The Chemical Potential of Components of Solutions

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1 CHEM 331 Physical Chemistry Fall 2017 The Chemical Potential of Components of Solutions We have shown that the Gibbs Free Energy for a solution at constant T and P can be determined from the chemical potential of each component of the solution via Additivity. G = And the 2 nd Law tells us that G 0 for systems under constant T and P constraints. Thus it is critical to be able to determine the chemical potential for each species in a solution in order to determine the equilibrium position ( G = 0) of the system. In particular, we want to know how the chemical potential depends on the composition of the solution? We will consider two cases, gaseous solutions and condensed phase (liquid or solid) solutions. Gaseous Solutions Gaseous solutions are the easiest to handle; mostly because the distance between the particles of the gas is large and hence the particles have limited intermolecular interactions. Of course we are talking about solutions that are of reasonably low pressure. If the gas is behaving Ideally, then we can simply modify our results for the pure gas: (T,P) = o (T) + RT ln All we need to do is replace the total pressure with the partial pressure P i. i(t,p,x i ) = i o (T) + RT ln By Dalton' Law, we have P i = x i P; where x i is the mole fraction of species i in the mixture. Hence,

2 i(t,p,x i ) = i o (T) + RT ln And now we have it; this gives us the dependence of i(t,p,x i ) on the solution composition x i. Of course, if the gas is non-ideal, then we must instead modify our equation involving the fugacity. If the gas is pure, we have: (T,P) = o (T) + RT ln So, it is natural to write the chemical potential for the gaseous solution as: i(t,p,x i ) = i o (T) + RT ln where f i is the fugacity of the i th component in the solution. As before, we need to be able to measure f i or this form for i(t,p,x i ) will not work. The extension from the pure case is obvious. ln = This does it. We can now determine i(t,p,x i ) for both Ideal and Real gases. Condensed Phase Solutions We now turn to the more difficult situation where the molecules are packed together closely and the intermolecular forces dominant the behavior of the solution. This might be a solution of Ethanol and Water (Alcoholic Beverage), a liquid, or Copper and Zinc (Brass), a solid. Again, we are trying to determine how i(t,p,x i ) depends on x i.

3 As a starting point, we will not directly attack the above problem, but will instead work with the case where the solution is volatile at the specified T and P. So we have a gas of composition y 1, y 2, in equilibrium with a condensed phase solution of composition x 1, x 2, (We use y i to denote the mole fraction of i in the vapor phase, allowing us to distinguish it from x i, the mole fraction in the condensed phase.) The vapor has a pressure P, which, assuming it is reasonably Ideal, a restriction that is not too severe for most practical cases, by Dalton's Law: P = Now to the crux of things; Raoult's Law will be used to determine P i. P i = x i P i *

4 where P i * represents the vapor pressure of pure component i. For a binary system: P 1 = x 1 P 1 * P 2 = x 2 P 2 * = (1 - x 1 ) P 2 * P = P 1 + P 2 = x 1 P 1 * + x 2 P 2 * = x 1 P 1 * + (1 - x 1 ) P 2 * = P 2 * + (P 1 * - P 2 * ) x 1 We see that each is linear in x 1. Raoult's Law works particularly well if the components are chemically similar; have similar intermolecular forces between unlike molecules as between like molecules. Solutions of Benzene and Toluene provide for good examples of solutions that follow Raoult's Law.

5 We now recall that the chemical potential of each species must be equal across a phase boundary. Thus, i,gas(t,p i,y i ) = i,liq(t,p i,x i ) i,gas(t,p i,y i ) = i o (T) + RT ln P i (Ideal Gas) = i o (T) + RT ln x i P * i (Raoult's Law) = i o * (T) + RT ln x i + RT ln P i This means: i,liq(t,p i,x i ) = i o * (T) + RT ln x i + RT ln P i If the condensed phase were pure: i,gas * (T,P * i ) = i,liq * (T,P * i ) This means: i,gas * (T,P i * ) = i o (T) + RT ln P i * Now we zero in on the result: i,liq * (T,P i * ) = i o (T) + RT ln P i * i,liq(t,p i,x i ) - i,liq * (T,P i * ) = RT ln x i To nail down the details, we must recall that G(T,P) for a condensed phase has almost no pressure dependence. G(T,P) = G o (T) + V G o (T) So, i,liq * (T,P i * ) i,liq * (T,P i ) Therefore: i,liq(t,p i,x i ) - i,liq * (T,P i ) = RT ln x i

6 This gives us the desired result: i,liq(t,p i,x i ) = i,liq * (T,P i ) + RT ln x i Simplifying the notation a bit, we write this as: i = i * + RT ln x i We must note that i,liq * (T,P i * ) is a Reference State for i in the liquid. It is not the Standard State; i o (T,P o ). Formal definitions of any Reference States we will be working with are provided for in the Appendix. Now what if the solution does not follow Raoult's Law? This is usually the case if the components of the solution are somewhat dissimilar. Two examples; Chloroform-Acetone solutions exhibit Negative Deviations from Raoult's Law and Methylal-Carbon Disulfide solutions exhibit Positive Deviations. It should be noted that P i for each component approaches the Raoult's Law prediction as the solution becomes pure in that component. Therefore, Raoult's Law is a limiting law, much like the Ideal Gas Law is a limiting law. P i = However, at the other end of the concentration scale, we see that P i begins to follow a different line. This is the Henry's Law line.

7 P i = Again, Henry's Law is a limiting law. K i is the Henry's Law Constant. If component i follows Henry's Law across the entire concentration range, then P i = K i when x i = 1. Think of this as a hypothetical Ideal-Dilute State. The chemical potential for this Reference State is denoted i **.

8 For real systems, the solvent will tend to follow Raoult's Law at the same time that the solute follows Henry's Law. In this dilute limit we will denote the solute as component j and the solvent as component i. This will help remind us as to which concentration limit we are working in. (It should be noted that this notational change is not universal.) Now, how does this help us to determine the form of j,liq(t,p j,x j ) when the component is dilute and following Henry's Law rather than Raoult's Law? We follow the same line of argumentation as before and start with the equality of j across the phase boundary. Thus, j,gas(t,p j,y j ) = j,liq(t,p j,x j ) j,gas(t,p j,y j ) = j o (T) + RT ln P j (Ideal Gas)

9 = j o (T) + RT ln x j K j (Henry's Law) = j o (T) + RT ln x j + RT ln K j This means: j,liq(t,p j,x j ) = j o (T) + RT ln x j + RT ln K j If the condensed phase were pure, but behaving as though it were Infinitely Dilute: j,gas ** (T,K j ) = j,liq ** (T,K j ) This means: j,gas ** (T,K j ) = j o (T) + RT ln K j j,liq ** (T,K j ) = j o (T) + RT ln K j Now to the result: j,liq(t,p j,x j ) - j,liq ** (T,K j ) = RT ln x j As before we assume j is relatively insensitive to small pressure changes: Therefore: j,liq ** (T,K j ) i,liq ** (T,P i ) j,liq(t,p j,x j ) - j,liq ** (T,P j ) = RT ln x j This gives us the desired result: j,liq(t,p j,x j ) = j,liq ** (T,P j ) + RT ln x j Again, simplifying: j = j ** + RT ln x j Summarizing, we now have an ability to determine how the chemical potential of species i or j, in a condensed phase solution, depends on its concentration. i = i * + RT ln x i (Ideal Solution)

10 j = j ** + RT ln x j (Ideal-Dilute Solution) It must be kept in mind that we have only considered condensed phase solutions which are in equilibrium with a vapor phase. This means that the Reference States i * and j ** are defined in terms of the vapor phase parameters P i * and K j. i * = i o + RT ln P i * j ** = j o + RT ln K j * (Ideal Solution) (Ideal-Dilute Solution) It is also important to keep in mind, the equations for i and j work over the entire concentration range, from x i = 0 to 1. In the former case, i governs over the whole concentration range so long as the solution is Ideal. In the later case, j works from x j = 0 to 1 if the solution is Ideal- Dilute. However, if the solution is Real, then we can only expect that i will be valid if i is acting as the solvent and x i 1. If we are dealing with the solute and the solution is Real, then j will only work if the solution is dilute; x j 0. Finally, in most practical cases the concentration of the solute is not reported in terms of its mole fraction x j, but instead in terms of its molality (m j ) or its molarity (c j ). Therefore, we should modify j such that its concentration dependence is given in m j or c j. We start by considering the modification to use m j for solute concentrations and follow an argument put forth by Castellan. m j = = where N is the number of moles solvent and M is the solvent's molar mass. We can now use this definition in our expression for the mole fraction of the solute: x j = = = M m j as m j 0 If we define the Standard State for molal based solutions to be Ideal-Dilute with m o = 1 molal, then: x j = So, now we can write the chemical potential in terms of m j : j,liq(t,p j,x j ) = j ** + RT ln x j = j ** + RT ln We now define a new Reference State as: = = j ** + RT ln + RT ln

11 j *** = j ** + RT ln This then allows us to write the chemical potential in the desired form: j = j *** + RT ln Something similar can be done to obtain a form for the chemical potential based on the solute's molarity. j = + RT ln We denote c j the solution's molarity as c j and take c o to be the Standard State molarity, set typically as 1M. In this case, our new Reference State is defined as: = j *** + RT ln being the solution density. Again summarizing, we now have four different forms for the chemical potential for a component of a condensed phase solution, each keyed off a different Reference State. i = i * + RT ln x i (Ideal Solution) j = j ** + RT ln x j (Ideal-Dilute Solution) j = j *** + RT ln (Ideal-Dilute Solution) j = + RT ln (Ideal-Dilute Solution) How does this address the problem we set out to solve? What is i,liq(t,p,x i ) for component i of a condensed phase solution at T and P, regardless of whether or not the solution is in equilibrium with its vapor.

12 The idea is that if the solution is Ideal or Ideal-Dilute, then we will carry over the previous forms for j to the current case. i = i * + RT ln x i (Ideal Solution) j = j ** + RT ln x j (Ideal-Dilute Solution) j = j *** + RT ln (Ideal-Dilute Solution) j = + RT ln (Ideal-Dilute Solution) Now, it looks as though the equations are the same as those above for our systems which have established a liquid-vapor equilibrium. Unfortunately this is not case. The Reference States for the case where we have a liquid-vapor equilibrium i *, j **, j ***, and are all defined in terms of vapor phase parameters P *, and K j. In the present case, we need definitions that are more general and that do not have a vapor phase dependence. This problem is handled quite naturally. For instance, we now define j ** as: j**(t,p) = How this plays out for a real system is outlined in the Appendix below. The full set of Reference State definitions is also provided in the Appendix below. And with this, we have partially answered the question we set out to address. Next we turn to the case where the solution is not Ideal or Ideal-Dilute, but is instead Real. This will then complete our discussion and give us a full set of 's that can be used to determine G.

13 Appendix - Reference State Definitions Ideal Reference State i*(t,p) = = Molar Gibbs Free Energy of the Pure Liquid/Solid Ideal-Dilute Reference Stat j**(t,p) = = Hypothetical Molar Gibbs Free Energy of Pure Liquid/Solid Behaving as if Perfectly Dilute Ideal-Dilute (molality) Reference State j***(t,p) = = Hypothetical Molar Gibbs Free Energy of Pure Liquid/Solid Behaving as if Perfectly Dilute at m o Ideal-Dilute (molarity) Reference State (T,P) = = Hypothetical Molar Gibbs Free Energy of Pure Liquid/Solid Behaving as if Perfectly Dilute at c o Notes: 1) Reference States are not Standard States. For example, the Ideal Reference State is the Pure Liquid/Solid at T and P: i*(t,p) = Reference State The Standard State is defined as the Molar Gibbs Free Energy of the Pure Liquid/Solid at T and P o : i o (T,P o ) = Standard State These are related as: i*(t,p) = i o (T,P o ) + For most cases, because the molar volume of a liquid or solid is small and the pressure difference is small, the integral term is negligible. i*(t,p) ~ i o (T,P o )

14 However, at high pressures, or when great accuracy is required, the integral term must be included. 2) A pictorial representation of the Ideal-Dilute Reference State: 3) Determination of AA** for Acetic Acid in a Toluene solvent at 69.9 o C. Vapor Pressure data is used to calculate AA - AA*. x AA P AA [Torr] AA - AA* [kj/mol] - RT ln x AA [kj/mol]

15 The sum of the last two columns is plotted and extrapolated to zero Mole Fraction. This is then: AA** - AA* = Determiniation of Ref. State for Acetic Acid in Toluene y = 3.744x x Mole Fraction Acetic Acid mu - mu* RTln(x AA) Sum Poly. (Sum) Poly. (Sum) For This Case: AA** - AA* = 3.14 kj/mol Reference States for a Volatile System Ideal Reference State i*(t,p) = i o (T) + RT ln Ideal-Dilute Reference State j**(t,p) = j o (T) + RT ln Ideal-Dilute (molality) Reference State j***(t,p) = j o (T) + RT ln + RT ln(m m o ) Ideal-Dilute (molarity) Reference State (T,P) = j o (T) + RT ln + RT ln where = c o [mol/m 3 ]

16 Notes: 1) For the Ideal Reference State, it is assumed the Total Pressure over the Liquid/Solid is P. However, this is usually not the case. It is instead typically P i *. Because the change in Gibbs Free Energy is negligible for Pressure changes for condensed phases, we have: i*(t,p) ~ i*(t,p i *) However, i*(t,p) is the preferred reference state. 2) For the Ideal-Dilute References States, it is assumed the Total Pressure over the Solution when K j, the Henry s Law Constant, is determined is P. However, this is usually not the case. Hence P is not well defined. Because the change in Gibbs Free Energy is negligible for Pressure changes for condensed phases, this is usually not of concern. 3) The difference between (T,P) and j***(t,p) is not very large for Aqueous systems at Room Temperature. For these systems: So, = 10 3 mol/m 3 m o = 1 mol/kg = kg/m 3 (T,P) - j***(t,p) = RT ln = (8.314)(298.15) ln = J/mol 4) A pictorial representation of the Reference States for volatile systems.

17 Appendix - An Application of the Gibbs-Duhem Equation Here we take a short excursion and apply the Gibbs-Duhem Equation to one of our results from above. We restrict ourselves to the case where the condensed phase solution is in equilibrium with its vapor and that the solvent obeys Raoult's Law over the entire concentration range. We will further restrict ourselves to binary systems and denote the solvent as component "1" and the solute as "2". The chemical potential of the solvent is given by: 1 = 1 * + RT ln x 1 Its differential is then: d 1 = By the Gibbs-Duhem Eq.: d 2 = - = - = - Since x 1 + x 2 = 1, dx 1 + dx 2 = 0. This means: d 2 =

18 Now we integrate: Or, 2 = RT ln x 2 + C where C is a constant of the integration. C can be determined by recognizing that if x 2 1, then 2 2 *. Therefore C = 2 *, giving us: 2 = 2 * + RT ln x 2 This means that the solute also follows Raoult's Law over the entire concentration range! In other words, if the solvent behaves Ideally over the entire concentration range, then so must the solute. Other results which can be obtained by similar arguments are: Powerful stuff. 1. If the solvent shows positive deviations from Raoult's Law, then so does the solute. 2. If the solvent shows negative deviations from Raoult's Law, the so does the solute.

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