Real Gases. Chapter Equations of State. 1. van der Waals: =

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1 Chapter 7 Real Gases 7.1 Equations of State 1. van der Waals: = ( ) 2 From this analytic function for we can (integrate at constant )toinfer() [Recall = ) ] The main assumption to recall is that there is ONE (molar volume) or its inverse (the density). This is often called the uniform density approximation. This is why the vdw s logic fails in the two-phase region. (van der Waals completely understood this! He knew when his eq. worked, when it failed and why!) This equation played an important role in the development of theory of real fluids. It captures two physical concepts and makes one major assumption. (a) represents the intermolecular attraction vdw EoS. Note the inset and the mechanical equilibrium condition connecting phases with a common tangent and thus equal pressures. (b) represents the hard-core repulsion and thus not all the actual volume is available for motion. If the hard core of the potential is denoted by = 2 the excluded volume per 4 mole is = 3 (2 ) 3 2=4 where is the volume of a mole of units. The vdw EoS does NOT deal with the shape of the potential, only the net influence of the a) attraction and b) hard core repulsion. There is a temperature at which these two effects cancel out, (see plot in virial EoS section.) This temperature is called the Boyle temperature T and has the analytic value of T = T % with % and & (c) The assumption is that the density must be uniform! This is clear as there is a (singular) molar volume. This is not the case when two phases coexist! (d) The vdw loop is composed of two parts. 5

2 One of the very nice features of the virial expansion is that the coeficients are derivable from the intermolecular potential V(r). For example, the coefficient B 2 (as can be shown in a proof suitable for a graduate class) is, 2 ( )= 2 R 2 ( () 1) i. The metastable branches, connecting the stable branches to the extremum. These branches can be studied for a finite period of time, as any metastable condition can, if thesystemisnotjolted(overthebarrier.) ii. The unstable branch with isothermal compressibility ( 1 ( ) = 1 ( ) where = 1). This branch does not exist in the real world. It is only a result of an analytic form of the vdw EOS. MOTHER NATURE KNOWS BET- TER! Exercise Show that NATURE s equal area construction connects the phases with equal Gibbs free energy per mole, =. (In the thought process, it is very helpful to draw a plot of ( ), draw vertical lines [horizontal in P(v)], and think about the areas under the curve.) 2. Other analytic EoS s (a) Berthelot: ( + )( ) = 2 or = [ (1 6( )2 )] (b) Redlich-Kwong: ( + 5 (+) )( ) = 3. Expansion EoS Virial: = + 2 ( ) ( ) 3 The first correction term varies from to + as the temperature increases. At low the attractive part of the intermolecular potential ( in vdw) is more important while at high the excluded volume or hard core ( in vdw) is more important. The temperature at which the two effects cancel (in ) is called the Boyle temperature. At both the attractive and hardcore features are important, they just happen to cancel leaving an approximately ideal EoS. Virial coefficient 2 ( ) Lim 2 =4 (4 times the hard-core molar volume.) Critical points All phase diagrams are characterized by ONE critical point and one or more triple points. Triple points (for one component, C = 1 systems) are fixedpointsinnature. (You will see why when we discuss the Gibbs phase rule, for now appreciate that with C = 1 and three phases in equilibrium, nature picks all the thermodynamics variables, you have no control!) The critical point is the terminus of the l-v equilibrium line. (There is no terminus to the s-l line.) This critical point is characterized by ( ) =and( 2 ) 2 = The first relations implies that that the isothermal compressibility is infinite (i.e. the system is infinitely compressible at T.) Approaching this point from below, on the l-v equilibrium line, the distinction between l and v disappears (.) Corresponding states The fact that simple analytic EoS, like the vdw equation, can explain the behavior of so many fluids, suggests that there is a greater truth. This truth is even more strongly suggested by creating a reduced ( =1) diagram for many fluids. (By fluids I mean both the gas and liquid phases.) By dividing the actual temperature and pressure by the values at the critical point, the corresponding states figure shown is obtained. The reason that all these substances look the same, in this reduced units plot, is that all the interactions look the same when plotted in reduced units, i.e. the energy scaled by T and the distance scaled by some critical molecular dimension. Allthispointsout,isthattheEOS(foraC =1system) is a function of two variables. 51

3 division by 1 bar.) At low pressure (where all gases become ideal) the fugacity becomes the pressure, lim =1 2. Thus for a pure material, = ( 2 ) ( 1 )= ln 2 1 andifthelowerpressureisthereference = = ln The EoS s of most fluids is idential if represented in reduced units. To get appropriate reduced units, the and (or ) are scaled by the critical point values. From another point of view, as there are only two parameters in the vdw s equation, it is not surprising that the values of two values that define the critical point, T and (or its inverse ) are sufficient to define a fairly universal EoS. The critical paramenters for the vdw EOS are = 27 2 = 3 = 8 27 Fig. potential scaling to universal form 3. The trends in, or the fugacity coefficient can be displayed on a universal figure and simply explained. = 1 ideal gas 1 decreased escaping tendency 1 increased escaping tendency 4. Exanding the CP + ln + ln = ref + ideal scaling + real corrections. See Fig. 7.9 pg What might reduce the escaping tendency? Consider a case where the intermolecular attraction is large (vdw term more important than the term.) The generalized force to leave all a molecule s friends behind and go into the gas phase is reduced. On the other hand, the hard-core part of the interaction enhances the tendency to escape from the condensed phase. 7.2 Fugacity of a RG G.N. Lewis wanted to preserve the functional forms appropriate for an IG when dealing with a RG. To do so, he defined a variable, a pressure-like -quantity, such that if we use this new quantity in the functional form appropriate for we get the correct generalized chemical force. 1. Following G.N., we DEFINE the fugacity 1 by, + ln = + ln (7.1) (Remember, arguments of ln and exp are unitless, when I write a form like the last, I am implying 1 Fugacity is from the latin fugere meaning to escape. Fugacity coefficient as a function of reduced pressure and temperature 6. The rate of change of with pressure, ( ) = For a vdw gas, = ( + 2 ) + 52

4 (a) The affect of is to reduce, the increment function. The real growth of with P is slower than the IG, i.e. we need a correction factor multiplying inside the ln that is 1 (b) The affect of is to increase v. Thus (P) grows faster than the IG ln(p) dependence. 7.3 Compressibility Fugacity The simpliest way to generate an approximate fugacity is from a plot like the one above ( ) To do this all one needs to know are the coordinates of the triple point Gross Compressibilities A family of universal curves, like the plot of ( ) can be drawn from the readily measurable gross compressibility ( ), = =1 (7.2) Such a plot again provides a verification of corresponding states and the trends are similarly explained. (That is, the vdw a dominates at low and thus 1 and 1atbothhigh and due to an increased importance of the vdw b term. See Fig. 7.5, pg 161.) f(p) or (Z) Both the gross and the differential compressibilities can be used to generate the fugacity coefficient. Lets run through the former. 1. We know both ( ) = and = + ln( ). 2. Consider both an real and an ideal gas: (a) = = + ln( ) (b) = = + ln( ) 3. Taking the difference and integrating from some low pressure 1 (= ) to some target 2 { } [ ]=( ) R 2 = 1 { = } [ ]= R 2 1 () { ( 2 ) ( 1 )} [ ( 2 ) ( 1 )] = R 2 1 () 4. As the real gas is ideal at low { ( 2 )} [ ( 2 )] = R 2 1 () 5. Plugging in for the and appreciating that the references are the same, ln 2 ln 2 = R 2 1 ( ) 6. Multiplying the RHS by and employing Z, R ln = 1 ( 1) 7. As = ln =ln = Z 1 (7.3) All you need to do to get the fugacity coefficient ( )isintegratethedifference of the compressibility from 1 (ideal) weighted by 1 If the attractive interactions dominate, the escaping tendency will be reduced, making = 1. If the hard core is more important (always the case if the target pressure is high enough), 1and the gas will have an enhanced escaping tendency. The fugacity coefficient and the activity coefficient (used for solutions) have the same physical content. I will make you show that ( ln ) = 1 and ( ln ) = 1 Therefore, using the later, one can relate the isothermal compressibility to the fugacity (with the gross compressibility as an intermediate result.) f(t) or ( ln ) What about the temperature dependence of the fugacity? First the logic. The attractive features of the potential become less and less important at high (You have to slow down to enjoy the attraction. Go too fast and all you do is bump into walls.) OK nice words - but WE WANT NUMBERS, numbers, numbers! The crowd is screaming for numbers. OK, to get numbers, we need equations. To get ( )wehavetofirst get a different form of the Gibbs-Helmholtz equation. Consider an isothermal process converting a mole of gas from low pressure (where = ) to some high target (where 6= ) 1. Thechangein is, = = ln = ln 2. The temperature dependence is captured by a form of the Gibbs-Helmholtz equation: the dependence with at fixed of [ ( ) ] [ ( ) ] = {( ln ) ( ln ) } = {( ln ) ( ln ) } 53

5 3. Now at very low pressure the fugacity equals the pressure and therefore the fugacity or ln does not change with anything at fixed pressure. Therefore, + 2 = {( ln 2 ) } or ( ln ) = 1 ( 2 ) 4. A result that can be integrated from one temperature ( 1 )toanother( 2 )toget, R ln =ln ( 2) ( = R 2 1) 1 1 ( 2 ) Recall that the quantity ( )isthedifference between the enthalpy at the target pressure to that at low pressure. 4. Integrating we find ln( ) = R ( 1 ) 5. Which leads to a fugacity coefficientattargetpressure P of ln = R ( 1 ) 5. Fortunately, we can get the ()[= (how changes with ) ] inside the integral above from two of our standard experiments. Considering H(T,P): 1 =( ) ( ) ( ). Indentifying ( ) = ( ) ( ) = 1 ( ) Therefore, measurement of a) ( ) and b) allows for the extraction of ( ) This partial can be integrated to get the difference needed to get the temperature dependence of R ( ) = 7.4 Mixtures of RGs You need to get for a pure gas, while one needs a quantity ( ) to get in a mixture. The differential is an example of a partial-molar quantity, in this case a partial molar volume 1. While for a pure material ( ) = 2. for a mixture ( ) = and = + ln As the first term is a constant ( ln ) = 1 ( ) = 3. We can insert the partial P (to ultimately extract = ) ( ln( ) ) = 1 (Do you see why one can use the total P on the RHS?) 54