Food Engineering Handbook Food Engineering Fundamentals Varzakas Theodoros, Tzia Constantina

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Fundamental Notes on Chemical Thermodynamics

Publication details https://www.routledgehandbooks.com/doi/10.1201/b17843-4 Tzias Petros Published online on: 02 Dec 2014

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Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 3.12 3.11 3.10 3.9 Mixtures 3.8 3.7 3.6 3.5 3.4 3.3 3.2 Scope 3.1 Introduction CONTENTS Equilibrium of Phases Equilibrium Liquid Mixtures 3.10.2 3.10.1 Fugacity Mixtures Gaseous and Gases 3.9.4 3.9.3 3.9.2 3.9.1 System Open One-Phase in Functions Relations of Thermodynamic Inequalities Fundamental System Closed Some Relations and Functions for One-Phase Thermodynamic 3.5.4 3.5.3 3.5.2 3.5.1 Laws Thermodynamic ofConcepts W, U, S Q, and KE, PE, 3.3.7 Molality 3.3.6 3.3.5 3.3.4 Process 3.3.3 Phase 3.3.2 3.3.1 System Terms of BasicGlossary Thermodynamic 3 ...... The Standard State of a Gas Component of State aGas Standard The Functions Dilution Thermodynamic Functions Reference and of States Thermodynamic Standard Excess Functions Functions Mixing Equilibrium Law of Thermal Law of Thermodynamics Third LawSecond of Thermodynamics LawFirst of Thermodynamics Mole Fraction Molar Quantities Extensive Intensive and Properties ...... Petros Tzias Petros Thermodynamics on Chemical Notes Fundamental ...... 44 60 46 46 46 48 48 48 48 48 54 54 50 56 58 58 45 45 45 45 42 42 57 47 43 55 55 55 41 51 Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 3.14 3.13 Solutions 42 mines the laws and relations governing and describing these energy transformations. laws energy the these relations and describing governingmines and physicochemical different phenomena the deter one another, to form in and from of (electrical, energy light energy, kinds work, of other heat, and etc.)transformation of study of is physical abranch the the with sciences concerned Thermodynamics 3.2 phenomena. relativistic nuclear not as and refer we and will statistical, to thermodynamics, cal chapter. beyond of scope are this the these tions, as cussion rela Similarly, about them. not prove we will mentioned thermodynamic the dis or philosophical description not enter we any analytical to and will cited, been so on. and reactions, chemical solutions, equilibriums, phase mixtures, with is that thermodynamics chemical with clauses to next dealing est we proceed will afew presenting clauses after of reason general inter for and this or chemists, neers volume much pleasant. easier and current study of the the of science make branch to so as important relations of this basic of the laws, reader the glossary, and concepts, reminding of thermodynamics, of what we read. understanding for better textbooks other open and seek to atany time need way the from released this we are because haveto fieldthermodynamics somenotes handy engineering of of chemical sciences. their for cornerstone the form they mathematics with engineers chemical for and chemists so on. Especially and geologists, engineers, chemical geophysicists, and mechanical such chemists, tool for as useful is avery engineers scientists and Thermodynamics 3.1 References 3.15 We will deal only with reversibleWe only chemi with is in interest deal an as will so far processes, have and quantities functions thermodynamic Simpledifferent the definitions of content whole of the as the volume, chapter is addressed, engi chemical to This chapter. is exactly intention of the this ToThis provide review abrief, elementary acertain from when chapters It we comprising useful is very study ahandbook INTRODUCTION SCOPE Electrolyte SolutionElectrolyte 3.12.2 3.12.1 3.15.3 3.15.2 3.15.1 Reaction Equilibrium Chemical Thermochemistry: 3.14.1 3.12.5 3.12.4 3.12.3 ...... Chemical Potential in Phase Equilibria Phase in Potential Chemical RulePhase Equilibrium Constant: Affinity of a Reaction Affinity Constant: Equilibrium of Enthalpy of Reaction the Determination of Enthalpy Reaction of and Enthalpy Formation LawDebye–Hückel Limiting aTwo-Phase in Entropy Change and Enthalpy Transition States of Corresponding Principle Vapor–LiquidBinary Systems ...... Food Engineering Handbook Engineering Food ...... 64 60 66 68 65 69 72 70 70 73 62 71 61 ------Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 the developmentthe thermodynamics. field, of the chemical which todaywe call foundation laying for the equilibriums phase and reactions relationsand chemical to thermodynamics. of expression andlawsecond the with first of the tion classical of the thermodynamics cycle, gas famous ideal his with N.L.S. Clausius to from Carnot founda who the laid Thermodynamics on Chemical Notes Fundamental below, [3]. SI units together their with defined of most common which the are terms, of thermodynamic is avariety There 3.3 engineer. or achemical chemist money. and saving time part other the from avoidingtities and and way, by this long difficult experiments and one part from values physical of of the other physicalcalculate data existing quan from quantities andaccurately. more easily ficulty or measured lowby others accuracy dif with measured are its relations, that quantities through possibility calculate, to is the of use thermodynamics important very One others. than measured easier be to but measurable, independently its some relations in of are are used them quantities result of a physical final or the reactions process. chemical possible of Using relations the we these predict direction can thermodynamics. in either by physics taken physicalence. introduced are The used quantities or are they experi from laws, derived directly which some are laws,from thermodynamic the way amathematical in deduced relations phenomena. are These or chemical cal well as of some physi as substances physical properties chemical interrelate and Historically [1]Historically scientists have many interconversion the studied work, of and heat Temperature Density volume Molar volume (m Specific Volume Pressure coming arePressure, Temperature concepts Volume, volume Pressure and substance of amount The tool a to useful avery be can Therefore, obvious it becomes thermodynamics that relations we can thermodynamic through often, very advantage is that, Another science [2]. physical experimental the is an or chemical All Thermodynamics which inequalities consists and of acollection of equalities Thermodynamics postulates Later, Gibbs application extended J. thermodynamic of the the Willard equal magnitude in systems where thermal flow not does systems in exist where thermal [3]. magnitude equal which has of matter property the of or as of hotness asubstance degree matter. tionHowever, difficult is avery the couldwe as simplyit describe Pa. is SI the of unit pressure The surface. of this physics. from is of the substance amount N number the to portional GLOSSARY OFBASIC THERMODYNAMIC [2–6] TERMS (m is the reciprocal of the specific specific ofvolume. the reciprocal is the is defined as the ratio of a perpendicular force to a surface by the area area the by to a surface force ratio the perpendicular of a as is defined 3 ) is the three-dimensional space occupied by space asubstance. three-dimensional ) is the is a fundamental concept used in thermodynamics and its defini and thermodynamics in concept used is afundamental is the volume is the of substance. by amount divided the 3 /kg) is the volume is the mass. unit per [2] N [2] B mol of entities B in the system. SI of of the entities unit the The Bin B . of an entity Bis entity aphysical of pro an quantity - - 43 - - - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 44 a certain number of them, which are called independent and all the rest (dependent) the all and independent called which of are number them, a certain but only its values properties, of know to the of a system state all the it is not necessary a system have they and any fixed given at values the define to In order system.of state with volume, related so on, which are and such pressure, temperature, quantities as cal or leaves enters energy it. we study. it is Otherwise it. with system world of interact the the whichings we can around consider part the world of the the are parts remaining the All By 3.3.1 T Temperature [4] scales The The A system is system

• • • • state S in thermodynamics we refer to any part of the real world real of we the we refer choose study. to any part to thermodynamics in y The CelsiusThe value (°C) scale by the 0.01°C is established assigning to lute temperature we use the degree K. we degree the use lute temperature system the for abso In points. the these between absolute temperature the to measure used instruments the as points as well fixed at different way. [4] accurate and easy literature the temperatures we find In can an in absolute the temperature measure to functions and instruments gives possibility by above-mentioned the the IPTS The thermometers. volume by constant gas have functions well determined the as been as fixed points The points.those between temperatures the we find by which functions, and instruments together some with interpolating Temperature (IPTS-68). Scale Practical International The absolute temperatures. thermodynamic the determine to used are absolute the volume with scale. Constant same is the thermometers gas 273.16 Tand temperatures atthe ter where P as defined scale temperature gas ideal The of is 273.15 water at1 atm K. 273.16 to of triple water is the equal this and point it scale is enough define to T isjust given Kelvin. this one degrees In in Celsius by relation: scale the which the to is ascale is related (T) scale absoluteThe temperature (760 atm Torr). of 1standard pressure atmospheric water atan of point value of point boiling the 100°C triple water and the to the s tem of asystem the is defined by values of properties. its This scale gives the temperatures at some reproducible points fixed gives scale temperatures the This closed T and P and if no matter enters or leaves enters any stage no process of matter if the it during 273.16 open T are the pressures of a gas trapped in gas thermome gas in of trapped agas pressures the are = 273.16 . A system is called . Asystem is called = t (°C) K P0 273.16 + lim surroundings 273.15 → K. The ideal gas temperature scale scale temperature gas ideal The K. ⎝ ⎜ ⎛ P 273.16 P T isolated Food Engineering Handbook Engineering Food ⎠ ⎟ ⎞ of the system. of the By surround K. The freezing point point freezing The K. if neither matter nor neither matter if Properties are physi are - - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 Considering that we divide a system into different parts, if for one property its value for if one property we that divideConsidering asystem parts, different into e 3.3.2 “equation of an state.” is called functions state thermodynamic between of a asystem state the it called is also is fixed by by values ones. independent of the Since aproperty the calculated be can Thermodynamics on Chemical Notes Fundamental relation:the From any extensive X intensity as Xofquantity it aphase is quantity defined m 3.3.5 irreversible. are is called process the Otherwise, amounts. tesimal by infini equilibrium system of from the differ properties the stage process of the called are process baric. metric isothermic, is called process the constant, system of the remains temperature the aprocess ing isochoric Process 3.3.4 opposite it system. case the is called the In within phase another with sive its have parts. value all as same properties not the it case, some is heterogeneous that of one and consist phase. In its inten of more than systemvalue, the is ahomogenous then a is called one and its have intensive all same properties the its parts, forIf all asystem, throughout 3.3.3 volume, molar pressure, density, so on. and is called property the then parts, energy, so on. and is called property this then for whole of the sum its values the to system parts, for different is equal the A process is called is called A process is called process the volume the constant, If system of the remains be can phase One whole value the in same system the well its has as different in as aproperty If If no heat enters or leaves no enters heat If asystem system the undergoing the and one process or or is the pathway which is the one another. to system one state through from possess P P and if the pressure remains constant the process is called is called process the constant remains pressure the if and isochoric xten rocess ha olar if the volume of the system remains constant, the process is called is called process the volume the constant, if system of the remains s e s ive Q adiabatic

, and if the pressure remains constant, the process is called is called process the constant, remains pressure the if , and uantities and open reversible extensive I nten when it exchanges matter either with its surroundings or when it exchanges either its with surroundings matter . s ive . Extensive properties are the volume, mass, internal volume, the internal . Extensive are mass, properties if it takes place slowly it if takes such in away and atany that intensive P X ro m p = state function erties ∑ . Intensive properties are the temperature, temperature, the are . Intensive properties X i n i

irreversible . A mathematical relationship . Amathematical phase . All natural processes processes natural . All . One system . One can isobaric. closed isometric . If dur m (3.1) iso iso by by or or 45 - - - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 The mole X The fraction m 3.3.6 where 46 dW x,then replacing it by adistance (ds) to it by force equals of adistance the and direction the in physics. from W (work), energy) borrowed concepts energy), PE are (potential (kinetic KE and 3.4 where n X quantity molar partial the Molality m 3.3.7 phase. this in stances where n By energy we mean the ability of asystem ability By work. the produce we to energy mean on asystem’s Pacting pressure of case auniform the In Sand of wall surface Work From any extensive intensive Xofan we aphase called quantity define quantity quantities molar Partial Molarity, It follows above the that from definition immediately ∑ CONCEPTS OFW, PE, U, KE, AND Q, S[6–8] a i i

n is the amount of substance of a and of of aand substance amount is the (W) is the energy produced by a force (F) acting on asystem replacing acting by and aforce produced energy (F) is the (W) ≠ m i is the sum of the amounts of the different substances contained in this phase. this in contained substances different of the amounts of sum the is the a means all n’s all a means except n is the number of number moles is the 1 kg of in solvent. of asubstance ole olality c is the number of number moles is the 1 Lof in solvent. of asubstance F raction a of a substance a in a phase is defined by the ratio: the is defined by aphase ain of asubstance

= a of the substance a in the phase by relation: phase the the ain substance of the

− P X ⋅ s a a in this phase. this in X ⋅ dW

= ∑ or dW dx a i ⎝ ⎜ ⎛ =

∂ = X ∂ n F ∑ X i a n = ⎠ ⎟ ⎞ ⋅ ds ∑ a i T,P, 1 n i n i

na i i

the sum of the amounts of all sub of all amounts of sum the the ≠ =

− Food Engineering Handbook Engineering Food P ⋅ dV

(3.5) (3.2) (3.3) (3.4) - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 system. height hthe of acceleration, and system, gravity the of mass the gthe where mis the PE Thermodynamics on Chemical Notes Fundamental given ones. most common the several expressions are equivalent other. each There with for all Below these are is no laws exception there experience and come from them. to Thermodynamic 3.5 dQ Salwaysprocesses and increases received heat where by is the dS it reversible in T. way natural atatemperature In as defined be It can system of capacity T heat the where Cis and the Q procedure. of a states final and followedon the original not on the and path of two of depends systems. heat amount the The temperature the in difference of the followed path the other. one the to from pass to (1) twobetween states (2). and absolute the Wetem. value measure of U, cannot but only differences of a sys by state the determined It state. is amagnitude asystem atacertain within ( velocity Potential energy (PE) energy Potential The SI for unit The Sis Joule/K. SI for Joule. unit The W, Qis the and KE, PE, ( entropy The For asystem Q, heat receiving (Q) Heat (U) energy Internal (KE) energy Kinetic

THERMODYNAMIC LAWS [2,5–7] υ ) and is equal to is equal ) and is the amount of energy, transferred from one system to another because because one system of from energy, amount another to is the transferred S ) (6) extensive is an of asystem. state on the property, depending is the energy possessed by of possessed asystem energy its because is the of m, mass is the total energy, except potential and kinetic, contained energy, contained total is kinetic, the except and potential is the energy possessed by possessed a system energy of is the its because position. Δ U depends only on the states (1) states only on the U depends (2) and not on and = KE dS C

= ≤ m ⋅ = = (T T m ⋅ dQ g ⋅ T 2 2 dS –T υ 2 ⋅ 2 , T h

1 ) its final and original temperatures. and original its final (3.8) Δ U = U 2 (3.10) –U (3.6) (3.7) (3.9) 47 - 1

Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 law can be expressedlaw as be can energy. or eliminate create can words, is no device other there that In is conserved. energy total the any process, In 3.5.1 48 Enthalpy Enthalpy W, Sfor last two and functions: W the P,We properties U, met have the functions Q, V, already thermodynamic the Tand 3.6 laws [2].thermodynamic them.” between equilibrium thermal in also are they one, then athird with equilibrium thermal in two systems“If are l 3.5.4 is zero.” entropy crystalline ofthe apure ics [2]. expression law As we for could give third the Planck’s [7]. “At postulate 0K law expression is notThe possible third of the without mechan reference statistical to t 3.5.3 isolated systeman increases. entropy of total the of words, anywork. process expenditure without other in the In It is not temperature possible alower ahigher to from heat transfer to temperature 3.5.2 as expressed process. pathway on the depending of not exact the system, differentials dW are but dQand transferred. heat dQis the and system, of work energy the dW is the produced, of internal variation where dU is the Furthermore, there are the following very important thermodynamic functions: thermodynamic following the are important very there Furthermore, andsecond law one but independent first is it not an This the is derived from By 3.5 3.9 substituting Equations and 3.11 Equation into be law first can the the of state final and original only on the depending exactdU is an differential of work to heat orversa transformation vice asystem first the in Considering the THERMODYNAMIC FUNCTIONS AND SOME RELATIONS FOR ONE-PHASE CLOSED SYSTEM [6,7] F S ir hird econd aw s t

L o L f aw T L aw hermal

aw o

o f

T f o T hermodynamic f T hermodynamic E hermodynamic q uilibrium dU

dU =

−

P = = dW H ⋅

s dV − s P = s U

⋅

+ dV and S dV and + dQ T + P ⋅ dS ⋅ V

Food Engineering Handbook Engineering Food = dQ/T. (3.12) (3.13) (3.11) - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 G Gibbs or Gibbs energy function the and Thermodynamics on Chemical Notes Fundamental following: the also and relations:useful dG dA dH reversible we processes have By applying the properties at exact differentials we can obtain the following the we very obtain atexact can By differentials applying properties the for and exact differentials are differentials their functions state Since are they have they and of dimensions functions energy. the state are functions three the All = Helmholtz free energy free Helmholtz H–T ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ∂ ∂ ∂ ∂ ∂ ⋅ ∂ A ∂ T S ∂ U G V P H S ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ = ∂ ∂ VT ⎠ ⎟ ⎞ = = ∂ SV ∂ TP ∂ ∂ A V T = PS V

T P S =− − − T =− == ⎠ ⎟ ⎞ == ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ S S + SV SP TV V P ⋅ TV ⋅ ⋅ dS S, =− P dT dT ,S = , = ⎝ ⎜ ⎛ , ⋅ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ V ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ∂ ∂ ⎝ ⎜ ⎛ ∂ ∂ ∂ ∂ +

∂ ⎝ ⎜ ⎛ ∂ + + G T ∂ A V H A ∂ P ∂ ∂ V U = S V V ∂ ∂ V S T P ⎠ ⎟ ⎞ U–T ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ P S ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⋅ ⎠ ⎟ ⎞ dP ⋅ ⋅ dP dP = =−

= U–T

−

T ⋅ P S

+ ⋅ P S

⋅ V

(3.25) (3.24) (3.20) (3.22) (3.23) (3.15) (3.16) (3.17) (3.21) (3.19) (3.18) (3.14) 49 Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 50 lowest value, except entropy maximum. which the gets the dG dA dH dU (3.11), (3.16), (3.17), (3.18) derive: we dQ processes for natural We (Section seen have 3.4) already for although areversible that dQ process 3.7 In other words, other forIn any closed isolated system [2]: above the get the functions atequilibrium process for that anatural means This

FUNDAMENTAL INEQUALITIES ai Ci Ci K P Ki V = S = =− = V 1 =− ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ∂ ∂ ∂ ∂ ∂ V ∂ 1 H T U T V T ≤ V 1 ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ T ⎠ ⎟ ⎞ ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ P ⎝ ⎜ ⎛ ∂ ∂ V P ∂ ∂ V P ∂ ⋅ ∂ T st P

V st dS. Similarly, for natural processes from equalities equalities from dS. Similarly, processes for natural P st ⎠ ⎟ ⎞ ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ he T ⎠ ⎟ ⎞ he he VP ∂ ∂ ⎝ ⎜ ⎛ S ≤ is ≤

P S heat ≤ ⎝ ⎜ ⎛ ≤

heat ∂ coefficien

∂ − th st T − T ⎠ ⎟ ⎞ ∂ ∂ S t S S V TP T ei he ⎠ ⎟ ⎞ ⋅ ⋅ capacity ⋅ dS ⋅ –P dS U,V, ⎠ ⎟ ⎞ =− capacity dT dT –P sotherma adiabatic ⎝ ⎜ ⎛ n + i

⎝ ⎜ ⎛ ∂ + ∂ V > V ∂ P to V ∂ V T ⎠ ⎟ ⎞ ⋅ 0 ft ⋅ at dV ⋅ dP ⋅ at T ⎠ ⎟ ⎞ dV dP

lc hermal constant compressibility constant =−

ompressibility

Food Engineering Handbook Engineering Food 1

expansio pressure volum e n

= T (3.26) (3.27) (3.36) (3.37) (3.30) (3.35) (3.34) (3.28) (3.29) (3.32) (3.33) (3.31) ⋅ dS dS Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 then the system equilibrium. the is in then if system and Sis increasing, that then in happens anything if that which means Thermodynamics on Chemical Notes Fundamental following the removal under above the of form: then matter, should written equation be of energy. systemaddition or is also consideration the we under in there If suppose that involvingfor a sionchange law first transformation ofthermodynamics the the of only We 3.12 Equation from seen have already dU 3.8 P. Tand place atconstant Gis content decreasing. then Pand asystem Tand in atconstant happens anything if from and decreasing, His content then Pand asystem Sand in atconstant happens anything is, if that from: His content decreasing then Pand asystem S and in atconstant happens anything if from: Uis content decreasing then Sand asystem Vand in atconstant happens anything if Inequality 3.42 is very important for chemists since most chemical reactions take take reactions for most 3.42 since chemical chemists important isInequality very Similarly, from RELATIONS OFTHERMODYNAMIC FUNCTIONS IN ONE- PHASE OPEN SYSTEM [2,6,7] dU =− Td ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ SP ∂ ∂ ∂ ∂ A U G H S T t t t t ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ U,V, S,V, T,V, S,P, T,P, dV n n n n n i i i i i = < = < < + <

− 0 ∑ 0 0 0 0 P i

⋅ dV μ ii dn

+ T

⋅

dS, which is the expresdS, which is the (3.40) (3.38) (3.39) (3.42) (3.43) (3.41) 51 - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 where dn 52 or where SI the system Joule in its mole. unit isof the per and substance, amount of dimension per it energy the has function, it intensive is an thermodynamic where n G(T,and P, n In a similar way, asimilar 3.16In Equations from 3.18 through we derive: Differentiation of Equation 3.50 ofDifferentiation Equation gives, 3.43 of Equation Integration by P, keeping T, n definition, By above-defined quantity The U(S, functions U, the Considering Gas and V, A, H, n 3.43Equations 3.46 called are through λ i k is called the absolute the is called ≠ i i μ is the amount of substance of species i transferred and and of of itransferred species substance amount is the indicates all the other species n species other the all indicates i = i ) we obtain, ⎝ ⎜ ⎛ ∂ ∂ n U i ⎠ ⎟ ⎞ S,V, n dG dA dH ki ≠≠ US =− =− = dG =⋅ activity =⋅ ⎝ ⎜ ⎛ μ Td Sd Sd ∂ ∂ i =+ is called the the is called n H ⋅+ ⋅− i ∑∑ μ Gn TP ⎠ ⎟ ⎞ SV TV TP i i

S,P, of the species i in the multicomponent the system. iin species of the = +⋅ – = nd R Tln n ii ∑ ki μμ i k ⋅+ Gibbs equations Gibbs except n except ⋅+ ⋅+ Vn dP = dV dP ii μ ⎝ ⎜ ⎛ chemical potential λ + i ∂ ∂ ∑ i (3.48)

n A ∑ i i ∑ ∑ constant [2] constant to, leads i i ii ⎠ ⎟ ⎞ i i i . T,V, n μ ii μ μ μ Food Engineering Handbook Engineering Food ii ii ii dn

n dn dn ki

≠≠ i . ), H(S, P, n

= ⎝ ⎜ ⎛ ∂ ∂ n μ G of the substance i; substance of the i i its energy. molar ⎠ ⎟ ⎞ T,P, i ), A(T, V, n n ki

(3.46) (3.50) (3.44) (3.45) (3.47) (3.49) (3.51) i ), ), Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 which is known as the Gibbs–Duhem equation. Gibbs–Duhem the which as is known Thermodynamics on Chemical Notes Fundamental following relations: useful or form, the takes Equating the expressions the 3.46 forEquating 3.47 Equations and in dG yields, From Equations 3.46 by simple mathematical manipulations we derive 3.46 the can From Equations manipulations by simple mathematical equation Gibbs–Duhem fugacitiesthe Usingandactivities above-defined the Sd Sd ⋅⋅ Sd ⋅⋅ ⎝ ⎜ ⎛ ⋅⋅ ⎝ ⎜ ⎛ ∂ TV ∂ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ TV ∂ ∂ nn μμ ⎝ ⎜ ⎛ T ∂ ∂μ ∂ ∂ –0 nn ∂ μμ ∂ k i –0 ∂μ μ μ ∂ V T T k ⎠ ⎟ ⎞ i P i i – i ⎠ ⎟ ⎞ T,V, i ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ T,P, ⎠ ⎟ ⎞ V T, V, P, dP T, n n n dP n ki n ik ik n ik ≠≠ ki ik ≠≠ =− =− dP =− +λ += = =− = ∑ ∑ ⎝ ⎜ ⎛ += i ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ i ⎝ ⎜ ⎛ ∂ ∂ ⎝ ⎜ ⎛ ∂ ∂ ∑ ∂ ∂ ∂ nR ∂ n ∂ V ∂ nR n n ∂ n i P ∂ S S ii i ii i i i ⎠ ⎟ ⎞ k i ⎠ ⎟ ⎞ nd ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ Tl T,P, k i ⎠ ⎟ ⎞ Tl T,V, T,P, ii T,V, ⎠ ⎟ ⎞ T,P, n T,V, μ n n nf n n ≠ n ≠ i ≠ ki ≠ n i i i

(3.56) (3.57) (3.54) (3.55) (3.60) (3.58) (3.59) (3.52) (3.53) 53 Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 tures, from which the extension which the from multicomponent to tures, ones is straightforward. mix or solid liquid, phase.exist gaseous, We ourselves in confine to binary shall may Amixture mixture. is called A system one substance consisting of more than 3.9 54 mixture for ideal an obtain respectively. B, and X xand and X where X Δ or as is defined function mixing V the respectively, such S, X, H, G, or A, as for quantity and any extensive thermodynamic (1 fractions of molar B with components A and mixture For a binary m 3.9.1 components. pure behave the in as mixture the components in the and irespectively, of pure and T. Pand same atthe mixture again the where i, respectively, of pure and T. Pand same atthe mixture both where mix B

From Equations 3.21From Equations 3.65 and P, atconstant nwe obtain 3.63From Equations 3.64 3.50 and 3.62 using Equations and for and X same the are species unlike and like between interactions the mixture, ideal an In for 3.61 which Equations A mixture or 3.62 a not is called valid are [2]: words, for other mixture if In any component the iin be to is said mixture By a definition, (T, atmole 1 fractions P, mixture this x) Bin of Aand functions molar the X m MIXTURES [2,6,7] μ λ

i i = , , m (1 µ λ (T, P, atT, P, x) mixture of the function molar is the x,X ixing 0 0 i i are the chemical potentials of component amole iwith potentials X fraction chemical the are

are the absolute the activities component of are a mole i with X the fraction − Δ m x) [X x) (T, P, 0), X mix F X unction m Δ A Δ

(T, P, 1 mix = mix X GR SR m m id m id (T, P, x) –(1 m (T, P, 1) components A pure of the functions molar the s =+ =− − x) –X T μ= [( ii [( 1x 1x −+ m λ= −− μ+ (T, P, 0)] ii 0 − )( ideal )( x) X x) ln ln X R Tln λ 1x if for if any component i[6]: 1x m 0 i − (T, P, 0) –xX +

[X X )] )] i B

(T, P, x) –X x lnx Food Engineering Handbook Engineering Food x lnx < > m 0 (T, P, 1) 0

(T, P, 1)] A (T, P, 1 real − m

x) x, and mixture.

= G i in the the in (3.66) (3.65) (3.64) (3.63) (3.62) (3.61) m − we we − x), x), i in in x - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 and from Equations 3.21 Equations from and 3.65: and Thermodynamics on Chemical Notes Fundamental is one, that nal the origi and state diluted final the between of mixing functions of thermodynamic dilution. is called process this mixture, the to is added this and one component excessspecies, is in others the to related of or multicomponent nonreacting homogenous one phase one binary in mixture If d 3.9.4 state. erence function. thermodynamic different of the 1 bar.as P, defined or solids be liquids atcertain Twhere Pcan pure solids the and liquids temperature. on the ( position, which is called function givenof iata com thermodynamic the pressure atafixed and temperature Weas define 3.9.3 is, one.ideal That By definition function excess e 3.9.2 = 101.325 From Equations 3.65,From Equations 3.66, 3.15: and The thermodynamic functions of the dilution process are given as the difference given difference the are as dilution of process the functions thermodynamic The or ref of its function expressed standard be in can function Any thermodynamic As at1 gas ideal pure the Usually, is accepted state for standard as gases fixed of1 pressure the at defined are states Historically, standard the reference state F S tandard unction xcess Pa) and in that case the standard thermodynamic functions depend only depend functions thermodynamic standard the Pa) case that in and ilution standard thermodynamic function F unction T

s and [8,9,12] hermodynamic is called one state that is used as reference as is used for calculation that the one state is called R standard state. standard Δ s ef X erence XV m,dil X m E m E

is the difference between the real real the between difference is the = =− ( S Δ Δ ΔΔ Δ F tates mix mix mix X unction m,mix H V X m m id id m

o ) = = f 2

T − 0 0 ( s hermodynamic of acomponent asystem iin atany

Δ mix X m,mix m id

) 1

Δ mix X atm and for and atm m and the the and (3.67) (3.68) (3.69) (3.70) atm atm 55 - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 equation: behaviorbe PVT the fluidexpressed ofby The can apure 3.10 56 C them. between interactions volume the account into (coefficient take to which tries the andthe b) moleculesof equation: ideality. from gas tion of areal P calculated fromb,c,… ture andonthetypeofchemicalspeciesgas.ThecoefficientsB,C,…canbe ficients b, c, … of Equation 3.72 virial coefficients and they depend on the tempera 8.3144 to is equal SI which in units constant, gas Ris the gas, and the where V or For an ideal gas we gas derive following ideal For the can an relations: Van is the gases behavior express to PVT real attempt of the Waals der One the z gas ideal For an behavior expressed PVT by compressibility is factor: also gases ofThe the the coefficients showdeviathe virial gas. The ideal of of equation an state is the This When 3.71Equations coef the of of and agas equations state 3.72 and virial called are GASES ANDGASES GASEOUS MIXTURES [5,6,8] m is the molar volume molar is the gas, V of the ⎝ ⎜ ⎛ ∂ ∂ H P = 1 and P 1 and ⎠ ⎟ ⎞ P PV TP ⋅= ⋅= == VR ⎝ ⎜ ⎛ m mp VT P → − + ⋅

V P 0 then RT V m a ⎝ ⎜ ⎛ m ⋅

2 ⋅+ = P T ∂∂ ⎠ ⎟ ⎞ z

R nRT/ = 1 () () ⎝ ⎜ ⎛ = C Vb 1 ⋅ T m T. V PV +++ RT

− + BP m P V m n R ⋅

b V ⋅+ = m ⎠ ⎟ ⎞

m V/n, volume mol Vthe nthe of and =⋅

= RT V R C V c m 2 2 ⋅ Food Engineering Handbook Engineering Food –0 T +…

… P ⎠ ⎟ ⎞ T

J ⋅ mol − 1 (3.77)

(3.76) (3.75) (3.72) (3.73) (3.74) (3.71) ⋅ K − 1 - - - . Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 we obtain: we Thermodynamics on Chemical Notes Fundamental is in equilibrium with its vapor is equal to the fugacity of fugacity its vapor. the to its with vapor is equal equilibrium is in is called or gas vapor. of fugacity the where fis the or d term [9] the originated Randall way and gas, Lewis ideal same To the an with in gas of areal express properties the 3.10.1 P for Tfrom and achange atconstant d or Gibbs–Duhem equation at constant T and for becomes gas Tand apure atconstant equation Gibbs–Duhem As we will see in the phase equilibrium, the fugacity of fugacity or aliquid asolid the which equilibrium, phase the in see weAs will ratio The So, 3.80 of Equation or gas vapor for or 3.81 instead areal [10]: we write 3.78 Equation substituting in is gas ideal, the If volume 3.73 the Equation from fugacity coefficient. fugacity F ugacity [7,10] Since ideal gas behavior gas P Since ideal as is approached μ− fugacity μ− 21 lim P0 →→ 21 n d μ nd μ Φ= μ

μ = 1

μ to P to = Φ= μ= R T d ln P R Tdln = R Tdlnf

(f), of dimensions pressure. the which has = = lim P0 RT V 2 RT nR Pd P f ⋅ dP P f ln

P T ln ⎝ ⎜ ⎛ =

P f P f 1 1 2 1 2

⎠ ⎟ ⎞

→ 0 then (3.85) (3.80) (3.84) (3.82) (3.83) (3.78) (3.79) (3.81) 57 Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 relation: 58 and to solid mixtures as well. as solid to mixtures and was Section 3.9 mentioned in that follow what itAll and will applicable are liquid to 3.11 pressure P where of is given gases by of component amixture agas state in standard The t 3.10.2 gas mixture. where (V relation: pressure P where x (d potential chemical of the terms in gas. of we of equation that if state know the The fugacity of a real pure gas at gas a given pure of P,fugacity a real The the evaluated [7] be through T can For f multicomponent fugacity the gases case, this In f fugacity partial For the gases, multicomponent real µ LIQUID MIXTURES [2,8] i B 0 is the mole fraction of component i and P is the total pressure (not the partial (not pressure partial mole the total of is the Pis component the fraction iand (g, T) is the chemical potential of the pure ideal gas, at temperature T and Tand gas, attemperature ideal pure of the potential chemical is the (g, T) m () i 0 ) g,T,P,x ). . Historically, P i he is the partial molar volume molar of of we of i, equation that if state know the partial is the S tandard cB = () ln g,T fl S ii =+ ⎝ ⎜ ⎛ 0 tate

++ ln = nx 1atm. RT P f

() o ⎠ ⎟ ⎞ f μ P ln

μ =− a i i ) P lim as follows: as ⎝ ⎜ ⎛ G T ⎣ ⎢ ⎢ ⎡ →

x ∫ P = 0 0 ⎣ ⎢ ⎢ ⎡ a R Td(lnf ⎝ ⎜ ⎛ xP ∫ P 0 s P f RT V P i C i ⎝ ⎜ ⎛ 0 m () ⎠ ⎟ ⎞ om V i RT = can be also evaluated [7] through the the evaluated [7] through also be can m 1 p ∫ P P 1 0

i onent ⎣ ⎢ ⎡ ⎠ ⎟ ⎞ i − ) V( ⎦ ⎥ ⎤ T (3.87) BC dP P 1 Food Engineering Handbook Engineering Food g,T,P,x ⎠ ⎟ ⎞ ⎦ ⎥ ⎥ ⎤ [2] ⎦ ⎥ ⎥ ⎤ i T of acomponent iis defined dP

⎦ ⎥ ⎥ ⎤ T

) − RT P ⎦ ⎥ ⎤ dP

(3.86) (3.90) (3.88) (3.89) Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 or Thermodynamics on Chemical Notes Fundamental particular, derived. In be can functions namic RT ln term the of difference the then between difference the a relation: for relativeand the corresponding the activity is what we call P, iatcertain liquid T,pure where For a liquid mixture using Equation 3.48 using Equation we obtain: For mixture aliquid After defining the standard state forstate the standard defining After mixture ideal Since of for case an the 3.92From Equations 3.93 and we get a When absoluteFor the activity, coefficient activity the f μ i , µ 0 i i

are the chemical potentials of component mole iwith x potentials fraction chemical the are =

1 from Equation 11.2 Equation 1 from we obtain relative activity γ i expresses deviation the of μ i from its standard state. its standard from μ i and its reference and state μμ λ ii i , == of i in the mixture. Some authors call a Some call authors mixture. of the i in λ 0 i μμ the absolute the activities of μ= HT ii 0 ii 0 i =+ + S =− 0 i RT a G μ+ λ 0 i μ i 0 μ =−

i 0 i

= = 0 i = i the standard states for the other thermody for other states the standard the ln

=μ f γ μ= RT λ i λ x+ R Tln i d x x ii dT μ 0 i ii μ i i 0 i i

μ= i (3.95) activity coefficient activity from ideality. from 0 i

ln d

RT dT ii μ μ+

λ λ 0 0 i µ a 0 i i ln 0 i i μ

at certain P, P if atcertain Tand i

is defined by the relation: the is defined by 0

γ , that is the term RTlna term is the , that R Tln μ

i and and x i (Equation 3.61), (Equation then µ

i 0 γ , respectively, and i is defined the is defined by i simply simply i and of the of the and activity. = i (3.97) (3.96) gives gives bar 1bar (3.94) (3.99) (3.92) (3.98) (3.93) (3.91) 59 - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 results can be derived from the treatment of liquid–solid and solid–vapor and equilibrium. of liquid–solid treatment the derived from be can results Similar states. liquid–vapor equilibrium clause treat we this will In at equilibrium. exact knowledge requires phases of processes the these in transfer mass study of the The not atequilibrium. are phases the other the to one phase from transfer is mass of there is acoexistence two there or When more phases. processes industrial many In 3.12 60 modified. must be rule phase the then azeotrope, an for phases, instance equilibrium ity of the phases. of three F and and P triplepoint freedom the of is there at degree no that finally T, and state P the curves havewe for the on points definition state. For of the two needed phases are for regionsthat of the P only one phase is given rule by following system the the ferent C. This in substances relation: of number dif the of Pand number phases the Fwith of freedom degrees the nects the pletely is called defined system oftions, etc.) is com aclosed, state isolated nonreacting the so that needed forderived every that, component i, the Gibbs–Duhemequationsforallphases, since dP,dT,anddn transfer ofmass,atmacroscopicobservation,fromonephasetotheother,writing hydrostatic, andchemicalequilibrium(fixedcomposition)wherethereisno For aclosed, isolated system consisting of b, several a, c, phases …inthermal, 3.12.1 phases. all in have we For phases. have example, value both in same properties the when these properties, = If a reaction is taking place in the system or we want to take in account apeculiar system account the in place or we in want take to is taking areaction If For example, P, the in 3.1 of Figure Tdiagram we for substance observe apure J.W. which con involving rule equilibriums phase Gibbs derived important avery mole frac of number temperature, (pressure, intensive properties minimum The where P, phases between states equilibrium with We T, here deal will • • • • • several intensive regarding We equilibrium in say two are or more phases that

2 and F 2 and

Diffusive equilibrium, when P,Diffusive equilibrium, T, all and when P,Osmotic equilibrium, T, several and phases. the of all constituents the is between no reaction when there equilibrium, Chemical phases. all in same Pis the pressure when the Hydrostatic equilibrium, phases. all in same Tis the temperature when the equilibrium Thermal =

EQUILIBRIUM OF PHASES [2,6,8,11,12] PHASES OF EQUILIBRIUM 0 which means only one pair of P, only one of pair 0 which coexistence state means the to Tcorresponds P ha =

s 1 which means that only one of that P,1 which means forthe definition of the Tis needed e R ule degrees of freedom. of degrees F μ= = a i C = + μ=

2–P 1 and F 1 and i b μ μ= i are the same. the are c i μ = i

are the same. the are 2, which means that both P, both that which means 2, T … Food Engineering Handbook Engineering Food i are zero it zero is are μ i are equal equal are (3.100) = 3 - - - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 Fundamental Notes on Chemical Thermodynamics on Chemical Notes Fundamental or i substance previous for the clause, everyIn we equilibrium b in saw two a, between phases that c 3.12.2 P only one single fluid phase. Beyond is no or liquid vapor disappears. but Cthere phase, line interphasial their and identical Finally, two become phases less Cthe less point distinct. and become atthe line separation volume molar interphasial density and in the and more similar and system of the more we which become phases liquid vapor to pass and temperature 3.1)(Figure the we gradually end C, point by up the to observe increasing that 3.1FIGURE C and the the and Point C is called the the Point Cis called point critical At the For system, two-phase asingle-component following curve vaporization the critical temperature temperature critical

hemical P–T diagram for a pure substance. for apure P–T diagram P P otential critical point critical Solid μμ 0, i a Fusion ⎝ ⎜ ⎛ ++ ∂ ∂ RT V P Sublimation

T in ⎠ ⎟ ⎞ C P Tc of the studying substance. studying the of ln and the corresponding P, corresponding the and Tthe ha T aa = μμ a i s a i 00 e =+ , E = ⎝ ⎜ ⎛ T ∂ q ∂ 0, i V Liquid 2 uilibria b i b P Gas 2

⎠ ⎟ ⎞ Tc R = [7,11] Tl Vaporization n

i b

C critical pressure pressure critical (3.102) (3.103) (3.101) 61 Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 and consequently from Equation 3.103 Equation consequently from and then 62 example of multicomponent system. system we binary atwo-phase study here of a single that ticomponent component. system an As than is more complicated evaporation. the It is changes obvious behavior during the of that amul temperature buta multicomponent in system constant, Pthe at constant remains temperature system. have phases liquid compositionvapor and same but the a multicomponent not in systemliquid amulticomponent asingle and in component one. the For instance, behavior the of between asingle differences component vapor– important are There b 3.12.3 phase. other of the properties the of we calculate one phase data can from since important very in equilibrium absolute activity, relative the activity, of fugacity a component a system the i in and for phase liquid while the usually since coefficients activity the that necessarily not does mean but this Since the Since the During the evaporation at constant pressure for evaporation pressure system the asingle-component atconstant the During are equilibrium of in phases function thermodynamic the relating equations The previous the From the relationsfollowing the connecting relation is obtained, we have of case vaporFor liquid equilibrium the for vapor seen that phase the inary µ xx i 0, a i a and and ↑ V a i b p . µ or µµ i 0,  i b – are the chemical potentials of the pure i at the same P, same iatthe pure of the potentials T chemical the are =+ L i q uid i 0, μμ  v i S =+ RT f y f i 0 i s μμ tem 0, i aa 0, i == ln v a a i λ = λ s = λ λ [7,9,11] 0 i i RT i 0, i   i b 0, i

b = a ln

i µ

f f i 0, i 0, i v  v RT

Food Engineering Handbook Engineering Food ln a i 

γ a i , γ i b will be equal equal be will (3.106) (3.107) (3.105) (3.104) (3.108) - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 Fundamental Notes on Chemical Thermodynamics on Chemical Notes Fundamental FIGURE 3.3 FIGURE X. Twith relating drawn 3.3 be can called curve vapor another form phase called acurve form compositions The liquid. phase liquid of the the with equilibrium whichphase, is in sition X phase liquid of the B, completelyand whole the of range acompo miscible X. At through any pressure P is mixture of the sure of vapor two pres 1the liquids. At 0and any mole between fraction mixture ideal an 3.2 FIGURE Figure 3.3 ofFigure two shows component A liquids P, the mixture of areal Xdiagram 3.2Figure shows avapor-pressure-composition (X

Vapor–pressure composition diagram for an ideal mixture of two liquids A and B. Aand of two liquids mixture ideal for an diagram composition Vapor–pressure PA Vapor–pressure composition diagram for a real mixture of two liquids. mixture for areal diagram composition Vapor–pressure P P 01 P A 0 bubble point line point bubble phase X Liquid liq liq corresponds to a different composition adifferent to X corresponds X mix

= P A phase X (1 and the corresponding compositions of the corresponding the and vap dew point line point dew

X − X) + P B Vapor X . A diagram similar to Figure Figure to similar . Adiagram

mole fraction) diagram formole fraction) diagram PB point line Bubble point line Dew 1 P B vap of the vapor of the (3.109) 63 - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 64 ferent gases of the same P and T.ferent Pand same of gases the it for and dif is different the ideality from gases expresses deviations real the of the V volumes molar the temperature and pressure same atthe gases For real different 3.12.4 considered. be to not necessary point. one critical Atization. and composition ABCDE each is acurve there called are process end of the this and start of the end temperatures C. point The critical the we as approach shorter passage becomes is changing. This temperature the pressure 3.1. of Figure line vaporization To vapor constant under the to liquid pure the from pass P, both need T(for single component the system either only Por only T). it is needed the systemof the state nentthe region system, define to order for vapor–liquid in we single compo the from of is freedom one more degree sition. there case that Since in components mixture. of two ofvapor the pure the pressure P with negative components and two of pure the pressure azeotropes one. ideal is an mixture the that mean mixture azeotrope. the and composition azeotropic vapor is phase. called and liquid This the both in 3.4 FIGURE m are different. The compressibility factor z =PV The different. are Similar behavior is observed for the processes of fusion and sublimation and it behavior of and is sublimation for fusion is observed processes and the Similar 3.4, place of Figure the the region is in In curve the included ABCDE inside the compo ataconstant 3.4Figure shows mixture liquid P, the of abinary Tdiagram vapor the than higher pressure azeotropic with positive are azeotropes There it not does compositionsequal, the two although are of cases phases the these In composition atafixed have composition mixtures liquid identical Some binary bubble point bubble P rinci

P–T diagram at constant composition for a mixture of two liquids. for composition amixture constant at P–T diagram P p le and and point line

o Bubble f A Liquid C dew point temperatures, point dew orresp Vapor + onding liquid S E B T tates respectively, of vapor pressure atthe [7,12] D Food Engineering Handbook Engineering Food m /RT defined in Section 3.10 defined /RT Vapor C line Dew

point az lower the than - - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 T perature Thermodynamics on Chemical Notes Fundamental vapor,where by and liquid vwe land mean respectively. state of transition of change called weproperty a define another to one phase from is transition there When e 3.12.5 gases. the for function all same where Fis the z constant. auniversal as ered consid be it and can 0.2–0.3, range narrow the in be to foundis also experimentally same. approximately the are where P However, P pressure reduced as by observed, defining it been has Knowing that atequilibrium that Knowing we of for case vaporization Thus, have the So,have we finally compressibility factor critical The Van the as is known Waals der This r , T r , and reduced molar volume molar reduced V , and r nthal , V m,r are the critical P, critical the are T, V and p y

and PT E rr ntro == by relation P P CC p ,, Δ y ∆ ∆ ∆ MM C z HH VV SS C i ab i i   hange i  = v v v = F (P ii  =− =− principles of corresponding states. corresponding of principles =− =− P V m,r = T T Cm R T m by relations the i i v µ

r i v that for equal P for that equal v i b , T in v C

,C r a ) S V H V M (3.112) T i m, i

a i r wo

= -P V V m, ha m C s C

e , T T C ran the V the s ition r , reduced tem reduced , m,r of all gases gases of all [2,7,12] (3.115) (3.116) (3.110) (3.113) (3.114) (3.111) 65 - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 or 66 vent is proportional to its to molevent fraction. is proportional of sol of fugacity pressure the solvent pure compositions the these partial near the respectively, its relative coefficient. activity and activity where of by relation liquids the solution solution. dilute is called ture concentration not does cover whole the 1. 0to from range mixture its the mole in fraction solubility and aliquid in when alimited oneor has for gas solid,ers, liquid, instance, it oth is convenient the some components from For several distinguish to mixtures 3.13 of vaporization. P where the dG and G When x When solvent of the solution potential chemical the For is given abinary mixtures in as by case, convention this In which liquid covers the we whole call the of range 3.120From Equations 3.121 and enthalpy change entropy and the we calculate can 3.117From Equations 3.119 through derived it is finally For atwo one phase, component, system phase ateach solution.

µ SOLUTIONS [2,5,11,13] SOLUTIONS i 0 is the reference chemical potential of pure solvent at certain P, solvent of pure reference potential is the chemical atcertain T, a i

sat → is the vapor pressure at the equilibrium of the two of phases. the equilibrium atthe vapor pressure is the When the solvent the When excess solute is in the low in and the concentration solvent

1 then also also 1 then , the component with the limited solubility limited component the with , the μμ ii =+ γ i

→ 0

1 and the solvent At the 1 and mixture. behaves ideal an as RT dP dP i

dT dT = i

GG i i S sa sa = ln t t ii H i dR dT –V = = ii = i –T S =+ ∆ ∆ ∆ ∆ H V V μγ S v i

i i  0  i i   v v v v i dP i (3.118)

Tl Food Engineering Handbook Engineering Food nx i

solute and the mix the and i , (3.120) (3.122) (3.117) (3.121) (3.119) γ i are, are, - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 f ture of the solution. of the ture where P Thermodynamics on Chemical Notes Fundamental * where P P,T. stant system or atcon real ideal solvent for fractions abinary molar of the function in pure ute behaves by convention, an, in the way ideal near which from is different solution. the of andpressure temperature the dilutionand at conditions of infinite state and where tion is given by of solution state hypothetical solute with the to mole 1. fraction f zero),to tend solute of fugacity the where the f dilution thethe solutionsolutesute atconditions in of all mole fraction infinite (the of state). (standard state)ence of 1 atm pressure fixed or atthe (refer temperature the and of pressure concentration soluteunit solution at certain solutions be solute higher cannot with concentrations. there solution. of the temperature the and what as we call i means pure solvent, pure means and 0 are the vapor pressure, fugacity of the pure solvent, pure of vapor fugacity the pressure, the respectively, are tempera atthe The standard chemical potential of asolvent potential chemical is given standard by relationThe the [2]: region the solutions of dilute very In where relation (3.124) sol the is valid solute of above the asoluFrom in the potential chemical mentioned analysis the Relation (3.124) is called is derived by extrapolation sol of state fugacity of the the the solute standard This ideal for solute state hypothetical, the the standard it as case, is adopted this In solute,For the however, since state it standard is not possible a define to similar solvent of the potential at1 liquid pure chemical of the is that standard The solvent region, the this In way behaves relations ideal the (3.123) an and in express f i solvent. 3.5 Figure shows of fugacities solute of variation the the the and 0 , γ

i i µ 0 , f , f , P are the standard pressure and the pressure of the solution, of the pressure the respectively, and pressure standard the , Pare i 0 are the fugacity and the chemical potential of pure i at the reference iatthe of pure potential chemical the and fugacity the are i i , x are the activity coefficient and the fugacity the of iata coefficient activity moleand x fraction the are i the partial pressure, fugacity, pressure, mole solvent of and the fraction partial the and Raoult’s law μμ μμ ii 2 0 V =+ () 2 T, ∗ Henry’s law Henry’s volume solvent. of pure (Figure 3.1). (Figure PP 0 PT ii 0 == RT =+ 0 xo ln ∗∗ 22 ii i ()

γ = and K is a constant called Henry’s called K is aconstant constant. and ii K x xR ,P rf =+ i (3.124) i μ is proportional to itsto mole x fraction is proportional P ∫ P 0 i 0 V( fx i 0 T,P)dP i Tl

n f f i 0 i

(3.126) (3.125) (3.123) atm atm P 67 i 0 ------, i i

Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 68 tively, molecule electrolyte, the of we write in the can M type cations of one the electrolyte. and anions the both physical meaning. ions, the no which of of has rest substance of amount all the by constant keeping ion would of of imply change this substance amount this of since the the potential, functions of one ion, for possibleexample, thermodynamic chemical its define to solution electrolyte an in it is not neutrality of electrical restriction ofBecause the constituted. ions which are they the from in dissociation completeunder or partial study, not found solutions thermodynamic in their oftions in nonelectrolytes. class of solute a special which substance, involve are Electrolytes several complica 3.14 way above the from mathematical relations. astraight derived in dilution.solute atconditions of infinite T, P, m where 3.5 FIGURE For example, of for case astrong, the completely of electrolyte the dissociated with functions Tothermodynamic the overcomeconsiderwe all difficulty this in solution the are electrolytes that the found fromfact arise difficulties The relations and for solution functions the be can thermodynamic other the All of asolute potential is given chemical by relation standard [2]:The

∞ 1 V ELECTROLYTE SOLUTION [1,2,8] , + means conditions at infinite dilution, conditions atinfinite means A m V 1 0 Pure solvent

− the standard molality, P standard the , where V Fugacities of solute and solvent in function of mole solvent ofFugacities solute fractions. and function in f 2 0 μμ 1 0 () T( 0 + =− and V and ⎣ ⎢ ⎡ 11 f 2 T,P, − f are the number of number positive the negative are and ion, respec 1 μ m) AB

= 0 the standard pressure, and and pressure, standard the

V RTln + X μ 1 +

+ m V m 1 0 1 μ −

f id 1 ⎦ ⎥ ⎤ μ 1 (T,P,m

∞ = K

−

(3.128) f 2

+ id f

= = i X Food Engineering Handbook Engineering Food P

∫

P 2 1

0 X 1

) the chemical potential at potential chemical ) the 2 1 ∞ T,P)dP f Pure solute 1 0 = K V 1 1 °

the volume the of (3.127) - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 molalities using and Thermodynamics on Chemical Notes Fundamental ln form: the takes mation ionic coefficient, activity mean which dilute the giving solutions for very by approxi solution. of dissociation in electrolytes the regarding viously by Arrhenius assumed of some pre ideas treatment was amathematical solutions. theory dilute in lytes This 1923,In Debye Hückel and behavior developed about the of strong electro atheory d 3.14.1 for same strong electrolytes. the are potentials cal state. ence solutions when where V molality mean the and μγ By this theory after a series of mathematical calculations we arrive ataformula we calculations arrive of mathematical aseries after theory By this expressions obtain to for chemi method of the the electrolytes case For weak the m The 3.132,From Equations 3.133, 3.131 and by and defining as coefficient activity defined is mean The 3.129Substituting Equations 3.130 and 3.128 Equation into we obtain MA μ =+ MA = R Tln ± (, V , and , and P μ μμ ebye + −− ++

+ TV () () γ V T P mm T P )( – γ ± ,, ,,

± =+ H − V +− → and and . + ückel

0 then 0 then μμ RTlnm mR mR V − 00 MA µ MA 0 =+ L =+ () R Tln T, (T, P) have a similar meaning as for (T, nonelectrolyte as the P) meaning have asimilar imiting γ ± Tlnm Tlnm

PV → ±±

=+ 0 and 0 and V +− mm VRTl L + γ γγ γμ ± ±+ V ±+ V +−

aw V0 = − = = C I µ +− R Tln [2,6] R Tln () MA 0 () T P T, V 1/2 nT V ,, + (T, to P) is equal + γμ γ z PV m − V γμ + γ − V − z m −− ++ −

− () ()

,, (3.136) T P T P P ,, ,, + μ m 0 VT ++ () )( mT mT T, + P + () + µ 0 MA

, MA 0 P 0 0 (T, P) refer atthe () () T P + , , ,) P P VT −−

0 () (3.130) (3.135) (3.134) (3.129) (3.132) (3.133) (3.131) , P 69

- - - - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 where C 70 from its elements at a certain temperature and pressure. When this formation is formation this When pressure. and temperature its elementsfrom atacertain compound that form to of heat of formation required heat The any component is the e 3.15.1 reaction. catalyst for this appropriate trouble, the allow the seek to usor release usfrom of it law areaction, can possibility of of realization the thermodynamic second the means. or other catalysts use they overcome hindrance, this to and not does start reaction this one to direction, shouldone reaction proceed that opposite atwhich extent orand the to stop state). (equilibrium it oneto will direction combustionthe of afuel. from released energy or the substances useful other various some substances, from produce, to needed energy the we calculate can For reaction. instance, a chemical tions gives exactly energy, the (endothermic) absorbed (exothermic) or released by condi lawcertain under thermodynamic Together first tant. the balance mass with systems impor is very reacting chemically to application ofThe thermodynamics 3.15 electrolytes. is given and ionic for strength a1–1I is called by electrolytes type z N But still, in these cases thermodynamics is useful, since through the prediction by prediction the through since is useful, thermodynamics cases these in But still, lawHowever, it results thermodynamic second the from several although in cases, proceed will reaction one chemical if law predict can thermodynamic second The with valid is still theory the electrolytes For mixed Debye–HückelThe for solution law dilute very accurate is very of strong m K T ε e ρ

2 + s = = 0

+ = , = =

, m

the charge on charge aproton the = z temperature of solution temperature ε R/N density of the pure solvent density pure of the Avogadro’s number Avogadro’s 2 − THERMOCHEMISTRY: CHEMICAL REACTION EQUILIBRIUM [11,15] 0

, of avacuum = − ε the electrical charges of the ions. of charges the electrical the , the molalities of the ions molalities of the , the r (2 (2 , nthal 0 ε , R r is the dielectric constant of the solvent of the constant dielectric is the and π N = p 0 gas constant gas

y ρ

s o ) 1/2 f F (e ormation 2 /4 /4 π

ε Im KT) =+ Im 2 1

and = () 3/2 2 1 ++ E ∑ zm i 22 nthal ii z 2 −− p z

f Food Engineering Handbook Engineering Food R eaction ε 0 is the dielectric constant constant dielectric is the [4,5] (3.137) (3.138) - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 its elements at the same standard conditions. its standard elements same atthe of 1 state its standard in compound the form to required of formation. enthalpy the to of heat is formation equal the then pressure constant considered under Thermodynamics on Chemical Notes Fundamental 25°C. of H where H H negligible enthalpy is given change no work in and the by is produced are energy P, kinetic constant For under and a reaction potential in where variations d 3.15.2 calorimeters. called are function of these instrument the and thermochemistry called of thermodynamics, branch the of for case a solution), the mixing by studied of heat are well reactions as the as effects. heat called Tare place atconstant taking pressure constant a single and compound oxygen. with compound oxidation the of from this ing 25°C. of 1 pressure place ataconstant taking reaction P, place atconstant enthalpy reaction. tion the to takes of heat is equal reaction the of thermodynamics. books many move zero, is therefore considered stable as state where v P0 These heat effects together with the heat mixing, the heat of heat solution the of (heat mixing, heat effects together heat the with These or fusion which vaporization, is for the during of transferred quantities heat The of heat resultthe heat reaction combustionof as The of is defined any compound from a in enthalpy the change as enthalpy of is defined reaction standard The reac the If by reaction. or rejected the absorbed heat of heat is the reaction The givenTables of enthalpies are of formation in compounds many standard with By of enthalpies conversion, h formation standard the enthalpy of formation, standard The , H r0 i enthalpies of products and reactants at a pressure of 1 atapressure reactants of enthalpies and products is the stoichiometric coefficient. is the P , H etermination r enthalpies of products and reactants at a pressure P, at apressure reactants of enthalpies and products respectively, and Δ HH RP Δ =−

Hh o P

F f −

H the 00 =− r

E = compound Δ H( (H nthal H r F P =h =−

− Δ H products p H ∑∑ compound y P0 ∑

F o of a compound is defined as the heat the heat as of is defined acompound ) i f + R (v vh) (H (3.140) ii eaction h) atm and constant temperature of temperature constant and atm r

− reactants elements H atm pressure and 25°C and pressure from atm r0 ) [4,5,7] (3.141) i (vh) of all elements of of all their

atm and temperature temperature and atm

(3.139) (3.142) 71 - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 behave gases. ideal as andproducts reactants the that or by simplificationconsidering, tally instance, for using relative existed tables. reactants and products of enthalpies of formation the standard the from calculated be can of and reaction enthalpy standard is the reaction, the in ofenthalpies compounds of formation the v’swhere the the stoichiometricand coefficientsmolar the h’s are the standard are 72 and was called by him as the affinity A affinity the as by him was called and tion reaction. of the sign of the the determines sign parentheses ofthe the where forand areaction dG where c T,at constant P usconsider reaction Let the e 3.15.3 reaction. of reactions which other we ofenthalpies know or subtracting the adding ply The quantity quantity The 3.144 on Equations Based 3.145 and 3.42 Equation and it follows: 3.46Equation atP, gives Tconstant by reaction sim enthalpy of acertain the several we determine cases, In can (H PP ξ − i is the extent is the reaction. of the are the constituents and v and constituents the are H) q 0 uilibrium and and μ dG 1 (H dn T, 1 P1 rr

C + − =+ −=

μ on H At equilibrium where At equilibrium μμ (d T,P dn 2 μμ v 11 dn 0 dn 1 s

) 1 = tant can be calculated either from known data, experimen data, known either from calculated be can vc 2 nd

11 μ − 12 += 1

dn μ −= : A i ++ 3 the stoichiometric coefficients. the dn dn 1 v 22

vc dn + 2 ff 2 3 22

μ nd inity − f . 2 23

→ ← dn μ −− dn vc 4 v dn 2 μμ 33

μμ

3 o + 3 33 4 dn

f was introduced by De Dander (16) by Dander De was introduced μ ==

nd a 3 dn 34 dG R dn vc − v 44 3 4 eaction 4 T,P

+ Food Engineering Handbook Engineering Food 44 ξ

dn μ = and consequently the direc consequently the and 4 0 n) d dn 4 ξ

d0 [6,7,11] (3.144) ξ≤

(3.146) (3.145) (3.147) (3.148) (3.143) - - Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 where Thermodynamics on Chemical Notes Fundamental REFERENCES K constant equilibrium the is called activities, not and on the and quotient. reaction is called 10. 9. 8. 7. 6. 5. 4. 3. 2. 1. From Equation 3.148From Equation using and when K and when K when K only when A Since proceeds one reaction have we Thus, where At equilibrium quantity The York, USA:McGraw-Hill, 1961,p.191. Ed. Singapore: McGraw-Hill BookCo,1987. University Press,1990. Co, 1985. Royal InstituteofChemistry, 1971. 1979. GN Lewis, M Randall. JM Smith, HC Van Ness. SE Wood, RBattino. JS Hsieh. CE Reid. P Rock. TE Daubert. ML MacGlashan. ML Mac Glashan. SW Angrist, LGHepler. A f 0 is the affinity the for affinity is the a a

< > Chemical Thermodynamics Q Q Chemical Thermodynamics Principles of Thermodynamics Principles a

a a = the reaction proceeds to the left, the to proceeds reaction the right, the to proceeds reaction the Chemical Engineering Thermodynamics Chemical Q a there is equilibrium. there Physicochemical Quantitiesand Units Chemical Thermodynamics Thermodynamics ofChemicalSystem A Thermodynamics AA f 0 Order andChaos ff Introduction to Chemical Engineering Thermodynamics = μμ =− µ 0 ii the Q the i 0 =+ ⎝ ⎜ ⎛ . AR AR aa aa 0 1 3 v v f f 0 0 3 12 . California,USA:University ScienceBooks,1983. = = ⋅ ⋅ TRTl a . Singapore: McGraw-Hill BookCo,1990. depends only on the temperature and on and temperature only on the depends RTln 2 v v 4 . Revised byKSPitzer, LBrewer. 2ndEd.New 4 . New York, USA:McGraw-Hill, 1975. Tl Tl ⎠ ⎟ ⎞ , New York, USA:BasicBooksInc.,1967. n n f

= nK > ⎝ ⎜ ⎛ a K Q 0 this means that means 0 this i aa aa Q (relation (3.91)) it is obtained: a a 1 3 v v a . London, UK: Academic Press Inc., 3 12 a

⋅ ⋅ 2 v v 4 . Singapore: McGraw-Hill Book 4 ⎠ ⎟ ⎞

. 2ndEd.,London,UK: The . Cambridge,UK:Cambridge a . (3.150) (3.152) (3.149) (3.151) . 4th A 73 f 0

Downloaded By: 10.3.98.104 At: 21:50 27 Sep 2021; For: 9781482261707, chapter3, 10.1201/b17843-4 74

14. 13. 12. Tassios. 11. DP 16. 15. Springer-Verlag, 1993. California, USA:StandardUniversity Press,1936. Publishers, 1978. Phase Equilibria Prentice-Hall Inc.,1997. with Applications toPhaseEquilibria JM Prausnitz,RNLichtenthaler, EGde Azevedo. JW Tester, MModell. HC Van Ness,MM Abbott. Th DeDonder, P Van Rysselberghe. MK Karapetyants. Applied ChemicalEngineering Thermodynamics . 3rdEd.New Jersey, USA:PrenticeHall,Inc.,1999. Chemical Thermodynamics Thermodynamics andits Applications Classical Thermodynamics of NonElectrolyte Solutions, The Thermodynamic Theory of Affinity The ThermodynamicTheory . New York, USA:McGraw-Hill Inc.,1982. . Trans. byGLeib. Moscow, Russia:MIR Molecular Thermodynamics ofFluid- Food Engineering Handbook Engineering Food . 3 rd Ed.New Jersey, USA: . Berlin,Germany: . Stanford,