2.3 molecular thermodynamics - treccani · 2019-12-28 · in a simulation of phase equilibria. ......

63VOLUME V / INSTRUMENTS

2.3.1 Introduction

Molecular thermodynamics investigates the equilibriumproperties of matter, exploiting knowledge of molecularstructure and the forces of molecular interactions. It is a fieldof study that is capable of providing answers to problems thatarise in the development of several modern technologies,including technologies of paramount importance in the oiland petrochemical industries. The study of thermodynamicequilibrium in heterogeneous systems, in particular, whichrequires the use of relations that allow the correlation and atleast prediction of the distribution of components among thedifferent phases present in a thermodynamic system, isapplied in the design and development of separationprocesses, in materials characterization, and in determiningthe distribution of hydrocarbons in oilfields.

The assumptions of this approach lie in statisticalmechanics, which was defined as the “art of predicting thebehaviour of a system with a high number of degrees offreedom, once the laws determining its microscopicbehaviour are known” (Binney et al., 1992). Molecularthermodynamics differs from statistical mechanics in theimportance given to the role of molecules in the solution ofproblems taken into consideration and because it is orientedto applications.

Three features concerning molecular structure andbehaviour are particularly important in the followingdescription:• Molecular geometry, describing the spatial distribution of

the nuclei of the atoms forming the molecules.• Dynamics, relating to the diverse molecular motions with

their associated energy values.• Interactions, relative to the nature and the magnitude of

intermolecular forces. If one refers to a biatomic molecule, for simplicity, it can

undergo different types of motion: translation (described bycomparing the molecule to a point mass concentrated at itscentre of mass), rotation (taking place around an axis passingthrough the centre of mass and perpendicular to the axis joiningthe two atoms) and finally vibration along this same axis.

Generally speaking, and neglecting reciprocalinteractions, molecular energy can be expressed as the sum ofdifferent mutually independent contributions, each

corresponding to the different types of motion previouslymentioned:

[1] e �etr�erot�evib�eel

where etr, erot, evib are the energies associated withtranslational, rotational and vibrational motions respectively;finally eel indicates the energy associated with the differentelectronic states of the molecule itself. Each of the terms inthe equation above is quantized since it can only take onwell-defined values. As an example, the relations obtained byusing quantum mechanics to calculate the values of the energylevels of biatomic molecules are summarized in Table 1.

The calculation of the energy values of rotational andvibrational quantum states in a polyatomic molecule isobviously more complex than that of the relationssummarized in Table 1. A polyatomic non-linear molecule ischaracterized by three principal inertial moments referring tothree mutually orthogonal axes passing through its centre ofmass. Moreover, other degrees of freedom associated withinternal rotations around simple bonds often exist, such asC�C bonds in hydrocarbons, as shown in Fig. 1.

A polyatomic molecule formed by n atoms has severalvibrational degrees of freedom, precisely 3n�5 if it is linear,3n�6 if is non-linear. Assuming that the variation of

2.3

Molecular thermodynamics

Table 1. Energy of the states associated with the internalmotions of a biatomic molecule A�B

Rotational motion h2

erot�11 J(J �1)8p2I

J �0,1,2,… quantum number

mAmBI �1131 ro2 moment of inertia

mA�mB

Vibrational motion 1 hnevib� hn�23�n��223�hnn

2 2

n �0,1,2,… quantum number

n �frequency of vibration

intramolecular forces for small displacements could bedescribed as a linear relationship, it is possible to decomposethe complex movements to which the different atoms aresubject to 3n�6 modes of independent harmonic vibration(called normal vibration). Each normal vibration ischaracterized by its specific frequency, and the vibrationalenergy of the molecule is given by the sum of the energiescompeting in each vibration.

Since, in ordinary temperature conditions, the energydifference between the fundamental electronic state and thefirst excited state, for almost all molecules, is much higherthan the average thermal energy, it follows that even atrelatively high temperatures, it can be assumed thatmolecules reside in their fundamental electronic state.

2.3.2 Statistical thermodynamicsassumptions

In a thermodynamic system at constant temperature T, it ispossible to identify different configurations related to thedistribution of its N molecules in the space contained in avolume V. A value Er of energy and a value Yr of a specificobservable variable correspond to each of these, the averagevalue of which can be obtained through the following relation:

[2] �Y��r

wrYr

where wr represents the probability that the system lies in the r configuration, expressed via the canonical distributionlaw:

e�bEr

[3] wr�11Z

where b�1/kBT, kB being the Boltzmann constant, while thepartition function Z is defined as

[4] Z(T,V,N )��r

e�bEr

Equation [3], which is the fundamental law of statisticalmechanics, is not derived here, while its relationship with theconcepts of thermal equilibrium and temperature aredescribed, and its application to various specific cases related

to the behaviour of fluids and their mixtures is also discussed. The partition function is directly linked to the Helmholtz

free energy function F:

1[5] F ��23 ln Z

b

which allows the evaluation of internal energy U, entropy Sand pressure P:

� ln Z[6] U ��131�

V,N�b

�F U �F[7] S��31�

V,N

�131�T T

�F[8] P ��31�

T,N�V

Therefore, once the partition function is known, it ispossible to derive the most important thermodynamic functionsand thus develop their application for the different problems.

2.3.3 Theory, simulation and experience

A direct application of equations [2] and [3] can be found insimulation techniques that have become widespread in recentyears due to the development of computer software. Forinstance, the Monte Carlo method takes into considerationmodel systems composed of hundreds, if not thousands, ofobjects which describe the molecules contained in a volumeV, and calculate the average of the properties associated withseveral configurations, randomly built by generating astochastic chain of events, called a markovian chain,produced by random motions of single molecules. Thecalculation of the average value of thermodynamic propertiesis carried out by [2], attributing to the probability adependence on energy given by equation [3].

A typical application of the Monte Carlo method isshown in Fig. 2, which refers to a bidimensional systemcomposed of six particles, for simplicity. It is possible tostart from a central configuration, surrounded by ‘phantom’

PHYSICAL AND CHEMICAL EQUILIBRIA

64 ENCYCLOPAEDIA OF HYDROCARBONS

A B

Fig. 1. Internal rotations of butane (A) and propylbenzene (B)molecules.

configurations in order to satisfy boundary conditions,which require the conservation of the number of particlesinvolved in the simulation. In fact, if one particle, followinga random movement, leaves the central zone, it is made tore-enter symmetrically at the side opposite to the one it left.

These simulation techniques make it possible to studysystems with two phases in equilibrium, proceeding forinstance according to the scheme shown in Fig. 3. This methodis called ‘statistical ensembles’, and it basically concernsfinding the minimum value of the Gibbs free energy function

at a given temperature, pressure, and the chemical potential ofthe components in the two coexisting phases. The two phasesare ideally separated into two boxes which are simulatedsingularly to make the calculations easier by virtue of theabsence of the interphase surface. The simulation involvesthree separate types of molecular movements respectively:internal equilibrium (by moving the molecules in each box),mechanical equilibrium (which imposes pressure equality inthe two phases, even following small volume variation) andfinally chemical equilibrium (which imposes the equality ofchemical potential, even following molecular exchangebetween the two phases). This method was successfullyapplied to several one and two-component systems by usingappropriate molecular models which take specific details intheir structure into account.

These simulation methods have now reached a considerablelevel of sophistication, both for their ability to describe largesystems and, above all, for the possibility to exploit molecularmodels of increasing complexity and realism. The resultsobtained in this way have proved to be of great importance inclarifying some conceptual aspects on matter behaviour.Nevertheless, the development of analytical expressions that areeasy and fast to apply still maintain great relevance. Thesemethods deal with the evaluation of thermodynamic propertiesof pure substances and their mixtures and the determination ofequilibrium conditions between different phases, operating atdifferent levels and linked together, which involve experience,theory and simulation.

Actually the solution of applicative problems oftenrequires the use of some effective parameters which reflect awhole group of specific factors. In principle, they can be

MOLECULAR THERMODYNAMICS


Fig. 2. Example of the application of the Monte Carlo method to a bidimensional system made up of six particles.

region I

region II

region I

initialconfiguration displacement

volumevariation

particletransfer

region II

vapour

T

liquid

Fig. 3. Scheme of applicationof the Gibbs’ statisticalensembles method in a simulation of phaseequilibria.

obtained from knowledge of the geometric and dynamiccharacteristics of molecules but unfortunately the currentstate of understanding is not at a high enough level as to giveresults of an accuracy compatible with the necessaryrequirements for the design of industrial units. It is thereforenecessary to use the information deriving from thecorrelation of experimental data in order to calibrate some ofthe parameters under examination semiempirically.

In the following, some fundamental concepts of molecularthermodynamics will be illustrated, obviously dedicatingparticular attention to its application in fluid systems, gas andliquid, with one or more components since, it is known thatthey are the key players in the oil industry. The goal is toprovide the basis for calculating the thermodynamic propertiesof hydrocarbons and their mixtures.

2.3.4 Intermolecular forces

A fluid whose molecules interact with an intermolecularpotential of the central type, which depends solely on thedistance between the centres of mass of the molecules, willbe referred to as ‘simple’. Strictly speaking, only the systemscomposed of monoatomic gases belong to this category;nevertheless, the fluids with relatively small and compactmolecules such as oxygen, nitrogen and methane can also be

considered simple. In this case, the intermolecular potentialu(r) is described by a curve with a shape similar to thatshown in Fig. 4 A. At short distances, repulsive forces appear,the potential of which increases rapidly with decreasingdistance, while attraction forces prevail as distance increases.As the distance goes to infinity the potential tends to zerocompatibly with the fact that the interaction between themolecules dies away. Repulsion occurring at short distancesis due to the overlapping of molecular electronic clouds,while attractive interaction (according to the interpretationfirst devised by Fritz Wolfgang London) is linked to theinstantaneous interaction between the molecules caused bythe dipoles, created by the motions of their electrons, thattend to oscillate in a synchronous or sympathetic mode. InFig. 4A two characteristic parameters can be identified: s,representing the distance at which the potential is equal tozero and u0, expressing the energy corresponding to itsminimum. An analytical form of intermolecular potential isgiven by the Lennard-Jones potential:

s s[9] u(r)�4u0��23�12

��23�6

�r r

Attractive interaction, called ‘dispersion interaction’, is,according to the above-mentioned London theory, inverselyproportional to the sixth power of the distance between thecentres of mass of the two molecules, and the parameterspresent in equation [9] can be determined by experimentalobservation on fluid behaviour. The data on state behaviour,or P, V, T behaviour, are used in particular. In Table 2 thevalues of these parameters for some common molecules arereported, derived by using the experimentally determinedvalues of the second virial coefficient. Obviously moleculeswith heteropolar bonds, and therefore with a permanentdipole electric moment, also display an electrostaticinteraction between dipoles that depends on the orientationbetween the molecules themselves. In these cases, thepotential has anisotropic characteristics since not only does itdepend on distance r, but also on the angles that describe themutual orientation of the two interacting molecules.



A

B

u(r)

�10

15 (e

rg)

u(r)

�1.0

�0.5

0

0

u0

s

s

0.5

1.0

1.5

r (A°)

r

2.5 3.5 4.54.03.0

Fig. 4. A, shape of the intermolecular potential. Numerical values of energy (erg) and distance (Å) refer to the case of helium. B, approximate shape of intermolecular potential corresponding to molecules considered similar to hard spheres with attractive interaction.

Table 2. Parameters of the intermolecular potentialof some substances

SubstanceIntermolecular parameters

s (Å) u0/kB (K)

He (helium) 2.576 10.2

H2(hydrogen) 2.915 38.0

N2(nitrogen) 3.681 91.5

O2(oxygen) 3.433 113

Cl2 (chlorine) 4.115 357

CH4(methane) 3.822 137

C2H6(ethane) 4.418 230

C3H8(propane) 5.061 254

C2H4(ethylene) 4.232 205

C2H2(acetylene) 4.221 185

C6H6(benzene) 5.270 440

Taking this into consideration, the intermolecularpotential for a pair of molecules can be expressed in thefollowing form:

[10] u(1,2)�u0(r)�uelet(1,2)�uind(1,2)�udisp(1,2)

Both the relative position of two molecules, and theirmutual and internal orientation are indicated by (1,2). Theterms present in the second member of the previousexpression indicate the following contributions respectively:a) u0(r) contribution depending only on distance; b) uelet(1,2)electrostatic interactions; c) uind(1,2) inductive interactionsdue to electronic clouds distortion; and d) udisp(1,2)dispersive interactions.

Even though theoretical and computational chemistry hasobtained several important results in the calculation ofmolecular energies in the last few years, a priori evaluationof intermolecular forces still presents some uncertainties.When studying equilibria between phases, effective potentialsare widely used, the parameters of which are evaluated in asemiempirical fashion, and which are essentially related tothe Lennard-Jones potential. An important aspect concernsthe transferability of interactions between single atoms, oratomic groups in the molecules. This issue is directlycorrelated with the group contribution models, which haveachieved particular relevance in several practical applications.

In its most simple form, the potential of two moleculescan be written in the following way:

qgqd[11] u(1,2)��

a,b

uab ��g,d

11

rgd

where a and b indicate the sites or groups in the molecules onwhich interactions are centred, while g and d are thoseinvolved in electrostatic interactions due to charges qg and qd.

Finally, a hydrogen atom in a molecule can act as abonding element (hydrogen bond) between an electronegativeatom, linked to it by a covalent bond, and anotherelectronegative atom, such as fluorine, oxygen and nitrogen.The hydrogen bond has an energy content lower than covalentbonds. The orders of magnitude of the energies involved are infact: non bond forces 0.2-2 kcal/mol; hydrogen bond 2-10kcal/mol; covalent bonds 50-100 kcal/mol.

Significant examples of hydrogen bonds can be found inhydrogen fluoride and in water. If hydrogen bonds are present,electrostatic forces obviously take on a dominant role.

Hydrocarbon molecules have a relatively complexstructure: elongated, ramified or cyclic; in this casedecomposing the interaction in the sum of the contributionscorresponding to the different groups of atoms present in themolecules is particularly advisable. In conclusion, in theanalysis that follows, it is reasonable to divide fluids intosimple and complex, relating to the properties of the

molecules of which they are composed, on the basis of theclassification shown in Table 3.

2.3.5 Calculation of the free energy function

The determination of the most important relations thatoperate in the solution of phase equilibrium problems stemsfrom the expression of Helmholtz free energy F(T,V,Ni), fromwhich it is possible to derive the pressure of the systemthrough equation [8] and the chemical potential of a genericcomponent through the following relation:

�F[12] mi��2323�

T,V,Nj�i�Ni

where the different components are indicated by i and j andtheir number of molecules by Nij.

In the following discussion, statistical thermodynamicswill be applied first to one-component systems, whilesubsequently many-component mixtures will be taken intoconsideration. Consider, then, a fluid containing N moleculesof mass m and introduce a typical length l�h�(2pmkBT)1�2,proportional to the molecular de Broglie wavelength, where his the Planck constant. It is possible to demonstrate that thepartition function Z can be expressed as

1[13] Z(T,V,N )�1 l3Nzint(T)NQ(T,V,N )

N!

The presence of the l term derives from the moleculartranslational motion, whereas

[14] zintzvib,rot��vib,rot

e�b(evib�erot)

is the molecular partition function related to internal motionssince it reflects the contribution due to vibrations and rotationsand its specific form is related to the properties of the moleculeunder consideration. In general, it does not depend on volumeand therefore it does not contribute to determine the statebehaviour of the system under examination. In the followinganalysis zint is not explicitly considered, since attention ismainly focussed on the investigation of state behaviour and ofphase equilibrium; for a thorough discussion of its role seeChapter 2.4, where the dependence of thermodynamic statefunctions on temperature is specifically examined.

Finally Q is called configuration integral and it is definedby the following relation:

[15] Q (T,V,N )Q �…e�b/(r1,r2,…,rN)dr1dr2…drN

where dr1, dr2,…,drN are elements of volume centred on thepositions of the different molecules as identified by vectors



Table 3. Subdivision of fluids into simple and complex

Fluids

simple Intermolecular forces can be described by a central potential

complex

– Intermolecular forces are acentric, above all due to dipolar, quadripolarinteractions, and so on

– The molecular structure is so complex that it is advisable to decompose theirinteractions in the sum of the contributions provided by the different atomicgroups present in the molecules themselves

r1,r2,…,rN, whereas / is the total potential energy due tointeractions between molecules.

Introducing [13] into [5] one gets

1 l3N zNint 1 1 Q

[16] F ��23 ln�1231��23 ln Q �Fid�23 ln12b N! b b VN

where

1 l3N zNint[17] Fid��23 ln�1231VN�b N!

is the free energy of ideal gas. Introducing [16] into [8] oneobtains

1 � ln Q[18] P �23 �121 �

T,Nb �V

which is identified with the equation of state of the system. Inorder for it to be valid even in proximity of the critical point,where the free energy function displays non-analyticalbehaviour, it is advisable to refer to the specific value of Qfor a single molecule, but making N and V tend to infinity,even though the value of molecular density r�N�V is keptfixed. This procedure is called the thermodynamic limit.

In reality, the application of equation [18] is severelypenalized by the calculation of the Q integral, which can beperformed only if appropriate approximations are introduced,the most drastic of which being the assumption that / is zeroand that molecules are punctiform. In this case, the integral isequal to VN and therefore one easily gets

[19] PV�NkBT

that is the ideal gas state equation.The potential energy of the group of molecules is usually

evaluated by intermolecular potentials through an additiverelationship, in which the interaction energies of the differentpairs of molecules are added:

[20] / ��i�j

uij(r)

2.3.6 Van der Waals theory

Before tackling the problem of calculating the configurationintegral Q, in general terms, it is advisable to spend sometime introducing a simplified approach to evaluate thepotential energy of molecules. This assumes that moleculeshave a uniform distribution, and therefore each of them issubject to an average field, and its potential energy is notinfluenced by fluctuations due to molecular motions.

For simplicity, it is also assumed that the intermolecularpotential is of the central type and has the shape shown inFig. 4 B, in which molecules are considered similar to hardspheres that cannot be overlapped, with a diameter equal to s,and subject to an attraction of the �u0s6�r6 type for r�s.Considering the interaction of a central molecule with thosesurrounding it, which are contained in a spherical shell with a4pr2dr volume, the potential energy of the fluid can bederived by calculating the following integral:

1 N 2 N2 aN2

[21] / �23Nu(r)4pr2 233 dr ��23p13 u0s3��122 V 3 V V

Since the distance in which the interaction appears isshort, the integral was extended to infinity in order to avoidthe tricky problem of boundaries. Therefore, the following

approximation �Q��exp(baN2�V) can be given. In addition,since molecules cannot get closer than a distance equal to s itis appropriate to define a covolume b�1�2(4�3pNs 3)representing the volume that is not accessible to moleculesdue to their impenetrability caused by the ‘hard’ portion oftheir potential. Accounting for this, the free energyexpression can be written as:

1 �Q�[22] F �Fid�

23 �ln13� ln VfN��Fid�b VN

1 Vf�23 ln��13�

N

ebaN 213

V �b V

where the free volume, or accessible to the molecules, isVf�V�b whereas parameter a, reflecting the influence of theattractive portion of potential, is defined by [21].

By replacing the previous equation into [8], thewell-known van der Waals equation can be derived

RT ã[23] P �11�1

V~� b~ V~ 2

whose isotherms have the typical shape shown in Fig. 5. Theprevious equation refers to one mole of fluid and therefore Nis the Avogadro number which, multiplied by kB, gives the gasconstant R, whereas V~, ã, e b~ refer to one mole. At a hightemperature, the shape of isotherms approaches that ofequilateral hyperboles, typical of an ideal gas, whereas acritical temperature Tc exists, at which the isotherm displays ahorizontal point of inflection. At lower temperatures,isotherms have the shape shown in Fig. 5. The presence of apart where pressure increases with increasing volumecorresponds to an unstable equilibrium state, since a smallalteration of the variables that identify it is sufficient toproduce a sharp spontaneous transformation towards a two-phase system, composed of a mixture of liquid and vapour inequilibrium. The identification of the characteristics of thetwo states in equilibrium can be performed through thegraphic method proposed by Maxwell, stating that the twoareas shaded in the figure must be equal. In point A and B the molar free energy must have the same value, and



P

T�Tc

T�Tc

Tc

V

AB

Fig. 5. Shape of isothermal curves in van der Waals equations.The region surrounded by the dashed line (Andrews’ bell curve)corresponds to the existence of a two-phase system made of liquid and vapour in equilibrium.

therefore its variation along the isotherm, given by theintegral of VdP, is zero only if the values of the two shadedareas are the same. In this way, a criterion to draw thehorizontal segment joining the two equilibrium states isderived. By applying this procedure to several differentisotherms, a series of points is identified. Joined together,these points define a typical curve with a bell-shape, calledan Andrews curve, which defines the zones of existencebetween the two phases. Therefore equation [23] provides acomplete description of the state behaviour of a fluid.

Parameters ã e b~ of the equation of state can be estimatedfrom the critical values of the state variables Pc, Vc, Tc. Anequation free from parameters specific to particular fluids isobtained, reflecting a general law called ‘of thecorresponding states’ whose validity surpasses the equationthat inspired it: all fluids can be described by the sameequation of state, as long as it is expressed by the reducedvariables (P�Pc, V�Vc, T�Tc). This only really happens withsimple fluids, whose molecules interact with a potential thatdepends solely on the distance between their centres of mass.

The van der Waals equation therefore satisfactorilydescribes the behaviour of fluids, including the occurrence of aliquid-vapour transition, and the existence of a critical point.However, from a quantitative point of view, its application isnot so satisfactory; this is demonstrated by the fact that theadimensional ratio PV�RT, evaluated at the critical point, takeson a universal value equal to 0.378, while experimentally avalue included between 0.27 and 0.29 is measured.

2.3.7 Virial development

A general method to calculate the integral of configurationsQ was proposed by Joseph Mayer and Maria Göpper Mayer(Mayer and Göpper Mayer, 1940). By introducing anappropriate function

[24] fij�e�buij(r)�1

it is possible to demonstrate that Q takes on the form

[25] Q(V,T,N )�…e�b�i�j

uij(r)drN�

�…�1��i�j

fij�� fij fkl�…�drN

where drN�dr1dr2…drN .The calculation of the different terms present in the

previous equation is made easier by the use of appropriatediagrams. Indeed, by indicating a molecule with • and the fijthat connects the molecules i and j with a, a diagramcorresponds to each term of development, as shown in Table 4.

It is then possible to identify a sequence of groups, orclusters, of molecules containing all possible diagramsconnecting a determined number of points directly or indirectly,as shown in Fig. 6. It is then possible to evaluate the contributionprovided by each one of them to the integral of configurations.

This approach produces an equation of state that has theform of a power series expansion of molecular densityr�N�V:

[26] P �rkBT(1�Br �Cr2�Dr3�…)

It is interesting to observe that the previous equation isknown as a virial equation of state, proposed on

phenomenological and empirical bases by HeikeKamerling-Onnes. The statistical approach previouslydescribed gives a precise physical meaning to the differentterms of the development that express the interactions ofdifferent groups of molecules. Moreover, it provides somerelationships that allow the calculation of the values of thecoefficients of the virial development (or virial coefficients)from the parameters of intermolecular potential. For example,by using the Lennard-Jones potential [9], the followingexpression of the second virial coefficient can be derived:



contributionsto virial

coefficientsmolecular clusters

second

third

fourth

.... (12 of the same type thatdiffer only in numeration)

1 2

2

2

2

2

1

4

3

1

3

3

3 4

2

3

3

44

1

4 3

2

1

4 3

2

1

2

1

2 3

321

1

1

1

�

�

.... (4 of this type)�




�

�

�

�

� (only this)

� �

Fig. 6. Example of diagrams representing groups of interacting molecules.

Table 4. Diagram connecting molecules

Integrand Diagram

1

f12 1 2

f12 f23

2

31

B(T*)[27] 121�3

0�1�e�u(r*)�T*� r*2dr*

b

where b is the covolume previously defined, r*�r�s thereduced distance, and T*�TkB�u0 is the reduced temperature.

In Fig. 7, a unified comparison is reported between thesecond virial coefficient, evaluated through equation [27] andthe experimental values for a series of molecules having acompact configuration, such as rare gases, nitrogen andmethane. It is possible to observe that, by adequatelycalibrating the parameters u0 and s, one can obtain anexcellent agreement between calculated and experimentalresults, confirming the relevance, but also the limits, of thecorresponding states law, which is valid for molecules subjectto a central potential.

As mentioned, the different terms of development [26]reflect the interactions among the groups that contain anincreasing number of molecules respectively, in agreementwith what is shown in Fig. 6. In principle, each one of themcan then be evaluated by the intermolecular potential;unfortunately, though, the corresponding expressions becomeincreasingly complex and thus cumbersome to evaluate.

Virial development is also applied to mixtures, providingadequate expressions of coefficients as a function ofcomposition given as molar fraction yi:

[28] Bmix��i, j

yi yj Bij

Cmix��i, j,k

yiyj ykCijk

……

where Bij , Cijk, etc. indicate a series of virial coefficientswhich depend only on temperature. The previous treatment isformally correct and elegant, but unfortunately thedevelopment through which the state behaviour is expressed,even though enriched by an increasingly higher number ofterms, does not converge when density approaches the highvalues typical of liquids. Its application therefore does notallow the description of condensation processes. Thiscircumstance pinpoints a severe limitation of the method,which forces us to take into consideration the problem of

continuity between the gaseous and the liquid state by using adifferent approach, knowing that in any case van der Waalstheory, even though apparently simpler, is capable of givingan answer to this problem.

2.3.8 Meaning and potentiality of the van der Waals theory

The condensation process is the subject of many studies andsome of their features deserve to be considered in detail, evenif only at a basic level. In order to do so, it is interesting torecall an investigation performed by Kac, Uhlenbeck andHemmer in 1963 (Kac et al., 1963; Uhlenbeck et al., 1963),which goes back to the calculation for the configurationintegral Q using an approach that is different from thoseconsidered until now. For this purpose, attention is focused ona monodimensional fluid, in which its N molecules arerandomly arranged along a segment of length L. Moreover, itis possible to suppose that they interact according to apotential expressed by the following relation:

r �d repulsiva forces[29] u(r)��age�gr r �d attraction forces

Even though it appears hardly realistic, it is easy to verifythat by using it, a value of potential energy of the systemequal to �a can be obtained, which is therefore constant andindependent from parameter g characterizing the radius ofinfluence of intermolecular potential. Having said this, it ispossible to demonstrate that, for a fluid whose molecules arearranged on a segment with length L, the configurationintegral Q can be calculated exactly, even within thethermodynamic limit. In this way, the correct equation ofstate of a monodimensional fluid can be derived, showingsome interesting singularities. Indeed, by examining thebehaviour of the fluid in the g��0 limit, which is in thesituation where attraction forces are weak but have a radius ofinfluence tending to infinity, even though the potential energyof the fluid remains unaltered, a monodimensional equationof state in the same form as van der Waals equation [23] canbe obtained where, however, the volume must be replaced bylength L. Meanwhile, it is possible to demonstrate that atransition of phase exists for which the zone of coexistence ofliquid and vapour is described by an isothermal horizontalsegment, without having to resort to the constructionproposed on heuristic bases by Maxwell.

The result obtained in this way obviously does not finddirect practical applications, since the situation taken intoconsideration is not realistic, but it has an interestingconceptual relevance since it emphasizes that thecondensation process is compatible with a physical situationwhere the ratio between the radius of repulsive forces andthat of attractive forces is much less than one. In this way it ispossible to clarify the apparent paradox by which therelatively simple van der Waals equation is capable ofdescribing the existence of both phases of a fluid, unlike themore complex virial development. Moreover, the way isshown to extend the theory adequately to more realistic andcomplex molecular systems. Indeed, it includes a separationof repulsive intermolecular forces, which determine thestructure of the fluid, and therefore its entropy, from attractiveforces that affect the value of internal energy. In other terms,this corresponds with observing that in the description of the



B*�

B/b

�4

�3

�2

�1

0

1

T*�kB/e

0.5 1 2 5 10 20

argonneonmethanenitrogen

50 100

Fig. 7. Comparison between calculated curves and experimental values of the second virial coefficient for a few simple molecules.

thermodynamic behaviour of fluids, it is possible to separatethe entropic effects associated with the distribution ofmolecules in space from the energetic ones.

2.3.9 Perturbative methods

Perturbative techniques are used in many different fields ofphysics such as, for instance, astronomy and quantummechanics, where some of the properties of a system areexpressed on the base of the known properties of a suitablereference state of the system. In thermodynamics they can beusefully applied to the calculation of the free energy function.

By taking into account the results obtained in theprevious paragraph – stating that the structure of a densefluid is essentially determined by the hard portion ofintermolecular potential, whereas the soft part reflectingattractive forces exerts a correcting action of minorimportance – it is possible to divide the potential energy ofthe system into the sum of two contributions

[30] / �/0�/p

The first corresponds to a suitable reference state of thesystem, whereas the second is a corrective term that can beconsidered as a perturbation. Indicating the free energy of thereference state with F0, it is possible to write

1 1[31] F�F0��23 ln 2333…e�b(/0�/p)drN�

b Q0

1 e�b/0 1��23 ln…e�b/p23133 drN��23 �ln e�b/p�

b Q0 b

The last expression represents the average value of theexp(�b/p) exponential of the perturbative energy, evaluatedby using the canonic distribution function [2] of thereference state of the system. It can be written in a form thatis easier to interpret and use, by developing F with respect toparameter b

b[32] F�F0��/p��23 [�/p

2��/p�2]�O(b2)

2

where the third term at the second member represents theaverage value of fluctuations of potential energy.

Before proceeding, it is important to observe that thethermodynamic properties of a fluid can be evaluated byusing a suitable function g(r) called ‘radial distribution’,which describes the average local variations of density thatare present around a molecule. As shown in Fig. 8, startingfrom the centre of a molecule, there are density variationswhich reflect the local structure of the system underexamination. It is possible to observe that when r increases,the value of the radial distribution function approaches 1since at high distances from the reference molecule theinteraction with the surrounding molecules tends to becomeuniform thus reaching the average value of the density of thefluid itself.

The radial distribution function can be experimentallydetermined by sending a collimated monochromatic beam ofX rays or neutrons through a specimen, and then measuringthe intensity of the diffracted radiation with a detector. Fromthis, by means of a suitable normalization operation, it ispossible to derive a S(k) structure factor which depends onthe wave number k�2p�l, where l is the wavelength of theincident radiation (in the case of neutrons it is given by h/mv).This factor is correlated to the radial distribution functionthrough the following relation:

[33] S(k)�1�rg(r)eik�rdr

where the integral at the second member is the Fouriertransform of g(r). From the radial distribution function it ispossible to calculate the potential energy of the fluid throughthe following relation:

2pN2

[34] / �131

0

g(r)u(r)r2drV

Furthermore, it is possible to demonstrate that the pressureof the system can be expressed in the following form:

1 �u[35] P �rkBT�23r24pr2g(r) 233 dr

6 �r

from which it is possible to obtain the equation of statewithout having to involve free energy.



A B

0

1

2

s3

g(r)

s 2s r0

1

g(r)

s s�l r

l

s

Fig. 8. Shape of the radial distribution functions g(r), for a simple fluid (A) and for a fluid made up of diatomic molecules (B).

Assuming that the distribution of the moleculescorresponds to that of the reference fluid, the averageperturbative potential energy can be written in the form:

2pN2

[36] �/p�0�131

0

g0(r)up(r)r2drV

where g0(r) is the radial distribution function of thereference fluid itself, whereas up is the perturbative part ofthe intermolecular potential. Generally speaking, if a suitablesystem, with a known distribution function, is available andcan be used as a reference, it is possible to calculate thevarious thermodynamic properties, among which freeenergy, which can be expressed in a first approximation bythe following relation:

[37] F �F0��/p�0

compatible with [32] if the perturbative terms of orderhigher than one are neglected.

The first and more immediate choice of a system to beelected as a reference is the hard sphere system, thecharacteristics of which were already well-known thanks tothe detailed studies of simulation performed using both theMonte Carlo method and molecular dynamics. Afterwards,knowledge was enormously enriched when hard objects withdifferent shapes were taken into consideration, so that they cansimulate the core of the most common molecules, or whenobjects interacting through different potentials were examined.

For some of these systems, specific and accurateequations of state were developed, among which closeattention should be paid to the equation for a hard spherefluid (Carnahan and Starling, 1972):

P 1�h �h2�h3

[38] 1333�1111123� f(h)kBTr (1�h)3

where h�(1/6)prs3. This equation was derived heuristicallywith the aim of reconciling the virial equation of state withthe results of the simulations obtained by using the MonteCarlo method and molecular dynamics.

Obviously, the most simple hypothesis that can beformulated on the characteristics of hard sphere systems isthat they are uniformly distributed in space, therebyassigning to the radial distribution function a step-functionshape, equal to zero below the sphere contact distance, andequal to one above it. By proceeding further in thisapproach, it is possible to derive the van der Waals equation,which therefore can be viewed as the first application of aperturbative method to the study of a fluid.

If, on the other hand, one takes correctly into account thefact that the radial distribution function cannot berepresented by a step function even in a hard sphere system,but should be described by a function having a shape likethat shown in Fig. 8, an equation of state having thefollowing general form is derived

PV~ 1[39] 13� f (h)�13 y(T,V~)

RT RT

where f (h) is given by [38] while the second term at thesecond member represents the perturbative contribution dueto attractive forces. The previous equation represents ageneralization of the van der Waals equation that is easilyrecovered by assuming

1[40] f (h)�121

1�4h

This method can be extended easily to different systemswhich can possibly be used as reference and which, asmentioned above, are composed of hard bodies with variousgeometric structures such as disks, ellipsoids, spindles andso on, that simulate in an increasingly accurate way theshape of the hard core of molecules.

The results obtained in this way are satisfactory indescribing many fluids. However, in order to apply theperturbative approach previously introduced moreaccurately, it is also necessary to take terms of decreasingimportance present in [32] into account. In this way, it ispossible to accurately describe the liquid-vapour equilibriumof many-component mixtures, in particular of hydrocarbons,as shown in Fig. 9, as a function only of the parameters of theintermolecular potential. Actually, this research hasunderlined the importance of intermolecular potential,whose shape influences noticeably the results of thesimulation calculations and so the importance of identifyingthe theoretical charcteristics intended to obtain greaterknowledge on the characteristics and form of intermolecularpotential.

The application can be extended to fluids made of polarmolecules, for which the chosen reference is represented bya fluid made of molecules interacting through aLennard-Jones potential, assuming the dipole attraction asthe perturbative term.

2.3.10 Application to complexmolecules

In the previous section it was observed how the original vander Waals approach can be improved by using appropriateexpressions describing the behaviour of the reference fluidsmade of hard spheres more accurately. In this approach,obviously, also the free volume available to molecules shouldbe modified. For instance, it is possible to demonstrate that ahard sphere state equation such as [38] is compatible withthe following expression of free volume.

[41] Vf �Veh(3h�4)�(1�h)2

which should replace in [22] the V�b term present in theoriginal version of van der Waals theory.

Actually, molecules have been compared to interactingobjects lacking structure in the models taken intoconsideration up to this point, since their rotation andvibration movements are not influenced by the surroundingenvironment, and therefore they do not have any influenceon the form of the state equation. This hypothesis isrigorously valid for monoatomic gases as it represents areasonable approximation for small polyatomic molecules,but it is inadequate for large dimension molecules such ashydrocarbons with more than ten carbon atoms, and more sofor polymers.

A molecule made of n atoms has 3n degrees of freedom,three of which are associated to their translation movementsin the volume in which the fluid is contained. The remainingdegrees of freedom are distributed between the threerotation motions taking place around three orthogonal axes,with their origin at the centre of mass of the molecule, andthe different vibration motions. In agreement with aproposal originally suggested by Ilya Prigogine, the fact thatthe rotations and vibrations of an extended molecule, due to



its flexibility, can be influenced by the presence of moleculesclose to it as a result of steric interactions, should be takeninto consideration. Obviously, when the density of the fluidtends to zero this effect disappears since the molecules are,on average, very far from each other, and so the ideal gasbehaviour is recovered. On the other hand, when the volumeof the system approaches maximum packing, all degrees offreedom associated with their external movements areblocked. By developing this approach, the followingexpression of free energy referred to one mole wasproposed:

V~f

[42] F~�F~

id�RT ln��13�c �1

ebã�V~�V~

where c is a suitable parameter, equal to one for monoatomic molecules, which reflects the amount by which rotations and vibrations are hindered by the molecules nearby.

The previous approach allowed the development of ageneral model, called PHCT (Perturbed Hard-ChainTheory), which provides a state equation which extends fromfluids made of simple molecules to fluids composed ofcomplex molecules, thus including a density interval thatgoes from the ideal gas to liquids, in which maximumpacking conditions are present. There are several versions ofthis model which depend on both the expressions used for Vfand on the method adopted to evaluate the attractiveinteraction between molecules. Parameter c is usuallydetermined on the base of experimental information; in thecase of hydrocarbons it is equal to 1 for methane, 1.91 forhexane, 2.55 for dodecane and it increases regularly with thelength of the molecules.

2.3.11 Mixtures

The problem of many-component mixtures, which isobviously of primary importance in the applications ofthermodynamics to phase equilibria, was only touched on inprevious sections, and some important aspects will beinvestigated in more detail in the following. It is advisable to

develop this approach starting from expression [5] of the freeenergy of a pure gas made by N molecules. In the specificcase of an ideal gas, the integral of configurations is equal toVN; by using Stirling’s approximation

[43] lnN! �N lnN �N

one gets

1 N V[44] F ��23 [ln(l3NzN

int)�N]�23 ln 23b b N

The second term at the second member gives thedependence of free energy from volume, so that its variationsare associated to those of the configurations that the systemcan reach. If it is derived with respect to temperature, atconstants N and V, and change of sign, it provides theexpression of configurational entropy

V[45] S �NkBln 23

N

Consider various samples of ideal gas formed by differentcomponents, each of volume Vi, where Ni molecules arepresent; if they are mixed together and the volume additivityrule V� iNiVi, is assumed valid, the following expression forentropy variation associated with the mixing process can bederived:

∆S V Vi[46] 13��i

Ni�ln�13�� ln�13��i

Ni ln/ikB Ni Ni

where /i�Vi �V is the volume fraction of component i of themixture. The previous relation is known as the Flory-Hugginsequation which was originally, and independently, derived bythese authors on the base of a lattice model. In a mixture ofideal gases, /i corresponds to the molar fraction yi of thedifferent components, therefore if one refers to one mole [46]becomes:

∆S[47] 13��

i

yilnyiR

Actually, [46] is more general since it can also be appliedto real gases and to liquids, the molar volume of which isdetermined by the volume of the molecules themselves, vi. In



A B C

P (

bar)

80

120

160

200

CH4 molar fraction0 0.40.2 0.6 0.8 1.0 0 0.40.2 0.6 0.8 1.0 0 0.40.2 0.6 0.8 1.0

10

0

30

40

20

50

60

171.1°C

104.4°C

37.8°C

37.8°C

4.4°C

70

10

0

30

40

20

50

60

70

C3H8 molar fraction

CH4-neo-C5H12

liquidvapour

C3H8-C6H6 CO2-C3H8

CO2 molar fraction

Fig. 9. Examples of phase diagrams calculated using perturbative methods (continuous lines) compared to experimental data (circles).A, methane-neopentane system; B, propane-benzene system; C, carbon dioxide-propane system.

this case, in fact, Vi�Nivi and the following expression ofentropy density can be derived:

∆S�kB /i[48] 113��i

1 ln/iV ni

from which it appears that bigger molecules contribute in asmaller part to mixture entropy. By indicating the densitywith ri (mass per unity volume) of the i-th component andwith wi the mass fraction corresponding to a mass M, vi canbe derived by the following relation:

wiM[49] ni�123

riNi

Obviously, for spherical molecules vi�(1�6)psi3.

2.3.12 Internal energy of a mixture

The potential energy due to interactions between moleculesof many-component mixtures can be expressed bygeneralizing [34] through the following relation:

2p[50] / �13�

ij

NiNjuij(r)gij(r)r2drV

where uij(r) is the potential energy of interaction of the pairof molecules i-j, whereas gij(r) is the corresponding radialdistribution function. Obviously, in order to apply theprevious expression it is necessary to know the radialdistribution functions of each pair of molecules and thecorresponding potential of interaction. A reasonablyapproximate method for tackling the problem consists ofassuming that each pair of molecules interacts with apotential compatible with a hard core, subject to an attractiveinteraction similar to that already used within the van derWaals model, that is:

r �sij

[51] uij(r)�� sij�u0

ij�13�6

r �sijr

where sij is the smallest approach distance of the i-j pair ofmolecules.

As one can observe in Fig. 10, radial distributionfunctions are deeply influenced by molecular dimension, and

that makes the calculation of the integrals present in [50]particularly cumbersome. One way to proceed consists ofassuming that the comparison between the potential and theradial distribution functions can be performed incorrespondence with a reduced distance given by the ratiobetween r and the molecular diameter; in other terms,attributing the following expressions to the potentials and tothe radial distribution:

r[52] uij(r)��u0

ij f �1�sij

r[53] gij(r)�g�1�sij

thereby [50] becomes:

2p[54] / ��13�

ij

NjNiu0ijs

3ij

0

f(x)g(x)x2dxV

where x�r/sij. The integral at the second member of theprevious equation can be considered as an effectiveparameter to be determined, for instance, semiempirically. Ina rougher approximation the radial distribution function canbe assigned the form of a step function:

0 r �s[55] g(r)��1 r �s

In this case, the expression of the energy per unit volumeexplicitly becomes

/ 2p 2p u0ijs

3ij

[56] 13��1343�ij

NiNj u0ijs

3ij��13�

ij

/i/j13233

V 3V2 3 ninj

where

V/i[57] Ni�

1343

ni

It is now advisable to define cohesive energy, indicatedby Pi

* for each component, which acts as a characteristicpressure operating internally in the fluid

2p u0iis

3ii e*

ii[58] Pi*��13 113�1�

3

3 ni2 sii

This equation also defines parameters e*ii. Similarly for

each pair



2

3

g11

r0

1

1 1 1 1

2

3

g12

r0

1

1 1 2

21 1

2

3

g22

r0

1

1 2

22 1

2

Fig. 10. Influence of moleculardimensions on the radialdistribution function.

2p u0ijs

3ij e*

ij s ij[59] Pij

*��13 113�131�3

3 ninj s ii s jj

Assuming

[60] ∆Pij*�Pi

*�P*j �2Pij

*

and substituting the previous relations into [56], it is possibleto derive the following expression of the internal potentialenergy of the mixture:

U[61] 1��P*��

i

/iPi*��

i� j

/i/j∆Pij*�V

and, consequently, the expression of parameter es(P*)1�3.One can observe that if the interaction parameters

between pairs of molecules are expressed by a geometricaverage, thereby assuming

241 2134

[62] e*ij ��e*

iie*jj sij��siisjj

one gets

[63] ∆Pij*�(Pi

*1�2�Pj*1�2)�di�dj

where di is the Hildebrand solubility parameter ofcomponent i.

2.3.13 Extensions and conclusions

The analysis developed in the previous sections was intendedto offer a general view of some of the significant aspects ofmolecular thermodynamics, particularly those concerningthe assumptions necessary to perform the calculations onfluid state behaviour and on phase equilibria. Without goinginto detail, for hydrocarbons and their mixtures (seeChapters 2.6 and 2.7) methods have been illustrated thatmake it possible to evaluate the most importantthermodynamic functions, particularly internal energy andentropy, from which it is possible to derive free energy andchemical potentials.

In order to provide a more general and flexible analysisit is appropriate to take into consideration a volume increasein a mixture by adding N0 moles of a zeroeth component

[64] V��i

niNi��V �V0

where V0�N0n0.The expression of mixing entropy of the mixture

becomes

∆S[65] 12��N0 ln(1�j)��

i

Ni ln(j/i)�kB

being

V[66] j �121

V0�V

Obviously, if V0 is zero, j�1. If the zeroeth componentswere assumed to be made up of cavities, or pseudoparticlesof v0 dimensions, they would reflect the presence of a freevolume that would influence the value of the thermodynamicproperties; entropy takes the form [65], which for n0�0, andthen j�1 identifies with [46], while for internal energy [61]becomes

U[67] 1��j P*

V

By combining the expressions of internal energyand entropy, it is possible to derive the expression offree energy which is fundamental in the calculations ofphase equilibria. This can be obtained by assigning toparameters j, e*, P* suitable values calibrated on thebasis of experimental data; for some hydrocarbons thevalues of these parameters are summarized in Table 5,e* and P* expressly, while j appears as the ratiodensity/j.

It is to be remembered, in any case, that in alargely semiempirical treatment such expressionsrepresent the basis to formulating some mixturemodels, regarding particularly hydrocarbons, such asthe models of regular solutions and athermalsolutions, which today are still being used to studyphase equilibria.



Table 5. Parameter values of e*, P*, density/jfor some hydrocarbons

Hydrocarbone*/kB(K)

P*(MPa)

Density/j(kg/m3)

methane 224 248 500

ethane 315 327 640

propane 371 313 690

butane 403 322 736

pentane 441 310 755

hexane 476 298 775

heptane 487 309 800

octane 502 308 815

nonane 517 307 828

decane 530 304 837

undecane 542 303 846

dodecane 552 301 854

n-C13H28 560 299 858

n-C14H30 570 296 864

n-C17H36 596 287 880

isobutane 398 288 720

isopentane 424 308 765

neopentane 415 265 744

2,2-dimethylbutane 455 275 773

2,3-dimethylbutane 463 289 781

cyclopentane 491 388 867

cyclohexane 497 383 902

benzene 523 444 994

toluene 543 402 966

p-xylene 561 381 949

m-xylene 560 385 952

o-xylene 571 394 965

Bibliography

Barker J.A., Henderson D. (1976) What is ‘liquid’? Understandingthe states of matter, «Reviews of Modern Physics», 48,587-671.

Beret S., Prausnitz J.M. (1975) Perturbed hard-chain theory: anequation of state for fluids containing small or large molecules,«American Institute of Chemical Engineers Journal», 21, 1123-1132.

Carrà S. (1990) Termodinamica. Aspetti recenti e applicazioni allachimica e all’ingegneria, Torino, Bollati Boringhieri.

Carrà S. (1998) Termodinamica molecolare, in: Enciclopedia delNovecento, Roma, Istituto della Enciclopedia Italiana, 1975-2004,13v.; v.XI: Secondo supplemento, 786-800.

Chandler D. (1987) Introduction to modern statistical mechanics,New York, Oxford University Press.

Gubbins K.E. (1994) Application of molecular theory to phaseequilibrium prediction, in: Sandler S.I. (edited by) Models forthermodynamic and phase equilibria calculations, New York,Marcel Dekker.

Prausnitz J.M. et al. (1986) Molecular thermodynamics of fluid-phase equilibria, Englewood Cliffs (NJ), Prentice Hall.

Rowlinson J.S. (1969) Liquids and liquid mixtures, London,Butterworth.

Sanchez I.C., Panayiotou C.G. (1994) Equation of state.Thermodynamics of polymer and related solutions, in: Sandler S.I.(edited by) Models for thermodynamic and phase equilibriacalculations, New York, Marcel Dekker.

Sandler S.I. (1985) The generalized van der Waals partition function.I: Basic theory, «Fluid Phase Equilibria», 19, 233-257.

Sandler S.I. (edited by) (1994) Models for thermodynamics and phaseequilibria calculations, New York, Marcel Dekker.

Vera J.H., Prausnitz J.M. (1972) Generalized van der Waals theoryfor dense fluids, «Chemical Engineering Journal», 3, 1-13.

Whalen J.W. (1991) Molecular thermodynamics: a statisticalapproach, New York, John Wiley.

References

Binney J.J. et al. (1992) The theory of critical phenomena. Anintroduction to the renormalization group, Oxford, Clarendon.

Carnahan N.F., Starling K.E. (1972) Intermolecular repulsionsand the equation of state for fluids, «American Institute of ChemicalEngineers Journal», 18, 1184-1189.

Kac M. et al. (1963) On the van der Waals theory of the vapour-liquidequilibrium. I: Discussion of a one-dimensional model, «Journalof Mathematical Physics», 4, 216-228.

Mayer J.E., Göpper Mayer M. (1940) Statistical mechanics, NewYork, John Wiley.

Uhlenbeck G.E. et al. (1963) On the van der Waals theory of thevapour-liquid equilibrium. II: Discussion of the distributionfunctions, «Journal of Mathematical Physics», 4, 229-247.

List of symbols

b covolumeBij second virial coefficient of the i, j molecule pairCijk third virial coefficient of the i, j, k molecule triad Er energy of the r configurationF Helmholtz free energy functionG Gibbs free energy functiong(r) radial distribution function h Planck constantkB Boltzmann constantN total number of molecules Ni number of molecules of the i componentPi

* characteristic pressure of the iP pressure Q configuration integral r distance between the centres of mass of two

molecules S entropy S(k) structure factor T thermodynamic temperature U internal energy u(r) intermolecular potential u0 parameter of intermolecular potential V volumeVf free volume vi,j molecular volumes of i, j componentswr probability of the r state of a systemY extensive thermodynamic variable Y molar value of variable YZ partition function of the system z molecular partition function l de Broglie wavelength F potential energy of a system r(N/V ) molecular densityri mass density of component is molecular diameter, parameter of the

intermolecular potential mi chemical potential of the i component/i volume fraction of the i component etr,rot,vib,el translational, rotational, vibrational and electronic

molecular energy

Sergio Carrà

Dipartimento di Chimica, Materiali eIngegneria Chimica ‘Giulio Natta’

Politecnico di MilanoMilano, Italy



2.3 molecular thermodynamics - treccani · 2019-12-28 · in a simulation of phase equilibria. ......

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