Fugacity, Activity, and Standard States

Transcription

1 Fugacity, Activity, and Standard States Fugacity of gases: Since dg = VdP SdT, for an isothermal rocess, we have,g = 1 Vd. For ideal gas, we can substitute for V and obtain,g = nrt ln 1, or with reference to a standard ressure,, we may write G G = nrt ln,or G m = G m + RT ln. Now, consider a real gas that obeys the equation of state which would give ( + a)(v m b)=rt, G m = G m + RT ln + a + a b( ). More comlicated equations of state will give rise to more comlicated exressions for G. In order to reserve the simlicity of the exression for G even for real gases, the concet of fugacity, f, was introduced by G.N. Lewis so that G m = G m + RT ln f. The defining equation for fugacity is d = RTd ln f with the condition that Lim = 1. 1

2 A working exression to calculate fugacity: For a constant temerature rocess, dg m = V m d, or,g m = G,m G 1,m = 1 V m d Let us add and subtract RT/ to the right hand side:,g m = RT 1 P + V m RT d = RT ln P 1 + P1 V m RT d. Now, by definition, for all gases, G m = Rtln(f /f 1 ). Therefore, we get RT ln f / P = f 1 / 1 P1 V m RT d. We now set 1 = 0 (recall that f = as 0), f = f and =, where is the ressure of interest, to get RT ln f = 0 V m RT d. The right hand side can be evaluated if V m can be conveniently exressed as a function of R, T and. However, this is often very difficult (consider the van der Waals equation of state, for examle). An alternative is to exress V m in terms of the comressibility factor Z, to get RT ln f = ZRT 0 ln f = P Z 1 0 V m = ZRT/, RT d, or d. The last exression can be conventiently and fairly accurately evaluated in most cases using Virial exansions.

3 Fugacity of Pure Condensed Phases The defining equation for fugacity of ure condensed hases is, like that of gases, with the condition that d c = RTd ln f c Lim f = 1, where µ c is the chemical otential of the condensed hase. Alying the criterion for hase equilibrium, dµ c = dµ g (where g stands for gas hase), we get RTd ln f c = RTd ln f g, from which we conclude that f c = kf g. Now, in theory, any solid or liquid taken to zero ressure would eventually become a gas for which f =, so that k = 1. The conclusion is that fugacity of a condensed hase is the same as the fugacity of the gas hase in equilibrium with it. Variation of fugacity with ressure The above statement (italics) and the relationshi ln f = RT 1 T G m = V m RT, T is sufficient to derive the Gibbs equation ln f = V m f 1 RT 1, where the left hand side refers to vaor fugacities (which could be relaced with vaor ressures) while the right hand side refers to total ressures acting on the liquid surface. Therefore, this equation rovides a way to calculate the variation of vaor ressures with changes in the total ressure. 3

4 Activity The defining equation for fugacity of the i-th comonent in a mixture is d i = RTd ln f i. This equation can be integrated to give i ln f d = RT i ln fi d ln f i i = i + RT ln f i f i, where and i f i are the chemical otential and fugacity in a reference or standard state. The ratio f i /f i is defined as the activity a i, so that the above equation becomes = i + RT ln a i. Standard States The standard state for a gas The standard state for a gas is taken to be the ideal gas at 1 bar (100 kpa or Ma) ressure. For such a gas, we have f = = 1 bar. For a real gas, in general, f 1 bar when = 1 bar but, with the above definition, we can define activity for a gas as a i = f i /, and set = 1 bar. Standard states for ure solids and liquids The ure solid or liquid under 1 bar external ressure is taken to be the standard state. With this definition, we have f = f (where f* is the fugacity at = 1 bar). Activity is still defined as a = f f = f f (1 bar), which means that for a ure substance at standard ressure, a = 1. Standard state of a solvent in a solution The convention is the same as above, with a i = f i /f i (1 bar). Raoult s law (to be covered later) states that for a volatile comonent of a solution, f i = x i f i. Therefore, we get a i = x i for solutions that obey Raoult s law. 4

5 In general, we may write a i =? i x i where γ i is called the activity coefficient. For solutions that obey Raoult s law, obviously, γ i = 1. Sometimes, the activity coefficient defined here is given the subscrit R to indicate that this refers to a Raoult s law standard state. Standard state of solute in a solution The standard state of a solute in a solution is defined as its activity in a solution whose concentration is 1 molal, i.e., 1 mole of solute in 1 kg of solvent. In general, a i =? H,i m with the condition that? 0. 5