Chapter 4. Fundamental Equations

Transcription

1 Chapter 4. Fundamental Equations wo ke problems in thermodnamics are to relate properties to each other and to develop was to measure them accuratel in the laborator. For eample, we have seen in Chapter 2 that the relationship between the heat capacities at constant volume and at constant pressure for an ideal gas is: C IG = C IG + R It is highl desirable to obtain a general relationship between the two heat capaci- ties, C = ( U / ) and C = ( H / ), for non- ideal gases, liquids and solids. In the present chapter, we develop a formal approach that allows us to obtain such relationships between thermodnamic de- rivatives in a sstematic fashion. An important consideration in developing thermodnamic relationships and methods for determining derived thermodnamic functions such as the energ, enthalp, or entrop, is the preservation of information content as one moves from one function to another. In the rest of this chapter, we demonstrate how all these functions, when epressed in terms of their natural variables, are essential- l equivalent to each other and encompass all available information about equilib- rium states of a sstem. hese equivalent, complete descriptions of thermodnam- ic properties are called fundamental equations and are the main topic of the pre- sent chapter. 4.1 hermodnamic Calculus As alread mentioned in Chapter 1, for a given quantit of a one- component ss- tem at equilibrium, all properties are completel determined b specifing the values of two additional independent thermodnamic variables. More generall, n+2 independent thermodnamic variables, where n is the number of components in a sstem, are needed to full characterie equilibrium states for multicompo- nent sstems. Common thermodnamic variables used to specif equilibrium Draft material from Statistical hermodnamics, (2012), A.Z. anagiotopoulos 1

2 2 Chapter 4. Fundamental Equations states in such sstems are the total volume, number of moles of each component, N 1, N 2, N n, temperature and pressure. he total entrop S can also be specified and controlled with some effort, namel b performing reversible processes and using the definition developed in Ch. 3, to measure entrop changes. ΔS = δqrev / It is clear from empirical observations that not all combinations of variables result in a unique specification of the state of thermodnamic sstems. his is es- peciall apparent when the possibilit of coeistence between phases of different properties is taken into account. For eample, at a given temperature and at the saturation pressure in a one- component sstem, there are infinitel man was in which two phases can coeist as equilibrium thermodnamic states differing in the relative amounts of the two phases. In other words, specifing the pressure and temperature at vapor- liquid equilibrium of a pure fluid does not uniquel specif the relative amounts of liquid and vapor. Similarl, a given pressure and volume (or densit) of a binar miture can be realied in man different was for a range of temperatures in a two- phase sstem. In addition to the issue of the uniqueness of thermodnamic states, certain privileged sets of variables are natural for spe- cific properties. hermodnamic functions epressed in their natural variables contain more complete information than the same properties epressed in other sets of variables. In order to eplain wh this is the case, let us consider the com- bined statement of the First and Second Laws for closed sstems derived in 3.4: du = ds d (4.1) his epression suggests that the volume and entrop constitute a good set of variables for describing the energ U of a closed sstem. More formall, we write: he derivatives are identified as: U = U(S, ) du = U U = and U ds + U S S d (4.2) = (4.3) o obtain the full epression for U as a function also of the amount of material in a sstem in addition to S and, we need to augment Eq. 4.2 b additional varia- bles, namel the amount of material of each component present in the sstem. For this purpose, we define an additional partial derivative, written here for a one- component sstem:

3 4.1 hermodnamic Calculus 3 µ U N S, (4.4) his derivative defines an important new thermodnamic variable, the chemi- cal potential μ of a component. he chemical potential is an intensive variable that has units of specific energ, [J/mol]. Its phsical interpretation in classical thermo- dnamics is somewhat obscure when compared to the two other more intuitivel understood first- order derivatives of the energ function given b Eqs How- ever, the chemical potential plas just as fundamental a role in thermodnamics as temperature and pressure. An eample of a relativel simple instrument that measures chemical poten- tials is the ph meter, which man of us have used in chemistr laboratories. he instrument measures the chemical potential of hdrogen ions, not their concentra- tion. Of course, the two are related, with the chemical potential increasing with increasing concentration but there is a clear distinction. After all, densit also increases with pressure, but pressure and densit are not usuall confused to be the same phsical quantit Another wa to develop a phsical feeling for the chemical potential is to consider our senses of smell and taste. What eactl is being measured when ou find that a food tastes sour, or when ou smell a fra- grance from a flower? One ma naivel suggest that it is the concentration of the molecules that correlates with the intensit of taste and smell, but in realit it is the chemical potential of the corresponding components that drives the biochemi- cal receptors responsible for these senses. In addition to chemical reactions, diffu- sion and evaporation are also driven b chemical potential (not concentration) differences. On a hot and humid summer da, sweating does not cool ou down because there are no chemical potential differences to drive evaporation, despite the large concentration difference for water between wet skin and air. Using the newl defined derivative of Eq. 4.4, we can now write a complete dif- ferential relationship for the energ as a function of entrop, volume and number of moles, U = U(S,,N): du = ds d + µdn (4.5) More generall, for a multicomponent sstem, one needs n+2 variables to full characterie the state of a sstem. When the function of interest is the total energ U, these variables can be selected to be the total entrop, S, volume, and amount of moles for each component present, {N} N 1,N 2, N n : n du = ds d + µ i dn i (4.6) i=1 he chemical potential μ i of a component in a miture is defined from a direct etension of Eq. 4.4, funda- mental equation

4 4 Chapter 4. Fundamental Equations µ i U (4.7) N i S,,N j i In this equation, N j i indicates that the number of moles of all components other than i are kept constant; more eplicitl, N 1,N 2, N i 1,N i+1, N n are constant while N i varies. Because the chemical potential definition implies an open sstem to which a component is added, its value depends on the reference state for energ for the corresponding component and thus is not directl measurable, unlike the first two derivatives of Eq. 4.6, temperature and pressure. Eq. 4.6 is the single most important equation of this book and is known as the fundamental equation of thermodnamics (abbreviated FE ). It links the ke quan- tities of the First and Second Laws and provides the starting point for understand- ing equilibrium and stabilit as well as for deriving thermodnamic relations. S, and {N}, the variables in which the function U is epressed in this representation, are all etensive, proportional to sstem sie (mass). he first derivatives,, and μ i's are all intensive, independent of sstem sie. An equivalent representation of the entrop S as a function S(U,,N 1,N 2, N n ) can be obtained b rearrangement of the terms of Eq. 4.6: n µ i ds = 1 du + d dn (4.8) i i=1 his representation of the fundamental equation is used in statistical mechanics. Eample 4.1 he fundamental equation for a pure substance is S = au N bn 3 U where a and b are positive constants. Obtain the equation of state, as a function of and, for this material. Starting from the FE in the entrop representation (Eq. 4.8), we obtain the pres- sure and temperature as: 1 = S U,N = a N + bn 3 U 2 and = S U,N = au N + bn 3 U 2 (i) aking the ratio of the two epressions in (i), Adapted from roblem in Callen, H. B, hermodnamics and Introduction to hermo- statistics, 2 nd Ed., Wile (1985).

5 4.1 hermodnamic Calculus 5 / 1/ = = au N + bn 3 U 2 a N + bn 3 U 2 = au bn 4 NU 2 au bn 4 NU 2 = U U = (ii) substituting (ii) into (i), ( ) = b = a N + bn 3 = a + b a 4 = b 3 a Manipulation of hermodnamic Derivatives Deriving relationships between thermodnamic derivatives is the ke objective of the present chapter. Manipulations of thermodnamic derivatives can seem quite daunting at times, but can be accomplished sstematicall with a small number of relativel simple mathematical tools. hermodnamic functions, after all, are mathematical objects that are subject to the same rules of multivariable calculus as an other functions, even if we frequentl do not have eplicit, closed- form e- pressions for them in terms of their independent variables. In the following, we will consider,,, and w to be mathematical objects with well- behaved functional relationships between them. We will also assume that there are two independent variables in this set and that an two of,,, and w can serve as the independent variables. his assumption is made for simplicit and without loss of generalit. he following general mathematical relationships are often useful for manipulating derivatives of functions of man variables in general and thermodnamic functions in particular: Inversion = 1 (4.9) ( / ) his relationship is for the underling functions rather than simple algebraic in- version; (,) and (,) that are being differentiated on the two sides of Eq. 4.9 are epressed in terms of different variables (see Eample 4.2 for an illustra- tion of this concept).

6 6 Chapter 4. Fundamental Equations Commutation = 2 = (4.10) In other words, mied second derivatives are independent of the order in which the are taken. his propert applied to thermodnamic functions leads to equali- ties between derivatives known as Mawell s relations. Chain rule = w w = inversion rule ( / w) / w ( ) (4.11) In Eq we have changed the independent variables from (,) in the left- hand- side to (w,) and (w,) in the right- hand- side. riple- product rule (also known as XYZ- 1 rule) his relationship links three cclicall permuted derivatives: = 1 (4.12) Here, the three functions being differentiated, (,), (,), and (,) epress the same underling relationship in terms of different variables. o prove this equalit, let us consider the differential form of the function = (,) : d = at constant, this epression becomes: d + d + d d = 0 Now let us obtain the derivative / ) of this relationship: + = 0 =

7 4.2 Manipulation of hermodnamic Derivatives 7 = 1 Eample 4.2 Demonstrate how the inversion, commutation, chain and triple- product rules ap- pl to the functions: (,)= 2 ; w = (variable transformation for the chain rule) For the function in the original representation, = (,) : ( ) = 2 / / o test the inversion rule, we obtain the function in the representation = (,) : = 2 2 = = ± = ± 2 Now we need to eliminate : ± = ± = / 2 = 1 / ( ) = 1 ( / ) For the commutation rule (independence on the order of differentiation), we have 2 = 2 = 2 2 = 2 2 For the chain rule, the new variable is defined as w =. hen: = w w = = 2 So that ( / w) = 1 = ± w ( / w) = ± 1 2 w ( / w) / w ( ) = 1/ = 2 ±1/2 w = ( / )

8 8 Chapter 4. Fundamental Equations For the triple- product rule we obtain: = 2 = ± = 2 = 2 / = ± = 2 2 = 2 = = = 1 2 One could have substituted variables other than and in the epressions for the three cclic derivatives with identical results. 4.3 Euler s heorem for Homogeneous Functions hermodnamic functions and derivatives have special properties resulting from the fact that the are either etensive (proportional to the sie of the sstem) or intensive (independent of the sie of the sstem). Mathematicall, functions that satisf the relationship f (λ) = λf () for an are called homogeneous of degree one. Functions can be homogeneous of degree one with respect to some, but not all the variables for eample, a function that is homogeneous of degree one with respect to variables 1, 2,, i and homogenous of degree ero with respect to variables 1, 2,, j has the propert: f (λ 1, λ i, 1, j ) = λf ( 1, i, 1, j ) foran 1, i, 1, j (4.13) Etensive thermodnamic functions are homogeneous of degree one with respect to their etensive variables, and homogeneous of degree ero with respect to their intensive variables. Euler s theorem for homogeneous functions provides a link between such func- tions and their derivatives, as follows: f f f f ( 1, i, 1, j ) = i (4.14) 2 i he proof of this theorem is as follows.

9 4.3 Euler s heorem for Homogeneous Functions 9 f (λ 1, λ i, 1, ) λ = f (λ 1, λ i, 1, ) (λ 1 ) = λf (,,, ) 1 i 1 λ (λ 1 ) λ = f ( 1, i, 1, ) + + f (λ 1, λ i, 1, ) (λ i ) = f (,,, ) 1 i f (,,, ) 1 i 1 1 i i (λ i ) Application of Euler s theorem to U(S,,N 1,N 2,,N n ), a homogeneous function of degree one with respect to all its variables, gives: λ = U = S U,{N } + U S,{N } n U + N i i=1 N i S,,N j i n U = S + µ i N i (4.15) i=1 Euler- integrated FE his Euler- integrated form of the fundamental equation is sometimes confusing, as it superficiall appears to suggest that U is a function of twice as man variables as previousl ( and S, and, μ i and N i). his is incorrect; despite appearances to the contrar, the same variable set as before, (S,,N 1,N 2,,N n ), is being used, with temperature, pressure and chemical potentials that appear in the equation also being functions of the same variables: (S,,N 1,N 2,,N n ), (S,,N 1,N 2,,N n ), and μ i (S,,N 1,N 2,,N n ). 4.4 Legendre ransformations Even though it is possible in principle to control the variables (S,,N) eperimen- tall, it is much more common to perform eperiments under constant- tempera- ture or constant- pressure conditions, rather than at constant volume or entrop. For eample, constant- temperature conditions are closel approimated when a sstem is immersed in a temperature- controlled bath. Constant- pressure condi- tions are imposed when a sstem is open to the atmosphere. ariables such as and appear as the first derivatives of the fundamental equation U(S,,N). Unfor- tunatel, while it is certainl possible to obtain functions such as U(,,N), the information content of such functions is not equivalent to that of the fundamental equation U(S,,N) this is because of the integration constants that will need to be introduced if one wants to go from U(,,N) to U(S,,N), as illustrated in E- ample 4.3.

10 10 Chapter 4. Fundamental Equations In Chapter 2, we have alread introduced an additional thermodnamic func- tion, the enthalp H = U +. Here, we will show that this and other similar de- rived thermodnamic functions are more than just convenient combinations of terms. he represent fundamental equations that pla the same ke role in ther- modnamics as the energ U, but are epressed in variables other than (S,,N). In mathematical terms, for a function f() of a single variable, we would like to obtain a new function g(ξ), with the same information content as the origi- nal function, but epressed in a new variable ξ = df/d, the first derivative of the original function. Equivalence of information content implies that the original function can be recovered from the transformed one without an ambiguit in the form of integration constants. It turns out that this can be done through a mathe- matical operation known as a Legendre transformation. For a function of one vari- able, the transformed function is obtained as g(ξ) = f ξ, which has the following properties: df = ξd dg = df dξ ξd = dξ (4.16) In the new function (referred to as the transform of the original function), the role of variables and derivatives has been echanged: the derivative of the original function is the variable of the new function and the variable of the original function is minus the derivative of the transform. One can recover the original function b simpl appling the Legendre transformation one more time: Eample 4.3 Consider the function: dg = dξ f = g +ξ (4.17) f ()= 2 +2 Construct the function f(w) and show that it cannot be used to full determine f(). Also obtain the Legendre transform of this function, g(w) and show how it can be used to reconstruct the original function f(). he derivative of f () is: ξ = df d = 2 he function f(ξ) can be obtained b substituting = ξ/2 in f():

11 4.4 Legendre ransformations 11 f (ξ)= ξ 2 +2 (i) 4 Given f(ξ), we can attempt to determine f() from: d df = 1 ξ = 1 ξ df df = ±2 f 2 = ± f 2 + c f = c ( ) 2 +2 (ii) An unknown integration constant c appears in epression (ii), so we have shown that f() cannot be full recovered from f(ξ). he Legendre transform g(ξ) of f() is: 2 ξ g(ξ)= f ξ = 4 +2 ξ ξ2 ξ g(ξ)= (iii) From Eq. 4.17, the variable can be obtained from (iii) from differentiation: = dg dξ = ξ 2 Now the function f() can be obtained from g(ξ) using Eq again: (iv) f = g +ξ = ξ ξ 2 w = ξ f = 2 +2 QED Eample 4.4 Consider the function f ( ) = in the interval [ 1,1]. Construct its first Legen- dre transformation g(ξ) and plot f ( ) and g(ξ), marking the points correspond- ing to = 0.7, = 0, and = 0.7 on the graph of g(ξ). Using the definition of Legendre transforms: df d = ξ ξ = 32 = ± ξ 3 g(ξ)= f ξ = ± ξ ξ3 ±ξ3/2 3 3/2+1 ξ = ξ 3/2 3 g(ξ)= 2ξ3/ he Legendre transform of g(ξ) is: dg dξ = ξ 3 =

12 12 Chapter 4. Fundamental Equations g ξ dg dξ ± ξ ξ 3 = ±ξ 3/ = ± ξ +1 = 3 +1= f () 3 = 2ξ3/2 lots of the original and transformed functions are shown in Fig Arrows mark the three points of interest (note that ξ = 3 2, so the ξ coordinate corresponding to = ±0.7 is ξ = = he transform is multivalued two values of the function g(ξ) eist for an given ξ. his is the price we have to pa for ensuring that the reverse transform is unique. In addition, the transformed function has a cusp at ξ =0, with a discon- tinuous first derivative. Multivalued functions arise naturall in thermodnamics when a sstem can eist in multiple phases; cusps are also present in thermod- namic functions for sstems with multiple phases. 3 Figure 4.1 A one- dimensional function and its transform. For functions of man variables, the Legendre transform procedure can be applied sequentiall, giving the first, second and higher- order transforms. Formall, we consider a general function of l variables, as a basis function (0), where basis means simpl the starting point of the transform: (0) ( 1, 2,, k, k+1, l ) (4.18) he first and second derivatives of the basis function are defined as:

13 4.4 Legendre ransformations 13 ξ i i (0) = (0) i hus, the differential form of (0) is: and (0) ij = 2 (0) = (0) i j i j j[i] i[ j] j[i] (4.19) d(0) = ξ ξ ξ k k + ξ k+1 k+1 + ξ l l (4.20) he k- th transform of (0) is denoted as (k) and is obtained from: (k) (ξ 1,ξ 2,,ξ k, k+1, l ) = (0) 1 ξ 1 2 ξ 2 k ξ k (4.21) he k- th transform is a function of a new set of variables, (ξ 1,ξ 2,,ξ k, k+1, l ). Its differential form is d(k) = 1 dξ 1 2 dξ 2 k dξ k + ξ k+1 d k ξ l d l (4.22) A common basis function for thermodnamics is the fundamental equation in the energ representation, U(S,,N 1,N 2,,N n ). Additional thermodnamic func- tions, such as H, are simpl transforms of the fundamental equation U that pre- serve its information content. Note that different Legendre transforms result if the order of variables of the original function is changed, for instance to U(,S,N 1,N 2,,N n ). Fundamentalequation: (0) = U(S,,N 1,N 2,,N n ) du = ds d + n i=1 µ i dn i (4.5) he first Legendre transformation of U(S,,N,N,,N ) gives a new thermo- 1 2 n dnamic function, the Helmholt free energ A: Fundamentalequation: (1) = A(,,N 1,N 2,,N n ) A = U S ; da = Sd d + µ i dn i n i=1 (4.23) he second transform of U, gives another fundamental equation, the Gibbs free energ G, named after American scientist Josiah Willard Gibbs ( ): Fundamentalequation: (2) = G(,,N 1,N 2,,N n ) G = U S + = A+ ; dg = Sd +d + µ i dn i i=1 n (4.24)

14 Function Internal energ U Helmholt energ A Enthalp H Gibbs free energ G,N = NC,N able 4.1 Fundamental equations (thermodnamic potential functions) for one- component sstems. ariables S N Τ N S N Τ N Differential Integral First de- rivatives du = ds d + µdn da = Sd d + µdn dh = ds +d + µdn dg = Sd +d + µdn U = S + µn A = + µn H = S + µn G = µn = U = U,N µ = U N S, S,N S = A = A,N µ = A N,,N = H = H,N µ = H N S, S,N S = G = G,N µ = G N = G,,N A second derivative 2 U 2,N =,N = 2 A NC 2,N = S,N = NC 2 H 2,N =,N = 2 G NC 2,N = S Mawell s rule,n = S,N,N = S,N,N = S,N S,N = 14

15 4.4 Legendre ransformations 15 Reordering the variables to U(S,,N,N,,N ) gives the enthalp H, first in- 1 2 n troduced in deriving First Law balances for open sstems as the first Legendre transform (1) : Fundamentalequation: (1) = H(S,,N 1,N 2,,N n ) H = U + ; dh = ds +d + µ i dn i n i=1 (4.25) he Legendre transforms of U are also known as thermodnamic potentials. able 4.1 (p. 14) summaries the information on variables and derivatives for the four thermodnamic potentials we have discussed thus far, for the case of a one- component sstem for simplicit. he table also lists a second derivative for each transform and an eample Mawell s relationship between second derivatives. he Euler- integrated form of each of the fundamental equations epresses them as sum of terms, each term being an etensive variable of the transform mul- tiplied b the corresponding derivative. he results are summaried in able 4.1 and are consistent with the Legendre transformations above. For eample, for the Gibbs free energ G, G = µn = U S + = H S = A+ (4.26) An important consequence of the integral relationship for G in the special case of a one- component sstem is that the molar Gibbs free energ is, in this case, equal to the chemical potential: for one- component sstems onl: G = µ (4.27) Additional transformations, not listed in the table, are possible as well. For e- ample, one could order the variables as U(N,N,,N,S, ) and perform a single 1 2 n transform to obtain a function of (µ,n,,n,s, ). he resulting function is also 1 2 n a valid fundamental equation, even if it is not frequentl used in practice. he final comment here is that the last transform, (n+2), ields a function of onl intensive variables,, and all the chemical potentials. B Euler integration we obtain that the resulting function is identicall ero, a result known as the Gibbs- Duhem relationship: n Sd +d N i dµ i = 0 (4.28) i=1 Gibbs- Duhem relationship

16 4.5 Derivative Relationships As alread stated, two ke objectives of classical thermodnamics are to obtain relationships between properties, and to develop techniques to measure them. Ob- taining thermodnamic relationships helps bring forward hidden connections between seemingl unrelated quantities and facilitates the measurement and vali- dation of phsical propert data. he Legendre transformation facilitates developing relationships between thermodnamic derivatives. For eample, the following relationships eist be- tween second derivatives of a transform and its basis function: 11 1i (1) = 1 (1) = (0) 1i (0) 11 (0) 11 (1) i j = (0) i j (0) (0) 1i 1j (4.29) i 1 (4.30) (0) 11 i, j 1 (4.31) For proofs of these relationships, as well as more comple general relation- ships between derivatives of the k- th transform and those of the basis function, see Beegle, Modell and Reid, AIChE J., 20: (1974) and Kumar and Reid, AIChE J., 32: (1986). For pure components, a major simplification of thermodnamic relationships is possible relative to the general multicomponent case, because composition is not an independent variable. Etensive properties can then be normalied b the amount of mass (moles) in a sstem to obtain intensive properties. hermodnam- ic derivatives at constant number of moles can be taken either on a total or on a molar basis, for eample: U,N = (NU) (NS) N,N = U = (4.32) N was eliminated in the last derivative of this epression because intensive prop- erties do not depend on the sie of the sstem, so that N is no longer a relevant constraint. he differential forms of the fundamental equations in one- component sstems for the intensive thermodnamic functions U, A, G, and H, are as follows: 16

17 4.5 Derivatives in erms of Measurable roperties 17 du = ds d d A = Sd d dh = ds +d dg = Sd +d (4.33) intensive FEs for one- component sstems First derivatives of these functions with respect to their natural variables are obtained directl, either from able 4.1 or from Eqs An eample of such a first derivative is given above in Eq temperature is clearl an eperimental- l measurable quantit. Other eamples include: U S,N = U S = and H S = G = (4.34) Some first derivatives of Fundamental Equations with respect to natural varia- bles cannot be measured directl: A = G = S and U N S, = µ (4.35) First derivatives of Fundamental Equations with respect to unnatural varia- bles or under other constraints can be obtained from the differential epressions of Eq For eample, to obtain the volume derivative of the energ at constant temperature in terms of measurable properties, we start from: du = ds d We now take the partial derivative ( / ), followed b Mawell s rule on A to obtain: U = S = (4.36) Second derivatives of fundamental equations with respect to their natural var- iables are important from a theoretical and eperimental viewpoint. For eample, differentiating A and G twice with respect to temperature, we obtain two im- portant eperimentall measurable quantities, the heat capacities defined in Chap- ter 3: 2 A 2 = S = 1 U = C (4.37)

18 18 Chapter 4. Fundamental Equations 2 G 2 = S = 1 H = C (4.38) he other two second derivatives of G with respect to its natural variables are also important eperimentall measurable quantities. he cross derivative with respect to temperature and pressure is: 2 G = G = = α (4.39) his derivative is proportional to the coefficient of thermal epansion, defined as: he second derivative with respect to pressure is: α ( / ) / (4.40) 2 G 2 = = κ (4.41) his derivative is proportional to the isothermal compressibilit, which is defined as: κ ( / ) / (4.42) Mied second derivatives of fundamental equations with respect to their natu- ral variables can be written in two equivalent was, using the commutation prop- ert (Mawell s relationships): 2 U S = 2 A = 2 H S = 2 G = S S = S = S = = (4.43) (4.44) (4.45) (4.46) It is clear from the last three epressions that derivatives of the entrop S with re- spect to volume or pressure are eperimentall measurable, even though the abso- lute value of entrop itself is not directl obtainable in classical thermodnamics.

19 4.5 Derivatives in erms of Measurable roperties 19 Other derivatives can be obtained using the tools described in 4.2. For eam- ple, the Joule- hompson coefficient, ( / ) H is the rate of temperature change with pressure when a fluid flows across a throttling valve, as can be confirmed from a differential First Law balance in an open sstem at stead state with d/q = d/w = 0 (see Eq. 2.18, p. 18). his coefficient can be epressed in terms of equation- of- state derivatives and heat capacities b first invoking the triple- product rule: H H H = 1 H = ( H / ) ( H / ) he denominator is simpl C ; the numerator can be obtained from the differential form of H, he final result is: dh = ds +d H Eample 4.5 Differential epressions for U H = S = 1 C + = + (4.47) Obtain differential epressions for U =U (, ) and U =U (,) in terms of eperi- mentall measurable properties. Note that these epressions are not fundamental equations because the natural variables of U are S and. We seek to construct epressions of the form: du = U du = U d + U d + U d d he temperature derivative ( U/ ) = C, and the volume derivative ( U/ ), have been obtained in Eq. 4.36, so the desired epression for U = U(,) is: du = C d + d (i) (ii) (iii)

20 20 Chapter 4. Fundamental Equations For the derivatives needed for epression (ii), we start again from the fundamental equation for U in terms of its natural variables, du = ds d. Differentiating with respect to at constant gives: U = S he temperature dependence of the entrop at constant pressure, ( S / ), can be obtained b differentiating dh = ds + d: H = C = S After substituting the result into (iv), we get: U = C he pressure derivative needed for (ii) is: U = S = We used Mawell s relationship on G to get rid of S. Substituting (v) and (vi) into (ii) we finall obtain: du = C d + d (iv) (v) (vi) (vii) One observation we can make here is that the functional forms of U in unnatural variables are more comple than the comparable epressions in its natural vari- ables. his is an additional disadvantage beond the loss of information content discouraging the use of functional forms such as U = U(, ) and U = U(,). Eample 4.6 Difference between heat capacities Calculate the difference between the heat capacities at constant pressure and con- stant volume, C C, for a general thermodnamic sstem, in terms of,, and their mutual derivatives. We use as starting point epression (iii) in Eample 4.5: du = C d + d

21 4.5 Derivatives in erms of Measurable roperties 21 Differentiating with respect to at constant and setting the result equal to e- pression (v) of Eample 4.5, we obtain: U = C + from(iii) C C = = from(v) C For ideal gases, one can simplif this relationship considerabl: ( / ) = R / and( / ) = R / C C = R2 /( )= R hus, the difference in heat capacities at constant pressure and constant volume for ideal gases is simpl the ideal- gas constant R, as alread derived in 2.4.