Thermodynamics

682 | ThermodynamicsH 26 kg+O 232 kgH 2 + O 238 kgFIGURE 13–1The mass of a mixture is equal to thesum of the masses of its components.H 23 kmol+FIGURE 13–2O 21 kmolH 2 + O 24 kmolThe number of moles of a nonreactingmixture is equal to the sum of thenumber of moles of its components.H 2 + O 2y H2 = 0.75y O2 = 0.251.00FIGURE 13–3The sum of the mole fractions of amixture is equal to 1.13–1 COMPOSITION OF A GAS MIXTURE:MASS AND MOLE FRACTIONSTo determine the properties of a mixture, we need to know the compositionof the mixture as well as the properties of the individual components. Thereare two ways to describe the composition of a mixture: either by specifyingthe number of moles of each component, called molar analysis, or by specifyingthe mass of each component, called gravimetric analysis.Consider a gas mixture composed of k components. The mass of the mixturem m is the sum of the masses of the individual components, and themole number of the mixture N m is the sum of the mole numbers of the individualcomponents* (Figs. 13–1 and 13–2). That is,m m ak(13–1a, b)The ratio of the mass of a component to the mass of the mixture is calledthe mass fraction mf, and the ratio of the mole number of a component tothe mole number of the mixture is called the mole fraction y:mf i m im mandy i N iN m(13–2a, b)Dividing Eq. 13–1a by m m or Eq. 13–1b by N m , we can easily show thatthe sum of the mass fractions or mole fractions for a mixture is equal to 1(Fig. 13–3):kka mf i 1and a y i 1i1i1m i andN m akThe mass of a substance can be expressed in terms of the mole number Nand molar mass M of the substance as m NM. Then the apparent (oraverage) molar mass and the gas constant of a mixture can be expressed asi1N ii1M m m m a m i a N i M k iN m N m N a y i M i andR m R um i1M mThe molar mass of a mixture can also be expressed asM m m m m m1N m a m i >M i a m i >1m m M i 2 1kmf iaMass and mole fractions of a mixture are related bymf i m im m N i M iN m M m y i M iM mi1 M i(13–3a, b)(13–4)(13–5)*Throughout this chapter, the subscript m denotes the gas mixture and the subscript idenotes any single component of the mixture.