Thermodynamics

698 | ThermodynamicsMixture:i h i,mixture Ts i,mixture g i,mixturePure substance: h Ts gFIGURE 13–18For a pure substance, the chemicalpotential is equivalent to the Gibbsfunction.where y i N i /N m is the mole fraction of component i (N m is the total numberof moles of the mixture) andm i a 0G0N ibP,T,N j g i h i Ts i1for component i of a mixture2(13–30)is the chemical potential, which is the change in the Gibbs function of themixture in a specified phase when a unit amount of component i in the samephase is added as pressure, temperature, and the amounts of all other componentsare held constant. The symbol tilde (as in v , h , and s ) is used todenote the partial molar properties of the components. Note that the summationterm in Eq. 13–29 is zero for a single component system and thus thechemical potential of a pure system in a given phase is equivalent to themolar Gibbs function (Fig. 13–18) since G Ng Nm, wherem a 0G0N b g h Ts 1pure substance2P,T(13–31)Therefore, the difference between the chemical potential and the Gibbsfunction is due to the effect of dissimilar molecules in a mixture on eachother. It is because of this molecular effect that the volume of the mixture oftwo miscible liquids may be more or less than the sum of the initial volumesof the individual liquids. Likewise, the total enthalpy of the mixture of twocomponents at the same pressure and temperature, in general, is not equal tothe sum of the total enthalpies of the individual components before mixing,the difference being the enthalpy (or heat) of mixing, which is the heatreleased or absorbed as two or more components are mixed isothermally. Forexample, the volume of an ethyl alcohol–water mixture is a few percent lessthan the sum of the volumes of the individual liquids before mixing. Also,when water and flour are mixed to make dough, the temperature of the doughrises noticeably due to the enthalpy of mixing released.For reasons explained above, the partial molar properties of the components(denoted by an tilde) should be used in the evaluation of the extensiveproperties of a mixture instead of the specific properties of the pure components.For example, the total volume, enthalpy, and entropy of a mixtureshould be determined from, respectively,V aiN i v iH aiN i h iandS aiN i s i1mixture2(13–32)Ay ABMixingchamberA + Bmixtureinstead ofV * aiN i viH* aiN i h i andS* aiN i s i (13–33)y BThen the changes in these extensive properties during mixing becomeFIGURE 13–19The amount of heat released orabsorbed during a mixing process iscalled the enthalpy (or heat) ofmixing, which is zero for idealsolutions.¢V mixing aiN i 1v i v i2, ¢H mixing aiN i 1h i h i 2, ¢S mixing aiN i 1 s i s i2(13–34)where H mixing is the enthalpy of mixing and S mixing is the entropy ofmixing (Fig. 13–19). The enthalpy of mixing is positive for exothermic mix-