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Real Vapors & Solutions University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014 Lecture 16 8/04/14 A. Real Vapors and Fugacity • Henry’s Law accounts for the properties of extremely dilute solution. As shown in Figure 1 Henry’s Law only accounts for the properties of solutions in the encircled areas. • For solutions that are not dilute, the physical properties of these solutions starting with their chemical potentials, is not accurately described by either Raoult’s Law or Henry’s Law. To accurately describe the free energy of a concentrated solution of a non-ideal vapor in contact with a non-deal, concentrated solution of a non-electrolyte, the concept of activity is introduced. Figure 1: The vapor pressure above mixtures of of CS2 and acetone deviate strongly from both Raoult’s Law (dashed lines) and Henry’s Law (encircled areas) for mole fractions between 0.2 and 0.8. • In the two simple solution models we have convered we assumed the vapor behaves ideally so that the chemical potential of each component in the vapor phase is obtained by a simple integration... RT dvVdPdPµiiii()== Pi (16.1) 0 ⎛⎞Pi ∴µµii()vRT−= ln ⎜⎟0 ⎝⎠Pi • If VRT VRT • ≠ or V == +α (16.2) nP nP where α is an empirical correction term intended to account for all effects (i.e. intermolecular interactions, finite molecular size etc. ) that cause deviations from ideal gas behavior. • The expression for the chemical potential of the vapor is now RTPPii dP dµαµµα v== VdP dP +⇒ dP v −=0 RT + dP iii() () ∫ ∫ PPPP00 • ii (16.3) ⎛⎞P Pi ∴µµvRTdP−=0 ln i + α ii() ⎜⎟0 ∫ Pi 0 ⎝⎠Pi • If we know the equation of state a can be determined explicitly. Otherwise a can be measured directl as a function of P and the integral can be evaluated numerically. See examples below. vap vap,0 ⎛⎞Pi • In analog with the ideal vapor expression µµii−=RT ln ⎜⎟0 , for a real ⎝⎠Pi vapor we define the fugacity f. The fugacity is the vapor pressure...or escape tendency... corrected for the fact that the gas is not ideal. RT dµαα==+=+ RTdfln dPdPRTdPdP ln P (16.4) α ∴RTdln f=+⇒=+ RTd ln Pα dP d ln f d ln P dP RT • The reference state can be defined as the limit of very low pressure where the vapor behaves ideally and so f0=P0. We obtain… ffPiii1 P i dfln≈=+ df ln dP ln α dP ∫∫∫ii ∫ fPP000RT P 0 iii i (16.5) ⎛⎞f ⎛⎞P 1 Pi ∴ln=+ ln i αdP ⎜⎟00 ⎜⎟ ∫ PPRTii0 ⎝⎠ ⎝⎠ Pi • We can rearrange equation 16.5 and after a little algebra we get ⎪⎪⎧⎫1 P f ≈=PdPPexp α γ f (16.6) ii⎨⎬∫ ii RT 0 ⎩⎭⎪⎪Pi where γf is the fugacity coefficient. The lower limit of the integration approaches zero in many practical applications. RT • The fugacity coefficient can be determined by measuring the quantity V −=α P P over a range of pressures and determining the integral ∫ αdP numerically. P0 • Note that γf measures the degree to which the component of the vapor deviates from 0 the ideal gas law at Pi and Pi . Therefore ⎛⎞fi f Lim=11 or Lim γ i = (16.7) PP→→00⎜⎟ () ii⎝⎠Pi B. Examples of Fugacity Calculations • Example 1: If the equation of state is known the fugacity can be determined by deriving the quantity α from the equation of state. Suppose the equation of state is P( V−= nb) nRT (16.8) where b is a constant that reflects the volume excluded by finite molecules in the gas VRT phase. Solve for V/n: ==Vb +. Then α=b and the fugacity coefficient is easy to nP 0 calculate. Assuming for convenience that PPii<< P P ⎧⎫⎧⎫i 0 ⎪⎪⎪⎪1 b bP()ii− P/ RT bP/ RT γα===≈expdP exp dP e e i (16.9) fii⎨⎬⎨⎬∫∫ RT00 RT ⎩⎭⎩⎭⎪⎪⎪⎪PPii All you need to calculate the fugacity is the pressure P and the value for b. • Note in some texts the fugacity is defined a little differently. Suppose we define the fugacity as RT α PV=+ RTα or V = + PP α ∴dRTfRTdPdPµ ==ln ln + ii iiP 1 α (16.10) ∴dfdPlnii=+ ln dP i RT Pi ⎡⎤1 Pi α ∴ fP= exp ⎢⎥ dP ii ∫ ⎢⎥RT0 Pi ⎣⎦Pi i • This alternative definition makes no difference to the result. Again for P() V−= nb nRT we now get with the alternative definition α = bPi . But we get the same final result as we did using the other definition: ⎡ 11PPiiα ⎤⎡bP ⎤ f ==PdPPdPexp⎢ ⎥⎢ exp i ⎥ ii∫∫ i i i ⎢ RT00 Pii⎥⎢ RT P ⎥ ⎣ PPiiii⎦⎣ ⎦ (16.11) ⎡⎤0 bP()ii− P =≈PPeexp ⎢⎥bPi / RT iiRT ⎣⎦⎢⎥ • Example 2: The fugacity can also be calculated from experimental data using the RT equation α =−Vi o r α =−PVii RT . Below is a study of the fugacity of Pi nitrogen gas as a function of pressure. Noteworthy features: V 11α • The quantity i −= is tabulated in the second column and plotted as a RTPRTPii function of pressure. • Note the area under the curve is 1 Patmi =800 α ln γ f = dP iiRT∫ P Pi ∼0 i • The third column is the fugacity coefficient explicitly and the degree to which F/P deviates from 1 measures the deviation from ideal gas behavior. • The fourth column tests the degree to which the data fits a model P() V−= nb nRT where bP<<RT . The closer the value in column 4 is to 1 the closer is the agreement to the model. C. Real Solution Equation of State • To account for non-ideality of the vapor above a real solution, we substitute the fugacity for the pressure. We also introduce the term activity, assumed to be the ratio of the fugacity at a given pressure to the fugacity at a reference pressure; fi 0 = ai (16.12) fi • If the solution were ideal the activity would just be the mole fraction of component I in the solution. But in analogy to the fugacity coefficient, deviations from ideality are reflected in a activity coefficient: fi 0 ==axiiiγ (16.13) fi • Equation 16.13 is the equation of state for real solutions. Like the fugacity coefficient, the activity coefficient can be measured directly or calculated if a model for the real solution is available. We will consider a real solution model in the next lecture, called the regular solution model. In the next section we show how activity and fugacity are introduced into equilibrium constant expressions. D. Summary of Equilibrium Relationships for Chemical Reactions with Real Vapors and Real Solutions (non-Electrolyte) • Consider a reaction ν ABAB++ννν C C D D where a, b, c, and d are stoichiometric coefficients. The condition for equilibrium is: ν AAµνµνµνµ+=+ BB CC DD •• • • ∴νµAA()()()()+++=+++RTaln A νµ BB RTa ln B νµ CC RTa ln C νµ DD RTa ln D • We now rearrange the equation: 0 •••• −∆GRTaRTa =νµAA() + νµ BB () − νµ CC( ) − νµ DD( ) = − ν A( ln A)( − ν B ln B ) ++ννCCDD()()RTln a RT ln a ⎡⎤νC ν D 0 −−ννABνC ν D aaCD ∴∆=−GRTa⎡⎤lnABCD + ln a ln aa ln =− RT ln ⎣⎦⎢⎥ννAB ⎣⎦aaAB νC ν D aaCD • In general K = where ai is the activity of species i. ν ABν aaAB ν ν νν C D 00AB PPiC⎛⎞⎛⎞⎛⎞⎛⎞PPPDAB • Ideal Gases: aKiP=⇒=000⎜⎟⎜⎟⎜⎟⎜⎟ PPPPPiCDAB⎝⎠⎝⎠⎝⎠⎝⎠ ν ν νν C D 00AB ffiC⎛⎞⎛⎞⎛⎞⎛⎞fffDAB • Real Gases: aKif=⇒=000⎜⎟⎜⎟⎜⎟⎜⎟ fffffiCDAB⎝⎠⎝⎠⎝⎠⎝⎠ • Non-electrolyte Solutions, Ideal, units of concentration: ν ν νν D D 00AB CCiC⎛⎞⎛⎞⎛⎞⎛⎞CCCDAB aKiC=⇒=000⎜⎟⎜⎟⎜⎟⎜⎟ CCCCCiCDAB⎝⎠⎝⎠⎝⎠⎝⎠ • Non-electrolyte Solutions, Ideal, units of molality: ν ν νν C D 00AB mmiC⎛⎞⎛⎞⎛⎞⎛⎞mmmDAB aKim=⇒=000⎜⎟⎜⎟⎜⎟⎜⎟ mmmmmiCDAB⎝⎠⎝⎠⎝⎠⎝⎠ • Non-electrolyte Solutions, Real, units of molality: ν ν νν C D 00AB γγiimm⎛⎞ CC ⎛⎞⎛⎞⎛⎞γ DDmm A m B aKim=⇒=000⎜⎟⎜⎟⎜⎟⎜⎟ mmmmmiCDAABB⎝⎠⎝⎠⎝⎠⎝⎠γγ where all activities correspond to equilibrium conditions. .
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