Chapter 1: Heat Capacities: Introduction, Concepts and Selected Applications

Published:16 Aug 2010

E. Wilhelm, in Heat Capacities, ed. E. Wilhelm and T. M. Letcher, The Royal Society of Chemistry, 2010, ch. 1, pp. 127.
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Once the fabric is woven it may be embellished at will.
Nero Wolfe in The Golden Spiders, by Rex Stout, Bantam edition, New York, NY, 1955.
1.1 Introduction
Heat capacities belong to the most important thermophysical properties of matter: they are intimately related to the temperature dependence of fundamental thermodynamic functions; they may be determined in the laboratory with great accuracy; and they are of key importance for linking thermodynamics with microscopic fluid structure and dynamics, as evidenced by the contributions to this book. They are thus indispensable in physical chemistry as well as in chemical engineering. For instance, as a classical example, consider the standard entropies of liquids at T = 298.15 K. They are evaluated from experimental heat capacities at constant pressure from low temperatures to 298.15 K and entropies of phase changes in between (assuming applicability of the third law of thermodynamics). The measured heat capacity of an organic compound can usually be extrapolated to 0 K by fitting a Debye heat capacity function to the experimental values at, say, 10 K.
The nature and the size of this monograph's topic make it impractical to cover the entire subject in one volume. As indicated in the title, the focus will be on heat capacities of chemically nonreacting liquids, solutions and vapours/gases (though polymers and liquid crystals are also covered). The individual specialised chapters have been written by internationally renowned thermodynamicists/thermophysicists active in the respective fields. Because of their topical diversity, in this introductory chapter I shall try to summarise concisely the major aspects of the thermodynamic formalism relevant to fluid systems, to clarify, perhaps, some points occasionally obscured, to indicate some ramifications into neighbouring disciplines, and to point out a few less familiar yet potentially interesting problems. The omission of any topic is not to be taken as a measure of its importance, but is predominantly a consequence of space limitations.
Calorimetric determinations of heat capacities of liquids have a long tradition, and many distinguished scientists have contributed to this subject. One can only marvel about the careful work of some of the early researchers, such as Eucken and Nernst,^{1,2 } who developed precursors of modern, adiabatic calorimeters. The adiabatic method for heat capacity measurements at low temperatures was pioneered by Cohen and Moesveld,^{3 } and Lange,^{4 } and became widely used. Indeed, during the following decades, many alternative designs of increasing sophistication have been devised and successfully used. A selection of adiabatic calorimeters which were described in the literature up to about 1970 is provided by references 5 through 19. For details, the interested reader should consult the classic IUPAC monograph edited by McCullough and Scott,^{20 } or the more recent ones edited by Marsh and O’Hare,^{21 } and by Goodwin, Marsh and Wakeham,^{22 } or the monograph on calorimetry by Hemminger and Höhne.^{23 }
More specialised reviews have been prepared by Lakshmikumar and Gopal,^{24 } Wadsö^{25 } and Gmelin.^{26 } A monograph focusing on differential scanning calorimetry has been presented by Höhne, Hemminger and Flammersheim.^{27 }
To date, the most widely used instruments for measuring heat capacities of liquids and liquid mixtures are based on the differential flow calorimeter designed by Picker,^{28,29 } which was commercialised by Setaram. Because of the absence of a vapour space, differential flow calorimeters are particularly useful. They may be fairly easily modified to be used at elevated temperatures and pressures, including the critical region. The first instrument of this type was constructed by SmithMagowan and Wood,^{30 } with improved versions being due to White et al.,^{31 } and Carter and Wood.^{32 } However, comparison of heat capacities measured by different types of flow calorimeters and differential thermopile conduction calorimeters shows small differences in measured heat capacities, which are attributed to conductive and convective heat losses. Conductive heat losses, the principal problem in flow calorimetric heat capacity measurements on liquids, have recently been analysed by Hei and Raal^{33 } for a fivezone model calorimeter.
Because of the importance of heat capacity data of liquids in chemical thermodynamics and chemical engineering, numerous critical data compilations have been published – starting at the end of the nineteenth century with Berthelot's Thermochimie,^{34 } and including such wellknown publications as the International Critical Tables,^{35 } Timmermans’ Physicochemical Constants of Pure Organic Compounds,^{36 } LandoltBörnstein,^{37 } and Daubert and Danner's Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, DIPPR® Database.^{38 } The most recent and the most comprehensive compilation of critically evaluated heat capacity data of pure liquids is the monograph on Heat Capacities of Liquids: Volumes I and II. Critical Review and Recommended Values, authored by Zábranský et al. (1996),^{39 } with Supplement I of 2001.^{40 } This monograph also contains a valuable survey of calorimetric techniques for determining heat capacities of liquids, and useful comments on terminology and criteria for the classification of calorimeters.
As concerns heat capacity data of mixtures, the situation is somewhat less satisfactory. Critically selected excess molar heat capacities at constant pressure of binary liquid organic mixtures have been included in the International DATA Series, SELECTED DATA ON MIXTURES, Series A,^{41 } and the Dortmund Data Bank (DDB) contains a large number of data sets on heat capacities of mixtures/excess heat capacities.^{42 } However, a monograph devoted to a reasonably comprehensive compilation of heat capacity data of liquid mixtures, though highly desirable, is not available.
For more than a century, experimental studies of realgas behaviour at low or moderate densities, have held a prominent position in physical chemistry. They were motivated, and still are, either by the need to solve practical problems – such as those encountered in the reduction of vapour–liquid equilibrium data – or by their usefulness as valuable sources of information on intermolecular interactions in both pure gases/vapours and gaseous mixtures. In this context, perfectgas (idealgas) state heat capacities are of central importance, say, in the calculation of property changes of singlephase, constantcomposition fluids for any arbitrary change of state. They may be determined by vapourflow calorimetry, or by speedofsound measurements. The statistical–mechanical calculation of perfectgas state heat capacities (they are 1body properties which do not depend on molecular interactions) has reached a high level of sophistication, with obvious great practical advantages. For instance, the calculations readily allow extension of experimental data to temperature ranges currently inaccessible to measurement. Data compilations of heat capacities of pure substances in the perfectgas (idealgas) state may be found in Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,^{43 } LandoltBörnstein,^{37 } in Stull, Westrum and Sinke's The Chemical Thermodynamics of Organic Compounds,^{44 } in the TRC Thermodynamic Tables,^{45 } in the book by Frenkel et al.,^{46 } and in the NISTJANAF Thermochemical Tables.^{47 } One should always keep in mind, however, that only comparison of experimental with calculated values leads to better approximations and/or new concepts.
1.2 Thermodynamics: Fundamentals and Applications
To set the scene for this monograph, a few selected basic thermodynamic relations will be summarised below. For further aspects and details the interested reader should consult a textbook close to his/her taste, perhaps one of those listed in references 48 through 58.
Convenient starting points are the fundamental property equations (also called the Gibbs equations) of a singlephase PVT system, either open or closed, where P denotes the pressure, V is the molar volume and T is the thermodynamic temperature. No electric, magnetic or gravitational fields are considered in such a simple system. For a multicomponent system, where the total amount of substance is given by , with n_{i} being the amount of substance of component i, the fundamental property equation in the energy representation is
and, equivalently, in the entropy representation
Here, U is the molar internal energy, S is the molar entropy of the fluid. The intensive parameter furnished by the firstorder partial derivatives of the internal energy with respect to the amount of substance of component i,
is called the chemical potential of component i. Its introduction extends the scope to the general case of a singlephase system in which the n_{i} may vary, either by exchanging matter with its surroundings (open system) or by changes in composition occurring as a result of chemical reactions (reactive system) or both. Corresponding to Equations (1) and (2), the primary functions (or cardinal functions, or Euler equations) are
in the energy representation, and
in the entropy representation.
In both the energy and entropy representations the extensive quantities are the mathematically independent variables, while the intensive parameters are derived, which situation does not conform to experimental practice. The choice of nS and nV as independent variables in the fundamental property equation in the energy representation is not convenient, and Equation (4) suggests the definition of useful alternative energybased primary functions. The appropriate method for generating them without loss of information is the Legendre transformation. These additional equivalent primary functions are the molar enthalpy
the molar Helmholtz energy
and the molar Gibbs energy
Substituting for U in Equation (6) from Equation (4) yields the alternative form
where x_{i}=n_{i}/n is the mole fraction. Substitution of U in Equation (7) yields
as alternative grouping, and substitution of U in Equation (8) yields the Euler equation as
The alternative primary functions H, F and G allow the development of alternative energybased fundamental property equations:
The four fundamental equations presented so far are equivalent; however, each is associated with a different set of canonical variables {nS, nV, n_{i}}, {nS, P, n_{i}}, {T, nV, n_{i}} and {T, P, n_{i}}.
A primary function which arises naturally in statistical mechanics is the grand canonical potential. It is the Legendre transform when simultaneously the entropy is replaced by the temperature and the amount of substance by the chemical potential:
with the alternative form
and the corresponding Gibbs equation
with the canonical variables {T, nV, μ_{i}}.
The complete Legendre transform vanishes identically for any system. The complete transform of the internal energy replaces all extensive canonical variables by their conjugate intensive variables, thus yielding the nullfunction
as final alternative primary function in the energy representation. This property of the complete Legendre transform gives rise to
as the corresponding alternative form of the fundamental property equation. It represents an important relation between the intensive parameters T, P and μ_{i} of the system and shows that they are not independent of each other.
While the extensive parameters of a simple phase are independent of each other, the conjugate intensive parameters are not, as shown above. For a given phase, the number of intensive parameters which may be varied independently is known as the number of thermodynamic degrees of freedom.
Treating the sum as a single term, the total number of equivalent primary functions and therefore the total number of equivalent fundamental property equations for a thermodynamic system is 2^{k}. Thus for nU=nU(nS, nV, n) there are but eight distinct equivalent primary functions [nU, Equation (4), plus seven alternatives] and eight distinct forms of the fundamental equation [d(nU), Equation (1), plus seven alternatives]. Of the seven Legendre transforms of the internal energy, five have been treated above (including the nullfunction). The remaining two, and , with the alternative forms X=TS−PV and Y=TS, respectively, have not received separate symbols or names. The corresponding fundamental property equations are and .
Since all the fundamental property equations are equivalent, alternative expressions for the chemical potential are possible, of which
is the preferred one, because temperature and pressure are by far the most useful experimental parameters. We recognise that the chemical potential of component i is just the partial molar Gibbs energy of i,
which quantity is of central importance in mixture/solution thermodynamics.
For a homogeneous fluid of constant composition, the following four energybased fundamental property relations apply:
It follows that
which relations establish the link between the natural independent variables T, P, V, S and the energybased functions U, H, F, G. In view of the definitions of F and G and Equation (29), the Gibbs–Helmholtz equations
are obtained.
A Legendre transformation of the primary function in the entropy representation, Equation (5), resulting in the replacement of one or more extensive variables by the conjugate intensive variable(s) 1/T, P/T and μ_{i}/T, defines a Massieu–Planck function. For instance, the molar Massieu function is
with its alternative form
Its differential form, an entropybased alternative fundamental property equation, is
From a secondorder Legendre transformation, the molar Planck function
is obtained, with its alternative form
Its differential form is another alternative entropybased fundamental property equation:
Note that
and
Another secondorder transform is the molar Kramer function
Its alternative form is
whence
The corresponding alternative entropybased fundamental property equation is
Again, the complete Legendre transform is identical zero, yielding in the entropy representation
Evidently, also the intensive parameters 1/T, P/T and μ_{i}/T in the entropy representation are not independent of each other.
Equations (22) through (25) are exact differentials, whence application of the reciprocity relation yields the Maxwell equations for a constantcomposition PVT system, of which the following two are particularly useful:
Two heat capacities are in common use for homogeneous fluids. Both are state functions defined rigorously in relation to other state functions: the molar heat capacity at constant volume (or the molar isochoric heat capacity) C_{V} and the molar heat capacity at constant pressure (or the molar isobaric heat capacity) C_{P}. At constant composition,
and
At this juncture it is convenient to introduce, by definition, a few auxiliary quantities commonly known as the mechanical and the isentropic coefficients. Specifically, these are the isobaric expansivity
the isothermal compressibility^{1}
the isochoric pressure coefficient
and the isentropic compressibility^{1} (often loosely called adiabatic compressibility)
where ρ=M/V is the density and M is the molar mass.
Note that
The isentropic compressibility is related to the thermodynamic lowfrequency speed of ultrasound ν_{0} (negligible dispersion) by
The ratio of the heat capacities and their difference may now be presented in several compact forms, where the most profitable are given below:
Since by definition the compression factor is given by Z≡PV/RT, alternatively
where R is the gas constant.
At low temperatures, where γ_{V} of liquids is large, direct calorimetric determination of C_{V} of liquids is difficult (it becomes more practicable near the critical point, where γ_{V} is much smaller). Thus most of the isochoric heat capacity data for liquids reported in the literature have been obtained indirectly through the use of Equations (55) and (56), that is to say from experimental molar isobaric heat capacities, isobaric expansivities and ultrasonic speeds. However, see for instance reference 59. Since also
Equations (54), (56) and (59) may be used for the indirect determination of isothermal compressibilities from densities, isobaric expansivities, ultrasonic speeds and molar isobaric heat capacities. All these quantities may now be reliably and accurately measured, whence the indirect method for determining the isothermal compressibility of liquids has become an attractive alternative to the direct method of applying hydrostatic pressure and measuring the corresponding volume change. For the difference between β_{T} and β_{S} one obtains, for instance,
A convenient way to derive the volume or pressure dependence of the heat capacities is via the differentiation of the appropriate Gibbs–Helmholtz equations. Starting from
these equations lead to
The pressure or volume dependence of the heat capacities may thus be determined from PVT data.
The molar thermodynamic properties of homogeneous constantcomposition fluids are functions of temperature and pressure, e.g.
Replacing the partial derivatives through use of Equations (48) and (62) yields
Entirely analogous procedures, using Equations (46) and (48), give
When T and V are selected as independent variables,
and
are obtained. All the coefficients of dT, dP and dV are quantities reasonably accessible by experiment. For some applications it may be convenient to treat S as a function of P and V. Using
one obtains
Finally we note the useful relations
and
where μ_{JT} is the Joule–Thomson coefficient. All three quantities C_{P}, (∂H/∂P)_{T} and μ_{JT} may be measured by flow calorimetry.^{60,61 } (∂H/∂P)_{T} is also known as the isothermal Joule–Thomson coefficient, and frequently given the symbol φ. For ideal gases Tα_{P} = 1 and thus μ_{JT} = 0. For real gases, the temperature T_{i} (at the inversion pressure P_{i}) where T_{i}α_{P} = 1 is called the inversion temperature. At that point the isenthalpic exhibits a maximum: for initial pressures P<P_{i}, μ_{JT} > 0, and the temperature of the gas always decreases on throttling; for initial pressures P > P_{i}, μ_{JT}<0, and the temperature of the gas always increases on throttling. The maxima of the enthalpics form a locus known as the inversion curve of the gas. There exists a maximum inversion temperature at P = 0. For pressures above the maximum inversion pressure, μ_{JT} is always negative.
Because of Equation (67) one obtains, for instance, for the isentropic compression or expansion of a gas
Since α_{P} of gases is always positive, the temperature always increases with isentropic compression and decreases with isentropic expansion.
In principle, the exact methods of classical thermodynamics are the most general and powerful predictive tools for the calculation of property changes of singlephase, constantcomposition fluids for any arbitrary change of state, say, from (T_{1},P_{1}) to (T_{2},P_{2}). For a pure fluid, the corresponding changes of molar enthalpy ΔH ≡ H_{2} – H_{1} and molar entropy ΔS ≡ S_{2} – S_{1} are, respectively,
and
where H^{R} and S^{R} are the molar residual enthalpy and the molar residual entropy, respectively, in (T,P)space, and C$Ppg$ = C$Ppg$(T) is the molar heat capacity at constant pressure of the fluid in the perfectgas (idealgas) state. The general definition for such molar residual properties is M^{R}≡M−M^{pg}, where M is the molar value of any extensive thermodynamic property of the fluid at (T,P), and M^{pg} is the molar value of the property when the fluid is in the perfectgas state at the same T and P. Given any volumeexplicit equation of state, these residual functions may be calculated from
and
respectively. Thus, application of Equations (75) and (76) requires PVT information for the real fluid as well as its isobaric heat capacity in the perfectgas state. We note that one may also define residual functions in (T,V)space: M^{r} ≡ M − M^{pg}, where the Ms are now at the same T and V. In general M^{R}(T,P) ≠ M^{r}(T,V) unless the property M^{pg} is independent of density at constant temperature, which is the case for C $Ppg$ and C $Vpg$ = C$Ppg$ − R. Since the perfectgas state is a state where molecular interactions are absent, residual quantities characterise molecular interactions alone. They are the most direct measures of intermolecular forces. In statistical mechanics, however, configurational quantities are frequently used. The differences between these two sets are the configurational properties of the perfect gas, and for U and C_{V} they vanish.
In actual practice, this approach would be severely limited by the availability of reliable data for pure fluids and mixtures. The experimental determination of such data is timeconsuming and not simple, and does not impart the glamour associated with, say, spectroscopic studies. Fortunately, statistical–mechanical calculations for C$Ppg$ are quite dependable for many substances, and so are groupcontribution theories, for instance the techniques based on the work by Benson and coworkers.^{62,63 }
The search for generalised correlations applicable to residual functions has occupied scientists and engineers for quite some time. The most successful ones are based on versions of generalised correspondingstates theory, which is grounded in experiment as well as statistical mechanics. The threeparameter corresponding states correlations, pioneered by Kenneth Pitzer and coworkers,^{64–67 } have been capable to predict satisfactorily the PVT behaviour of normal, nonassociating fluids. They showed that the compression factors of normal fluids may be satisfactorily expressed as
where
is Pitzer's acentric factor, T_{r} = T/T_{c} is the reduced temperature, P_{r} = P/P_{c} is the reduced pressure, P_{σ}_{,r} = P_{σ}/P_{c} is the reduced vapour pressure, here evaluated at T_{r} = 0.7, P_{σ} is the vapour pressure of the substance, T_{c} is the critical temperature of the substance and P_{c} is its critical pressure. In fact, this method is a thermodynamic perturbation approach where the Taylor series is truncated after the term linear in ω. The generalised Z^{(0)} function is the simplefluid contribution and applies to spherical molecules like argon and krypton, whose acentric factors are essentially zero. The generalised Z^{(1)} function (deviation function) is determined through analysis of highprecision PVT data of selected normal fluids where ω≠0. One of the best of the generalised Pitzertype correspondingstates correlations for Z^{(0)}, Z^{(1)} and the derived residual functions is due to Lee and Kesler.^{68 }
An alternative to the direct experimental route to highpressure PVT data and C_{P}(T,P) is to measure the thermodynamic speed of ultrasound v_{0} as a function of P and T (at constant composition), and to combine these results, in the spirit of Equations (50), (54) and (60) with data at ordinary pressure, say P_{1} = 10^{5} Pa, i.e. ρ(T,P_{1}) and C_{P}(T,P_{1}). For a pure liquid, upon integration at constant temperature, one obtains^{69,70 }
The first integral is evaluated directly by fitting the ultrasonic speed data with suitable polynomials, and for the second integral several successive integration algorithms have been devised. The simplicity, rapidity and precision of this method makes it highly attractive for the determination of the density, isobaric expansivity, isothermal compressibility, isobaric heat capacity and isochoric heat capacity of liquids at high pressures. Details may be found in the appropriate chapters of this book, and in the original literature.
From experimentally determined heat capacities of liquids, relatively simple models have been used to extract information on the type of motion executed by molecules in the liquid state. In general, they are based on the separability of contributions due to translation, rotation, vibration and so forth. Though none of them is completely satisfactory, they have provided eminently useful insights and thereby furthered theoretical advances. Following the early work of Eucken,^{71 } Bernal,^{72 } Eyring,^{73 } Stavely,^{74 } MoelwynHughes,^{75 } Kohler,^{18,76 } Bondi^{77 } and their collaborators, one may resolve the molar heat capacity C_{V} of simple, nonassociated liquids into the following contributions:^{78,79 }
The translational (tr) contribution arises from the motion of the molecules under the influence of all molecules (translational movement within their respective free volumes), the rotational (rot) contribution arises from rotation or libration of the molecules as a whole, the internal (int) contribution arises from internal degrees of freedom, and the orientational (or) contribution, for dipolar substances, results from the change of the dipole–dipole orientational energy with temperature. C_{int} can be subdivided into a part stemming from vibrations (C_{vib}) which usually are not appreciably influenced by density changes (i.e. by changes from the liquid to the perfectgas state), and another part, C_{conf}, resulting from internal rotations (conformational equilibria), which does depend on density. Preferably, all these contributions to C_{V} are discussed in terms of residual quantities in (T,V)space.^{78,79 } The residual molar isochoric heat capacity of a pure liquid is defined by
For liquids composed of fairly rigid molecules, such as tetrachloromethane, benzene or toluene, to an excellent approximation C$intr$ ≈ 0, whence
where C$trr$ = C_{tr}–3R/2, and C$rotr$ = C_{rot}–3R/2, for nonlinear molecules, represents the excess over the perfectgas phase value due to hindered rotation in the liquid of the molecules as a whole. Using corresponding states arguments to obtain reasonable estimates for C$trr$, values for the residual molar rotational heat capacity C$rotr$ may be obtained, which quantity may then be discussed in terms of any suitable model for restricted molecular rotation.^{61,78,79 }
The resolution of the variation of C_{V} of pure liquids along the orthobaric curve (subscript σ), i.e. for states (T, P_{σ}), into the contributions due to the increase of volume and to the increase of temperature, respectively, is a highly interesting problem.^{78–80 } It is important to realise that due to the close packing of molecules in a liquid, even a rather small change of the average volume available for their motion may have a considerable impact on the molecular dynamics: volume effects may become more important in influencing molecular motion in the liquid state than temperature changes. Since
evaluation of (∂C_{V}/∂T)_{V} requires knowledge of the second term of the righthand side of Equation (85). At temperatures below the normal boiling point, the saturation expansivity α_{σ} = V^{−1}(∂V/∂T)_{σ} is practically equal to α_{P} of the liquid [see below, Equation (109)]. In principle, the quantity (∂C_{V}/∂V)_{T} is accessible via precise PVT measurements, see Equation (63), but measurements of (∂^{2}P/∂T^{2})_{V} are not plentiful. Available data^{70,81 } indicate that it is small and negative for organic liquids, that is to say, C_{V} decreases with increasing volume. Alternatively, one may use^{18,78,79 }
where the last term in parenthesis on the righthand side can be evaluated by means of a modified Tait equation,^{82 } that is
This equation holds remarkably well up to pressures of several hundred bars, and for many liquid nonelectrolytes m ≈ 10. For liquid tetrachloromethane at 298.15 K,^{78 } the calculated value of (∂C_{V}/∂V)_{T} amounts to –0.48 J K^{−1} cm^{−3}, for cyclohexane^{78 } –0.57 J K^{−1} cm^{−3} is obtained, and for 1,2dichloroethane^{83 } it is −0.60 J K^{−1} cm^{−3}. These results indicate a substantial contribution of (∂C_{V}/∂V)_{T}Vα_{σ} to the change of C_{V} along the orthobaric curve as well as to the corresponding change of C$Vr$.
Equation (56) is a suitable starting point for a discussion of the temperature dependence of κ≡C_{P}/C_{V} of a liquid along the orthobaric curve:
Usually, the second term in parenthesis on the righthand side of Equation (88) is positive and the third term is negative; the fourth term may contribute positively or negatively. Thus κ may increase or decrease with temperature.
The importance of the heat capacity in the perfectgas state has been stressed repeatedly. Flow calorimetry is a commonly used method for measuring C_{P} of gases and vapours,^{84 } and allows straightforward extrapolation to zero pressure^{85 } to obtain C$Ppg$. The virial equation in pressure
where B′ is the corresponding second virial coefficient and C′ the third virial coefficient, may be used to calculate the residual heat capacity of a pure fluid according to
Since the second virial coefficient B′ of the pressure series is related to the second virial coefficient B of the series in molar density (1/V) by
one obtains from the twoterm equation in pressure
Thus the pressure derivative of C_{P} is given by
thereby providing an experimental route to the determination of the second temperature derivative of B.
Flowcalorimetric measurements of deviations from perfectgas behaviour, particularly via the isothermal Joule–Thomson coefficient φ≡(∂H/∂P)_{T}, have the advantage over compression experiments that adsorption errors are avoided, and that measurements can be made at lower temperatures and pressures.^{86,87 } Specifically,
where
Here, C is the third virial coefficient of the series in molar density, P_{2} − P_{1} is the pressure difference maintained across the throttle, and (P_{1}+P_{2})/2 is the mean pressure. The zeropressure value of the isothermal Joule–Thomson coefficient is thus given by
Integration between a suitable reference temperature T_{ref} and T yields^{61 }
This relation is of considerable importance for obtaining virial coefficients (of vapours) in temperature regions where conventional measuring techniques are difficult to apply. The isothermal Joule–Thomson coefficient of steam, the most important vapour on earth, was recently reported by McGlashan and Wormald^{88 } in the temperature range 313 K to 413 K, and values of φ_{0} derived from these measurements were compared with results from the 1984 NBS/NRC steam tables,^{89 } with data reported by Hill and MacMillan,^{90 } and with values derived from the IAPWS95 formulation for the thermodynamic properties of water.^{91 }
The thermodynamic speed of ultrasound (below any dispersion region) is related to the equation of state, and hence to the virial coefficients. For a real gas, v$02$ may thus be expressed as a virial series in molar density 1/V,^{92 } i.e.
where
For constantcomposition fluids, the acoustic virial coefficients B_{ac}, C_{ac}, … are functions of temperature only. They are, of course, rigorously related to the ordinary (PVT) virial coefficients. For instance,
Since pressure is the preferred experimental parameter, one may also write a virial expansion for v$02$ in powers of the pressure with corresponding virial coefficients B$ac\u2032$, C$ac\u2032$, … The coefficients of the density and pressure expansion are interrelated; for example
Thus, measurements of the speed of ultrasound as function of density (or pressure) will yield information on B together with its first and second temperature derivatives, and C$Vpg$ (or κ^{pg}) through extrapolation of v$02$ to zero density. The principal advantages of the acoustic method are its rapidity and the greater accuracy at temperatures where adsorption effects become important.^{93 }
All this valuable thermophysical information can then be used to obtain reliable second virial coefficients over large temperature ranges. For a fluid with spherically symmetric pair potential energy u(r),
where N_{A} is the Avogadro constant and k_{B} is the Boltzmann constant. Inversion^{94 } then yields the fundamentally important potential energy function u(r) for a pair of molecules.
While a discussion of experimental acoustical methods is way outside the scope of this introductory chapter, the following comment is indicated. For gases/vapours at low to moderate pressures not too close to saturation, the highest experimental precision, when measuring v_{0}, is obtained through use of a spherical resonator, a technique which was pioneered by Moldover, Mehl and coworkers.^{95,96 }
So far, the focus was on homogeneous constantcomposition fluids, of which pure fluids are special cases. I will now briefly consider the case where a pure liquid is in equilibrium with its vapour. Such a situation is encountered, for instance, in adiabatic calorimetry, where the calorimeter vessel is incompletely filled with liquid in order to accommodate the thermal expansion of the sample (usually, the vapour space volume is comparatively small). One has now a closed twophase singlecomponent system. The heat capacity of such a system is closely related to C$\sigma L$, i.e. the molar heat capacity of a liquid in equilibrium with an infinitesimal amount of vapour (as before, the saturation condition is indicated by the subscript σ). For a detailed analysis see Hoge,^{97 } Rowlinson and Swinton,^{56 } and Wilhelm.^{98 }
The molar heat capacity at saturation of the substance in the equilibrium phase π (denoting either the liquid, π = L, or the vapour, π = V) is given by C$\sigma \pi $≡T(∂S^{π}/∂T)_{σ}, whence one obtains, for instance,
Here, γ_{σ}≡(∂P/∂T)_{σ} is the slope of the vapourpressure curve, and
denotes the expansivity of a pure substance in contact with the other equilibrium phase (i. e. along the saturation curve). As already pointed out, below the normal boiling point, the difference α$PL$ – α$\sigma L$ is usually negligibly small. At the critical point
Neither C$P\pi $ nor C$\sigma \pi $ is equal to the change of enthalpy with temperature along the saturation curve. From Equation (65) one obtains
Since U = H – PV,
Thus for the saturated liquid at [T, P_{σ}(T)] at temperatures where Tα$PL$<1, the following sequence is obtained:
The differences between the first four quantities are generally much smaller than between C$VL$ and (∂U^{L}/∂T)_{σ}.
While the general equations apply also to the saturated vapour (π = V), the inequality does not. Since α$PV$V^{V} is always large, for saturated vapours the difference C$\sigma V$–C$PV$ is always significant [see Equation (104)]. In fact, for vapours of substances with small molecules, such as argon, carbon dioxide, ammonia and water (steam), α$PV$V^{V} may be large enough to make C$\sigma V$ even negative. Finally we note that the difference between the saturation heat capacities in the vapour phase and the liquid phase may be expressed as^{98 }
and the difference between the isobaric heat capacities in the vapour phase and the liquid phase as
where Δ_{vap}H denotes the molar enthalpy of vaporisation, and Δ_{vap}V≡V^{V} − V^{L} is the volume change on vaporisation. In deriving these equations, use was made of the exact Clapeyron equation
and the exact Planck equation.^{99 }
There are, of course, many additional details and fascinating topics, in particular when mixtures and solutions are considered, which fact is amply evidenced by the contributions to this monograph. Enjoy!
1.3 Concluding Remarks
Calorimetry and PVT measurements are the most fundamental and also the oldest experimental disciplines of physical chemistry. Although simple in principle, enormous effort and ingenuity has gone into designing the vast array of apparatus now at our disposal. In this introductory chapter, I did not cover design of experiments beyond the bare rudiments – the reader is referred to the relevant articles and books quoted, and to the chapters of this book focusing on this aspect. Let it suffice to say that the advances in instrumentation during the last decades have greatly facilitated the highprecision determination of caloric and PVT properties of fluids over large ranges of temperature and pressure. At the same time crossfertilisation with other disciplines, notably with ultrasonics and hypersonics, and with biophysics, is becoming increasingly important, as is the close connection to equationofstate research and, of course, chemical engineering.^{56,61,79,98,100–103 } The discussion presented here and in the chapters to follow may perhaps best be characterised by a statement due to Gilbert Newton Lewis (1875–1946) on the practical philosophy of scientific research:
The scientist is a practical man and his are practical aims. He does not seek the ultimate but the proximate. He does not speak of the last analysis but rather of the next approximation. … On the whole, he is satisfied with his work, for while science may never be wholly right it certainly is never wholly wrong; and it seems to be improving from decade to decade.
By necessity, this introductory chapter is limited to a few topics, the selection of which was also influenced by my current interests. In conclusion, I hope to have:
formulated concisely some important aspects of the thermodynamic formalism needed in this area of research;
discussed and made transparent some key aspects of experiments;
shown how to apply and to appropriately extend wellknown concepts to perhaps less familiar, yet potentially important, problems;
stimulated some colleagues to enter this fascinating and important field of research.
Success in any of these points would be most rewarding.
In this chapter the isothermal compressibility is represented by the symbol β_{T}and not by κ_{T} as was recently recommended by IUPAC. Similarly, the isentropic compressibility is represented by the symbol β_{S} and not by κ_{S}