9THERMODYNAMICSINUNITPROCESSESEquilibrium Compositionof Nonideal Gases.In ordertoprovideequa-tionswhich would be useful for all substances,Lewis and Randallidefinedaterm,fugacity.This is an arbitrarilydefined term thatmaybesubstitutedforpressure in the preceding equation and which will make it applicableforall substances.The mathematical definition of fugacity f isdG-RTdInfSince the numericalvalue of the fugacity of a Bubstance is often dificultto calculate, the ratio of the fugacity at one state to that at some otherarbitrarilydefined stateis often used.The second such condition is calledthe standard state.The ratio is called the activity and is defined thus:a-fwhere f' refers to the fugacity of the substance at the standard state.Although anyarbitrarystandard statemaybe chosen,customhaslimitedthis choice.In the study of chemical equilibria involving gas reactions,the standard state is chosen as that in which the fugacity of the pure gasisequaltounity.ThisformostgasesmeansthepuregasatlatmpressureThestandard-statetemperaturemustbethesameasthatof thesystemforwhich wearecalculating the equilibrium composition.(Thismustnotbe confused with the standard thermochemical temperature of 25°C298.15°K.)Calculation of theFugacityof Gases.In the application of the chemicalequilibrium equation to nonideal gases, it is the fugacities and not thepartial pressures that are important.Therefore, we must have a meansof relating thefugacity to the partial pressure in order to use the equation.Fugacity is really snother term which is a measure of the deviation of agas from ideality.Since this is the case, there must be a relationshipbetween fugacityand the compressibilityfactorPV.-RTwhichis anothermeasureof thisdeviation.The Law of Corresponding States.When a gas is nonideal, PV doesnot equal RT.However, the quantity RT can bemultiplied by a correc-tion factor so that for1mole of gasPV-2RTThis correction factor (the compressibility factor)varies with both tem-perature and pressure for each gas,If there were no relation between thefactors fordifferent-gases,it wouldbe necessarytohave charts ortablesshowing zas a function of temperature and pressureforeach individualILewisand RANDALL,"Thermodynamics,MeGraw-Hill Book Company,Inc.,NewYork,1923
10UNITPROCESSESIN ORGANICSYNTHESISgas.This would make the use of the compressibility factor impractical.However, it has been found by experiment that when gases are at thesamereduced temperature (T,=.T/T.)and reduced pressure (P,=P/P.)their compressibility factors are approximately the same. The degree towhich a gas deviates from ideal behavior, therefore, depends upon itsreduced temperature and pressure.At equal reduced conditions the devia-tion is the same for all gases. This rule is an empirical one which has beenfound to hold with a fair degree of accuracy for most gases.It is calledthelawof correspondingstates.The principal value of the law of corresponding states is that it makespossible the construction of charts or tables which give the value of thecompressibilityfactor for any gas as a function of reduced temperatureand pressure.The data are usually presented in chart form, with zplottedagainst P, and with T, as the parameter.1 Such a chart is shown in Fig. 1-2.Thelawof corresponding statesmay alsobeused in the constructionof charts or tables which give the deviation of enthalpy, entropy, or heatcapacity from that of the ideal gas as a function of reduced temperatureandpressure.One of the most important uses of this law is the development of amathematical relationship between the fugacity of a nonideal gas and itscompressibility factor z. From this relationship, charts have been made?which permit the calculation of the fugacity of those nonideal gases whichobey thelaw of corresponding states.A chart of this type is shown inFig.1-3.The ordinate is the ratio of fugacity to pressure (f/P)and iscalled the fugacity coeficient (v).This isplotted against reduced pressure,with reduced temperature as a parameter, The fugacity-coefficient chartmay also be used to calculate the fugacity of individual components ofgaseous mixtures and also to calculate the fugacity of a pure liquid at itsvaporpressure.Equilibrium Constant for Nonideal Gases. When the equilibrium con-stant is applied to nonideal gas mixtures, the fugacities are used insteadof partial pressures. Thus, the equilibrium equation for the reactionaA+bB=cC+dDKmJebecomesfafebutsinceJA=PAPA"-铃where1NeL8oNandOBERT,Chem.Eng.,61 (7),203-208(1954)*HouGEN and WArsoN, "Chemical Process Principles," part II, p. 622, John Wiley& Sone, Inc., New York, 1947
CompressiblityFactor,z3.0412=1.0Reducedtemperature.41L.6Tr2.02O1515元1.020L8160.80.70-1.40.751.30.800.60.85-L20.90.二1.1530.9510.41.051.03T,oL0.21.0=Reducedtemperoture,Tr.All charts in this section odoptedtromRef.2,withpermission0.10l20810O.10.22460.40.60.81:0ReducedPressurePrFic.1-2.Compressibility factor plotted against reduced pressure with reduced temperature parameters
21.5金1.00.9150.81.51.30.71.3L0.6P0.51.10.47121.00.310.980.20.960.940.90.940.1300.20.31520400.40.50.60.70.80.91.01.52340689102Reducedpressure,,'p/peFrG. 1-3. Generalized fugacity coefficient.ve-reduced-pressure
13THERMODYNAMICSINUNITPROCESSESthismaybewrittenK-epovbpsVipiuapiKwovipepspepD2=KkorpipevAvepipsKvavswhereby definitionvveHere Kis a direct measure of the effect of nonideality of the gases uponthe equilibrium composition.Calculation of the Equilibrium Constant. The equilibrium constantK,for a chemical reaction can be calculated from the standard free-energyfunctions orfromchangesin enthalpyand entropyfortheprocess.Datawill be found in the references listed at the end of this chapter. Since,AGRTIKandTAfor a process occurring at constant temperature and pressure, data mustbe available for the temperature at which the reaction takes place.Ifvalues of G are not available, calculate them from enthalpy and entropydata.H was calculated on p.5.AS can be calculated in a similarmanner:TAC,odTATASZ(S)produeu-(S)reaetantThe standard entropy change of the reaction is calculated from the valuesof the absolute entropy of the products and reactants at the standardtemperature Ts=25°C.The calculation of absolute entropy of many substances ispossible be-cause the substances follow the third law of thermodynamies.This statesthat the absolute entropy of a pure substance in the crystalline state iszero at the temperature of absolute zero.Thus, if the latent heats X ofall the phase changes are known and if the heat capacity isknown, theabsolute entropy of the pure substance may be calculated by the followingequation:aT+zSoEntropies calculated in thismanner were compared.with absolute entropiescomputed backward from measurable equilibrium constants and werefound to check quite closely.If the standard free energy of the substances involved in a chemical re-