Simple Mixtures. Chapter 7 of Atkins: Section

Transcription

1 Simple Mixtures Chapter 7 of Atkins: Section Colligative Properties Boiling point elevation Freezing point depression Solubility Osmotic Pressure Activities Solvent Activity Solute Activity Regular Solutions Prof. Mueller Chemistry Fall 2003 Lecture 17-1

2 Osmosis There are many times in nature when a solvent will pass spontaneously through a semipermeable membrane, which is a membrane permeable to solvent, but not solute The osmotic pressure, Π, is the pressure that must be applied to stop the influx of solvent - one of the most important examples is the transport of fluids through living cell membranes - also the basis of osmometry, the determination of molecular mass by measurement of osmotic pressure To treat osmosis thermochemically, we note that at equilibrium, chemical potential on each side of the membrane must be equal This equality implies that for dilute solutions the osmotic pressure is given by the van t Hoff equation Π = [ B]RT where [B] = n B /V is the molar concentration of the solute. Prof. Mueller Chemistry Fall 2003 Lecture 17-2

3 Osmosis, 2 On the right, the opposing pressure arises from the head of the solution that osmosis produces. Equilibrium is established when the hydrostatic pressure of the solution in the column is equal to the osmotic pressure Prof. Mueller Chemistry Fall 2003 Lecture 17-3

4 Osmosis, 3 Equilibrium is established when the hydrostatic pressure of the solution in the column is equal to the osmotic pressure. Pure solvent side: µ A* (p) at pressure p Solution side: x A <1, so µ A* is lowered by solute to µ A. However, µ A is now increased by greater pressure, p + II µ A Account for the presence of solute µ A ( x A, p + Π)= µ A Take pressure into account: µ A Combining equations, we have ( p)= µ A x A, p + Π ( p +Π)= µ A ( ) ( p + Π)+ RT ln x A ()+ p V m dp Dilute solutions: lnx A replaced by ln(1-x B ) -x B and V m constant: Prof. Mueller Chemistry Fall 2003 Lecture 17-4 p RT ln x A = RTx B = ΠV m p +Π p p +Π V m dp

5 Osmometry Osmotic pressure is easily measured, and is quite large. Osmometry can be applied for the determination of molecular weights of large molecules (proteins, synthetic polymers), which dissolve to produce less than ideal solutions. The Van t Hoff equation can be rewritten in virial form: II = [B]RT {1 + B [B] +...} where B is the empirically determined osmotic virial coefficient Osmotic pressure is measured at various concentrations, c, and plotting II/c against c is used to determine the molar mass of B Consider the example of poly (vinyl chloride) PVC, in cyclohexanone at 298 K Pressures are expressed in terms of heights of solution, D=0.980 g cm -3 in balance with the osmotic pressure Prof. Mueller Chemistry Fall 2003 Lecture 17-5

6 Osmometry, 2 c(g L -1 ) h(cm) Use II = [B]RT {1 + B [B] + } with [B] = c/m, where c is the mass concentration and M is the molar mass. The osmotic pressure is related to the hydrostatic pressure by II = ρgh, where g = 9.81 m s -2. Then: h c = RT ρgm Bc 1+ M + L = RT ρgm + RTB L ρgm 2 c + Plot h/c vs. c to find M, expecting a straight line with intercept RT/ ρgm at c = 0. Data set: c(g L -1 ) h/c(cm g -1 L) Prof. Mueller Chemistry Fall 2003 Lecture 17-6

7 Osmometry, 2 The data give an intercept of 0.21 cm g -1 L Thus: Prof. Mueller Chemistry Fall 2003 Lecture 17-7

8 Osmometry, 3 The data give an intercept of 0.21 cm g -1 L, which is equal to RT/ ρgm Thus: M = RT ρg cm g 1 L ( ) 298 K ( ) 1 = J K-1 mol -1 ( 980 kg m -3 ) 9.81 m s -2 = kg mol m 4 kg 1 Prof. Mueller Chemistry Fall 2003 Lecture 17-8

9 Activities Just like real gases have fugacities to describe deviations from perfect gas behavior, components of real solutions have activities. Fugacity: µ = µ 0 + RT ln f p 0 = µ 0 + RT ln p p 0 + RT lnφ Now for solutions, chemical potential of component A as described so far: If ideal: µ A = µ A + RT ln p A p A µ A = µ A + RT ln x A Activity is a kind of effective mole fraction, like fugacity is an effective pressure Prof. Mueller Chemistry Fall 2003 Lecture 17-9

10 Solvent Activities All solvents obey Raoult s law, (p A /p A * = x A ) as the concentration of the solute approaches zero µ A = µ A + RT ln x A The standard state of the solution is the pure liquid at 1 bar and is obtained when x A =1. Just like with fugacity, we would like to preserve the form of the ideal equation and thus we define an activity a A such that: µ A = µ A + RT ln a A We can define an activity coefficient a A then by comparing the two equations for chemical potential, finding a A = p A p A Prof. Mueller Chemistry Fall 2003 Lecture 17-10

11 Solvent Activities The activity of the solvent approaches the mole fraction as x A 1. a A x A as x A 1 Just as for real gases we used the fugacity coefficient, φ, we can introduce the activity coefficient: a A = γ A x A and γ A 1 as x A 1 The chemical potential of a real solvent is then simply: µ A = µ A + RT ln x A + RT lnγ A The standard state of the solvent is obtained when x A = 1. Prof. Mueller Chemistry Fall 2003 Lecture 17-11

12 Solute Activities Solute activities must be defined keeping in mind that ideal-dilute (Henry s Law) behavior is obtained not when x B approaches 1 (pure solute), but rather as x B approaches 0 (no solute!). Ideal dilute solutions: if the solute follows Henry s law behavior then and: µ B = µ B + RT ln p B = µ B + RT ln K B + RT ln x B p B p B Combine the second term with the first to give a NEW standard chemical potential (since K B and p B * are solute dependent only): and then we can comfortably write: p B = K B x B µ 0 B = µ B + RT ln K B p B µ B = µ B 0 + RT ln x B Prof. Mueller Chemistry Fall 2003 Lecture 17-12

13 Solute Activities So now for deviations from ideal-dilute solutes: We now write, analogously to the real solvent case, the chemical potential in terms of an activity (replacing the mole fraction): but now: µ B = µ B 0 + RT lna B a B = p B K B We can also define yet again an activity coefficient for the solute: a B = γ B x B and γ B 1 as x B 0 The chemical potential is then: µ B = µ B 0 + RT ln x B + RT lnγ B Prof. Mueller Chemistry Fall 2003 Lecture 17-13

14 Solute Activities - Units In the chemical world, we often speak of compositions not in terms of mole fractions, but in terms of molalities (b) in units of mol kg -1. Fortunately, we are free to choose our standard state in order to take this into account and simplify the calculations and expressions in future chapters. Now write: But now the standard chemical potential is DIFFERENT than earlier. Here the chemical potential has its standard value when the molality is equal to a standard molality b o = 1 mol kg -1. What happens as b approaches 0? µ B = µ B 0 + RT lnb B What does such a chemical potential mean in practical situations? Prof. Mueller Chemistry Fall 2003 Lecture 17-14

15 Solute Activities - Units Introducing non-ideality in these units is done as follows: Activity (a) and activity coefficient (γ) are now defined by at all T and p. a B = γ B b B b 0 where γ B 1 as b B 0 There is also a biological standard state (see section 7.7(c)) where a standard state is defined in more biologically relevant conditions - where ph = 7 rather than ph = 0!). Prof. Mueller Chemistry Fall 2003 Lecture 17-15

16 Summary of Activities Prof. Mueller Chemistry Fall 2003 Lecture 17-16

17 Regular Solutions Thermodynamic properties of real solutions are described with excess functions, X E For example, the excess entropy, S E, is the difference between the observed thermodynamic function and the function for an ideal solution: S E = mix S mix S ideal So, the further excess energies are from zero indicates the degree to which a solution is non- ideal a regular solution has H E 0, S E =0, with two kinds of molecules distributed randomly (like ideal solution), but having different interactions with one another. A model function for excess enthalpy might look like: H E = nβrtx A x B = nwx A x B Where w = βrt is an indication of the energy of A-B interactions relative to A-A or B-B interactions Prof. Mueller Chemistry Fall 2003 Lecture 17-17

18 Regular Solutions For different values of β: H E = nβrtx A x B = nwx A x B endothermic exothermic Prof. Mueller Chemistry Fall 2003 Lecture 17-18

19 Regular Solutions G E = H E mix G = nrt( x A ln x A + x B ln x B + βx A x B ) For different values of β: For β > 2 get spontaneous phase separation (two minima)! Prof. Mueller Chemistry Fall 2003 Lecture 17-19

20 Activities of Regular Solutions mix G = nrt( x A ln x A + x B ln x B + βx A x B ) What does this equation imply about the activity coefficients for the two components? mix G = nrt x A lna A + x B ln a B These two equations for mix G are equal if: ( ) ( ) = nrt x A ln x A + x B ln x B + x A lnγ A + x B lnγ B lnγ A = βx B 2 and lnγ B = βx A 2 These equations are called the Margules equations. Then: ( ( ) 2 ) a A = γ A x A = x A exp( 2 βx B )= x A exp β 1 x A Prof. Mueller Chemistry Fall 2003 Lecture 17-20

21 Activities of Regular Solutions Going back now to a previous definition of the activity of A: β = 0, ideal solution Raoult s Law behavior β > 0, endothermic mixing unfavorable A-B interactions higher vapor pressure β < 0, endothermic mixing favorable A-B interactions lower vapor pressure ( ( ) 2 ) a A = γ A x A = x A exp( 2 βx B )= x A exp β 1 x A ( ) p A p A = a A p A = x A exp β( 1 x A ) 2 Prof. Mueller Chemistry Fall 2003 Lecture 17-21