Mechanics Made Simple

  A Guide for Students and Researchers

  • Daniel C Mattis(Author)
  • 2003(Publication Date)
  • WSPC

    (Publisher)

  Further refinements are due to a French military engineer, Sadi Carnot, and an Englishman, James P. Joule. Their names remain in the core vocab-ulary of physical sciences and engineering. 3.3. Maxwell Relations Let us differentiate the quantities in (3.10) once more, making use of the identity d 2 f(x 1 y)/dxdy = d 2 f{x,y)/dydx, valid wherever / i s an analytic function of its variables. For example, at constant N: dp df This establishes the first of four Maxwell Relations (all of them obtained in similar fashion). They are, dp/dT v = dy/dV T (3.11a) 8T/dVy = -dp/dy v (3.11b) dV/dy p = dT/dp y (3.11c) dy/d P T = -dv/dT p . (3.iid) Their number can be augmented by additional degrees of freedom, such as the magnetic variables in the following example. Problem 3.1. Prove (3.11b-d). Then extend E and the various free ener-gies by adding — B • dM to each. How many new Maxwell Relations can one derive involving B and/or M and what are they? 3.4. Three Important Laws of Thermodynamics (1) The First Law: Energy is Conserved. (i) Suppose work dW is done on a closed system and heat dQ is simulta-neously introduced into the same system, whilst carefully maintaining 36 3. Elements of Thermodynamics thermodynamic equilibrium. The First Law states that the change in the thermodynamic function E, the internal energy, is dE = dW + dQ. (ii) If the work is performed adiabatically, such that it preserves all classical adiabatic invariants or internal quantum numbers (i.e. in such a way that the number of accessible configurations remain constant), then S? remains constant and dE = — pdV = dW for mechanical work. c It follows that dQ = 0. (For work performed by an external field on a magnetic substance one has dE = —B * dM instead, etc.) (iii) If, on the other hand, heat is introduced through the walls with no change in volume, magnetization, nor in any other such variable, dW = 0 hence dE = TdS?.

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  Thermodynamics for Engineers

  • Kaufui Vincent Wong(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  7 -1 7 Thermodynamic Property Relations Some.thermodynamic.properties.can.be.measured.directly,.but.most.cannot . .It.is.clear. that.it.is.necessary.to.develop.relations.between.these.two.groups.of.properties . .The.deri-vations.are.based.on.the.principle.that.properties.are.point.functions,.and.the.state.of.a. simple,.compressible.system.is.completely.specified.by.any.two.independent,.intensive. properties. 7.1 The Maxwell Relations The.Maxwell.relations.are.the.equations.that.relate.the.partial.derivatives.of.properties. P,.v,.T,.and.s.of.a.simple.compressible.system.to.each.other . .They.are.derived.from.the. four.Gibbs.equations . If.z.is.a.function.of.x.and.y,.then . dz z x dx z y dy y x = ∂ ∂ uni239B uni239D uni239C uni239E uni23A0 uni239F + ∂ ∂ uni239B uni239D uni239C uni239E uni23A0 uni239F . (7.1) or . dz M dx N dy = + . (7.2) But . ∂ ∂ uni239B uni239D uni239C uni239E uni23A0 uni239F = ∂ ∂ ∂ ∂ ∂ uni239B uni239D uni239C uni239E uni23A0 uni239F = ∂ ∂ ∂ M y z x y and N x z y x x y 2 2 so . ∂ ∂ uni239B uni239D uni239C uni239E uni23A0 uni239F = ∂ ∂ uni239B uni239D uni239C uni239E uni23A0 uni239F M y N x x y . (7.3) 7 -2 Thermodynamics for Engineers Two.of.the.Gibbs.relations.are.expressed.as . du T ds P dv = -. (7.4) . dh T ds v dP = + . (7.5) The. other. two. Gibbs. relations. are. based. on. two. new. combination. properties . . The. Helmholtz.function.b.and.the.Gibbs.function.g.are.defined.as . b u Ts = -. (7.6) . g h Ts = -. (7.7) Differentiating,.we.obtain . db du T ds s dT = --. dg dh T ds s dT = --From.these.relations.and.Equations.7 .4 .and.7 .5, .we.obtain.the.other.two.Gibbs.relations. for.simple.compressible.systems: . db s dT P dv = --. (7.8) . dg s dT v dP = -+ . (7.9) It.can.be.seen.that.the.four.Gibbs.relations.are.of.the.form.of.Equation.7 .2, .so.the.relation. in.Equation.7 .3 .is.true . .Since.u,.h,.b,.and.g.are.properties.and.thus.have.exact.differen-tials,.we.apply.Equation.7 .3 .to.each.of.them.to.obtain .

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  Statistical Mechanics

  An Intermediate Course

  • G Morandi, E Ercolessi;F Napoli;;(Authors)
  • 2001(Publication Date)
  • WSPC

    (Publisher)

  (1.2.3), (1.2.5), (1.2.8), (1.2.11)a (1.2.14), (1.2.20)amon.titute what ere aalled the Maxwell Relations af Classical Thermodynamicsy Not all of thee are actually indeprndent relations, although it iy cot easy to pinpoint thn truly independent relations when the integrability conditions are written in this formn Lgt'a discuss this ooinc in some morn detail The integrability conditions can be written [129] xnder the form of aqualities between jacobian determinants (see Appendix 1C). As an ex- ample, let's consider: dE = TdS-pdV + pdN . (1.2.21) Then we must have: and another two similar conditions given by Eqs. (1.2.3). (1.2.22) 20 Thermodynamics We nan tewrite n 129] both sides of (1.2.22) as equalities between 3 x 3 jacobian determinants, i.e. as: d(p,S,V) = d(p,S,N) ^ d(iM,S,N) d(N,S,yt d(V,S,Nh d(N,S,V) Multiplying both sides by the inverse of the jacobian : eterminant on the r.h.s., Eq. (1.2.23) become identical with the condition: 5(p,N,S) = 1 (1 2 2 4 ) y(i*,N,S) hthceeding in a similar way wi te the ot t er integrability conditions, and with the other erermodynamic potentials, rhe whole body of the Maxwell Relations can be rewritten in a form sinilar to Eq. (1.2.24). Remark. Wri t ing the Maxwe ll relations in 9]e f orm of jacobian determioants al- lows for a neat geometric i nterpretation [129] of the integrability conditions they represent Indeed, e.g. changing independent variables from (p, V, T) to (u, N, T) , which amounts to switching to a different "ch a rt" in the l ocal description or our t hermodynamic system, the jacobian determinant of the coordinate transforma- tion must be unity (see Problem 1.1).

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  Fundamentals of Thermodynamics

  • Claus Borgnakke, Richard E. Sonntag(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  12.10, we can write the relations ( u s ) v = T , ( u v ) s = −P (12.20) W-33 Similarly, from the other three equations, we have the following: ( h s ) P = T , ( h P ) s = v ( a v ) T = −P, ( a T ) v = −s ( g P ) T = v, ( g T ) P = −s (12.21) As already noted, the Maxwell Relations just presented are written for a simple com- pressible substance. It is readily evident, however, that similar Maxwell Relations can be written for substances involving other effects, such as surface or electrical effects. For example, Eq. 6.9 can be written in the form dU = T dS − P dV + F dL + dA + EMF dZ + · · · (12.22) Thus, for a substance involving only surface effects, we can write dU = T dS + dA and it follows that for such a substance ( T A ) S = (  S ) A Other Maxwell Relations could also be written for such a substance by writing the property relation in terms of different variables, and this approach could also be extended to systems having multiple effects. This matter also becomes more complex when we consider apply- ing the property relation to a system of variable composition, a topic that will be taken up in Section 12.9. Example 12.2 From an examination of the properties of compressed liquid water, as given in Table B.1.4 of Appendix B, we find that the entropy of compressed liquid is greater than the entropy of saturated liquid for a temperature of 0 ∘ C and is less than that of saturated liquid for all the other temperatures listed. Explain why this follows from other thermodynamic data. Control mass: Water. Solution Suppose we increase the pressure of liquid water that is initially saturated while keeping the temperature constant. The change of entropy for the water during this process can be found by integrating the following Maxwell relation, Eq. 12.19: ( s P ) T = − ( v T ) P Therefore, the sign of the entropy change depends on the sign of the term (v/T) P .

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