Chapter 5: The Thermodynamic Description of Mixtures
• Partial molar quantities • Volume • Gibbs Energy (Chemical Potential) • Gibbs-Duhem Equation
• Thermodynamics of Mixing • Henry’s Law • Raoult’s Law
PHYSICAL CHEMISTRY: QUANTA, MATTER, AND CHANGE 2E| PETER ATKINS| JULIO DE PAULA | RONALD FRIEDMAN ©2014 W. H. FREEMAN D COMPANY Partial Molar Volume
• The partial molar volume is something like an effective molar volume,
VJ = (∂V/∂nJ)p,T,n′
and is often denoted by 푉ഥ퐽.
• Partial molar properties of a mixture depend on what else is in the mixture as much as the substance in question.
• For a two-component solution, from slope equation at constant T and P, change in V is
휕푉 휕푉 푑푉 = d푛퐴 + d푛퐵 휕푛퐴 휕푛퐵 푃,푇,푛퐵 푃,푇,푛퐴
= 푉ത퐴푑푛퐴 + 푉ത퐵푑푛퐵 • 푉퐽s are constant at constant concentration
• Can change amount of solution, but not concentration, by increasing moles of components by dn, proportional to the amount already there:
d푛퐴 = 푛퐴푑휆 and d푛퐵 = 푛퐵푑휆
• The change in V becomes: 푑푉 = 푉ത퐴푛퐴푑휆 + 푉ത퐵푛퐵푑휆
• Integration gives V of solution: 푉 = 푉ത퐴푛퐴 + 푉ത퐵푛퐵 Partial Molar Volumes
• Why is partial molar volume different from molar volume?
If a bushel of oranges is mixed with a bushel of
apples, get ~2 bushels of fruit, i.e., DmixV = 0
If a bushel of oranges is mixed with a bushel of
sand, get less than 2 bushels, i.e., DmixV < 0 Partial Molar Quantities
Partial Molar Property: Thermodynamic quantity that indicates how an extensive property of a solution or mixture varies with changes in molar composition at constant T, P. • For a phase of one component, partial molar quantities are identical with molar quantities.
• For an ideal gas or liquid solution, certain partial molar quantities (푉ത푖, 푈ഥ푖 , 퐻ഥ푖) are equal to sum of the respective molar quantities for the pure components, while others (푆푖ҧ , 퐺ҧ푖 , 퐴ҧ푖)--all related to entropy-- are not. • For a non-ideal solution, all partial molar quantities differ from the corresponding molar quantities.
• Differences are of interest because they arise from intermolecular interactions
• Every extensive property has a corresponding partial molar property, but , 퐺ҧ푖, gets a special name and symbol: the chemical potential, .
By the same argument we found the total volume as a function of partial molar volumes, the Gibbs energy of a binary solution is given by 퐺 = 푛퐴휇퐴 + 푛퐵휇퐵 Chemical Potential
With T,P, and n, use slope formula for G:
푑퐺 = 푉푑푃 − 푆푑푇 + 휇퐴푑푛퐴 + 휇퐵푑푛퐵 + ⋯ which is the fundamental equation of chemical thermodynamics!
Chemical potential is the slope of a plot of G vs. nA at constant T,P,n’: 휕퐺 휇 = 퐽 휕푛 퐽 푇,푃,푛′
At constant T and P, slope equation becomes 푑퐺 = 휇퐴푑푛퐴 + 휇퐵푑푛퐵 + ⋯
Recall that at constant T, P Gibb’s energy gives maximum non-PV work: 푑퐺 = 푑푤푛표푛푃푉,푚푎푥. What is Chemical Potential? • shows how G of system changes when a substance added. If > 0, G ↑ as n ↑.
• For a pure material, is just free energy/mole, Gm = G/n.
• (For a mixture ≠Gm)
Not only does show how G changes when n changes, but how U & H change (but under different conditions): G U PV TS U G PV TS
U dU dG PdVVdP TdSSdT At constant V and S, n S,V dG VdP SdT dn
dU PdV TdS dn
In fact, shows how U, H, A, and G depend on composition. For example, can show:
H n S,P • For this reason, is central to chemistry. Potential Displacement Pressure Volume Gravity Height • Can think of contributions to G in terms of pairs of Voltage Charge variables: Chemical potential Moles
• If in system there is a difference in column A, column B will change until the difference is 0.
• Can think of as the potential for moving material.
• If the chemical potential in one region of a system is different from that in another region, material will be transferred until the potential difference = 0. All chemical potentials in a system are equal at equilibrium. Gibbs-Duhem Equation Describes the relationship between changes in chemical potential for components in a mixture.
Since 퐺 = 푛퐴휇퐴 + 푛퐵휇퐵, the total derivative of G is: d퐺 = 푛퐴푑휇퐴 + 푑푛퐴휇퐴 + 푛퐵푑휇퐵 + 푑푛퐵휇퐵
But at constant T and P, 푑퐺 = 휇퐴푑푛퐴 + 휇퐵푑푛퐵 Setting the two equations for dG equal (since G is a state function):
푛퐴푑휇퐴 + 푛퐵푑휇퐵 = 0
This is a specific case of the Gibbs-Duhem Equation:
푛퐽푑휇퐽 = 0 퐽 푛퐴 푑휇퐵 = − 푑휇퐴 푛퐵 The chemical potential of one component in a mixture cannot change independently of the chemical potentials of the other components! True of all partial molar quantities. Used in practice to determine the partial molar volume of one component in a binary mixture when the partial molar volume of another component is known. Thermodynamics of mixing, Gibbs Free Energy 푃 푃 From 퐺 푃 = 퐺휃 + 푅푇푙푛 , 휇 푃 = 휇휃 + 푅푇푙푛 푚 푚 푃휃 푃휃 where 휇휃 is the standard chemical potential (1 bar) • Before mixing (initially at same T & P)
Ginit AnA BnB
n ( RT ln(P / P )) n ( RT ln(P / P )) A A B B • After mixing G n ( RT ln(P / P )) n ( RT ln(P / P )) fin A A A B B B
• Free energy of mixing for an ideal gas:
DGmix nART ln(PA / P) nB RT ln(PB / P)
nRT(xA ln xA xB ln xB )
Because mole fractions (xi) are always ≤ 1, for a perfect gas, ∆mixG < 0, i.e., perfect gases mix spontaneously.
Thermodynamics of mixing, Entropy and Enthalpy • Entropy of mixing:
DGmix Since 휕퐺Τ휕푇 푃,푛 = −푆, then DSmix T
DSmix nR(xA ln xA xB ln xB ) • Enthalpy of mixing (at constant T, P):
DHmix DGmix TDSmix 0
• The DmixV (V of solution - Vs of components): * * D mixV V n1Vm1 n2Vm2 • Combining equations:
* - molar Vol. of pure liquid Vm,i
V n1V1 n2V2 * * D mixV n1V1 Vm1 n2 V 2 Vm2 Calculate DmixG and DmixS for mixing 3 moles of H2 and 1 mole of N2 at 298.15 K.
If the initial pressure of N2 is p, then that of H2 is 3p. Thus, the initial Gibbs energy is:
oo Gi (3.0mol )[ ( H22 ) RT ln 3 p (1.0 mol )[ ( N ) RT ln p ] After the partition is removed, the initial partial pressures of both gases are halved, and the Gibbs energy changes to: The Gibbs energy of mixing is then ∆mixG = Gf – Gi or:
DmixG nRT(xA ln xA xB ln xB ) 3 / 2pp 1/ 2 DG (3.0mol ) RT ln (1.0 mol ) RT ln mix 3pp (3.0mol )RT ln 2 (1.0 mol ) RT ln 2 (4.0mol )RT ln 2 6.9 kJ
DmixG DmixS nR( x A ln x A x B ln x B ) T p,, n n AB All of the driving force for mixing comes from DmixG the increase in entropy of the system, because 23.1 J/K T the entropy of the surroundings is unchanged. Chemical potentials of liquids
Pure A (l) in A(l) in a mixture in equilibrium with its P* P equilibrium with vapor. vapor. (g) RT ln A (g) RT ln A A P A P (Pure substances are denoted with a *.) * (l) (l) A A
Pure A Mixture of A & B P* P * (l) (g) RT ln A (l) (g) RT ln A A A P A P
At equilibrium, the two chemical potentials of A are equal:
P (l) * (l) RT ln A A A P* A A solution for which the partial vapor pressure of each volatile component is directly proportional to its mole fraction in the liquid phase obeys Raoult's Law, and is said to be an ideal solution: P x P* A A A
(l )* ( l ) RT ln x AA A
Raoult’s Law can be applied quantitatively for solutions in which the molecules of the various components are similar in size, shape, and intermolecular forces.
Raoult’s Law should not be confused with Dalton’s Law, which relates partial pressures of gases to their mole fractions in the vapor phase:
Pi YiP Solutions of chemically similar liquids Solutions of chemically dissimilar liquids behave nearly ideally deviate from ideality Henry's Law In ideal solutions the solute, as well as the solvent, obeys Raoult’s Law. But at low concentrations of solute, although vapor pressure of solute is proportional to its mole fraction, the constant of proportionality is not the vapor pressure of the pure substance, but is the so-called
Henry’s Law constant: PB = xBKB where PB is the vapor pressure of the solute, xB is its mole fraction, and the empirical Henry’s Law constant,
KB has the dimensions of pressure.
Mixtures for which the solute obeys Henry’s Law and the solvent obeys Raoult’s Law are termed ideal-dilute solutions. Henry’s Law is essentially a limiting law, since
푃퐵 퐾푋 = lim 푋퐵→0 푋퐵 and can also be expressed in other concentration units:
푃퐵 푃퐵 퐾푚 = lim or 퐾푐 = lim 푚→0 푚 푐→0 푐 Henry’s Law behavior arises from the fact that in very • Henry’s Law works well when the gas is only dilute solutions, the environment of the solvent molecules sparingly soluble in the solvent, behaves ideally, is virtually the same as that in the pure liquid. However, and does not ionize or otherwise react with the the environment of the solute molecules is very different solvent. from that in the pure state, since they are surrounded by solvent molecules. • Similar to Raoult's Law, and is exactly the same for * an ideal solution with PB = KB. (Fig. 5A.14) • Henry's Law applies only to dilute solutions.
• Raoult's Law is applicable at high concentrations of solvent. Vapor Pressure Diagrams for Real Solutions Thermodynamics of Mixing of Ideal Solutions
In an ideal solution, the chemical potential of each component, J, is given by the same expression as that used previously for the mixing of two gases: (l )* ( l ) RT ln x JJ J The total Gibbs energy before the liquids are mixed is: ** Gi n A A n B B where the * denotes the pure liquid.
After mixing, the total Gibbs energy (assuming an ideal solution) is:
** Gf n A( A RT ln x A ) n B ( B RT ln x B )
The Gibbs energy of mixing is then Gf – Gi:
DmixG n A(ln) RT x A n B (ln) RT x B nRT (ln x A x A x B ln) x B As for gases, the ideal entropy of mixing two liquids is then given by:
DmixS nR( x A ln x A x B ln x B ) and, as was also true for gas mixtures, the enthalpy of mixing in an ideal solution is zero, since
DmixGHTS D mix D mix then
DmixHGTS D mix D mix 0
It should be noted that the change in volume upon mixing (∆mixV) is also zero because ∆mixG for an ideal mixture is independent of pressure, i.e., (∂Gmix/∂p)T = Vmix.
Thus, the thermodynamic conclusions for mixing in an ideal solution are similar to those for perfect gases, although the assumptions are different. In a perfect gas, it is assumed that there are NO intermolecular interactions. In an ideal solution, it is assumed that A-A, B-B, and A-B interactions are the same.
Deviations from Raoult's Law
Raoult's Law is purely statistical - chemical behavior is found in deviations from statistical behavior. E ideal Deviations discussed in terms of excess thermodynamic functions: X = DmixX – DmixX
Regular Solution Theory
E E Excess Enthalpy S =0; H = nRTxAxB, where depends on energy of A-B interactions, relative to A-A interactions
< 0: mixing is exothermic, i.e., A-B interactions are more favorable than A-A or B-B interactions, e.g., the mixture forms H-bonds
> 0: mixing is endothermic, i.e., A-B interactions are less favorable than A-A or B-B interactions, e.g., H-bonds are disrupted; extreme case: immiscibility
Thus, ∆Gmix becomes: DGmix nRT( x A ln x A x B ln x B x A x B )
Colligative properties
Colligative Property: A solution property that depends only on the number of solute molecules or ions present and not on any specific chemical property of the solute (i.e., on the mole fraction of the solute, not on its chemical identity.)
• vapor pressure lowering • boiling-point elevation • freezing-point depression • osmotic pressure
• These changes in solution are in response to the changes in chemical potential of the liquid solvent by the addition of solute molecules to the pure solvent.
• These methods historically used for determining molecular weights. (modern methods: mass spectrometry and gel-permeation chromatography)
The phenomenon of colligative properties is essentially an entropy effect, i.e., the addition of solute particles (molecules or ions) increases the disorder and, hence, the entropy of the solvent compared to the pure solvent.
Boiling Point Elevation
For solvent vapor in equilibrium with a solution of a non-volatile solute:
** AAA(g ) ( ) RT ln x Rearranging: **()()g D G ln x AAvap A RT RT Taking the derivative of both sides with respect to temperature: dxln 1 (/)DDGTH A vap vap dT R dT RT 2 where the Gibbs-Helmholtz equation [((∂G/T)/∂T)p] was used to express the term on the RHS. Then multiply both side by dT and integrate from xA = 0 to xA
ln x A 1 T D H dln x vap dT A RTT * 2 ln(xA 1) where T* is the normal boiling point of the pure solvent. The LHS integrates to ln xA , which equals ln(1 - xB). Thus, assuming that ∆vapH is constant over the typically very small temperature range between T* and T,
DDvapHHT 1 vap 1 1 ln(1x ) dT B * 2* RTRTTT
* It can be shown (Justification 5B.1), using the approximations T ≈ T and xB << 1, that the above equation results in RT *2 * T – T = ∆Tb = KbxB = xB Dvap H
where ∆Tb is the elevation in boiling point of the pure solvent caused by the presence of the solute B at a mole fraction of xB . The so-called boiling point elevation constant, Kb, is an empirical constant that is related to ∆vapH of the pure solvent and is unique for each solvent.
Because the mole fraction xB is directly proportional to the molality b (moles of solute per kg of solvent) of the solution, the latter concentration unit is usually used for boiling point elevation measurements.
Freezing Point Depression
Using an analysis analogous to that for boiling point elevation, it can be shown that for a solution in equilibrium with a pure solid solvent, the melting point of the solid solution is lower than that of the pure solid solvent: ** AAA(s ) ( ) RT ln x which leads to: RT *2 ∆Tf = KfxB = xB D fus H where T* is the normal melting point of the pure solid solvent, and Kf is the freezing-point depression constant.
As was the case for the boiling point elevation constant, Kf is an empirical constant that is related to ∆fusH of the pure solid solvent. Osmotic Pressure
Osmosis is the spontaneous passage of a pure solvent into a solution of the solvent separated from it by a semipermeable membrane, i.e., one that allows passage of solvent molecules through it, but not solute particles (molecules or ions). The pressure that must be applied to the solution to prevent passage of the solvent through the membrane is referred to as the osmotic pressure, Π. Because the chemical potential of the pure solvent (A) is larger than that of the solvent in the solution, the driving force for osmosis is to decrease μA (i.e., decrease the Gibbs energy and increase the entropy) in the pure solvent until it equals that in the solution, and the system is at equilibrium.
In essence, the applied osmotic pressure, Π, on the solution side of the membrane increases μA in the solution until it equals that in the pure solvent. Hence, at equilibrium,
** AAA()(,)p x p
As shown in Justification 5B.3, the osmotic pressure needed to equalize μA on both sides of the membrane is given by the van’t Hoff Equation: Π = [B] RT where [B] = nB/V is the molar concentration, i.e. the molarity, of the solute, B, in the solvent A. When expressed in the following form, the van’t Hoff Equation clearly resembles the perfect gas equation:
Π Vm = RT xB since, when the solution is dilute, xB = nB/nA. and nAVm = V, the total volume of the solution. Use of Osmotic Pressure
Because osmotic pressure effects are very large compared to freezing point depressions and are relatively easy to measure, they are used to determine molecular weights of macromolecules, such as synthetic polymers and proteins.
Example: For a solute having a M.W. of 40,000 g/mol, in an aqueous solution with a concentration of 5 x 10-6 molar.
• Freezing Point Depression Method: -6 ∆Tf = Kfb = 1.86 (5 x 10 ) = 9 μK very difficult to measure!
• Osmotic Pressure Method: Π ≈ cRT ≈ (5 x 10-6) (0.082) (298) ≈ 1.2 x 10-4 atm
= 0.093 mm Hg = 1.3 mm H2O
• Problems with osmotic pressure measurements: - Must use correct membrane - Times needed to reach equilibrium may be very long (hours to days!) Vapor Pressure Diagrams
For a two-component ideal solution containing the volatile liquids A and B, the partial vapor pressures are given by Raoult’s Law:
* * pA = xAp A and pB = xBp B
* * where p A is the vapor pressure of pure A and p B that of pure B. Therefore, the total vapor pressure p of the mixture is:
* * * * * p = pA + pB = xAp A + xBp B = p B + (p A – p B)xA
Thus, at constant T, the total vapor pressure of the * * mixture changes linearly from p B to p A as xA varies from 0 to 1, as shown in Fig. 5C.1. Vapor Pressure Diagrams
The partial pressures of A and B in the gas-phase that is in equilibrium with the liquid mixture are given by Dalton’s Law:
pA = xAp and pB = xBp
Thus, the mole fractions, yi, in the gas phase are: pp yyABand ABpp
Thus, if the solution is ideal, the partial pressures, pi, and the total pressure, p, can be expressed in terms of the mole fractions, xi, in the liquid: * xpAA yABA*** y 1 y (Eq. 5C.4) pBABA() p p x As shown in Fig. 5C.2, because A is the more volatile * * component, yA > yB at all compositions, for pA ≥ pB . Vapor Pressure Diagrams
Because the composition of the liquid can be related to the composition of the vapor via * Raoult’s Law (pi = xip i) and yi = pi/p, the total vapor pressure, p, can be related to the composition of the vapor: ** ppAB p *** pABAA() p p y As shown in Fig. 5C.3, the total vapor pressure is largely determined by that of the more volatile component, A. Vapor Pressure Diagrams
Figs. 5C.2 and 5C.3 can be combined into a single Figure that shows the total system composition (both liquid and Tie Line vapor) for one of the components, represented by zA. In the region labeled “Liquid” the applied pressure is sufficiently * high (higher than p A) that only the liquid phase exists, so zA = xA.
Similarly, in the region labeled “Vapor” the pressure is sufficiently low that only the vapor phase exists, so zA = yA.
Between the two lines, both liquid and vapor phases exist, where, at any point, a horizontal line (called a tie line) connects the mole fraction of A in the liquid phase, xA, with that in the vapor phase, yA. Vapor Pressure Diagrams
a1, a1‘ a3, a3’ a4
Isopleth Vapor Pressure Diagrams Vapor Pressure Diagrams The Lever Rule
Any point in the two-phase region of a vapor-liquid phase diagram represents quantitatively the relative amounts of the two phases. The amounts, nα and nβ, of the phases α (vapor) and β (liquid), that are in equilibrium, for example, can be determined by measuring the distances lα and lβ along the horizontal tie line and then applying the so-called lever rule:
nα lα = nβ lβ
For example, the two lengths lα and lβ from the central point on the tie line in Fig. 5C.9 are approximately equal, indicating that the relative amounts, nα and nβ, of the α (vapor) and β (liquid) phases at this composition are also approximately equal at equilibrium. Distillation Distillation
Vapor Vapor Composition Composition
Boiling Temp. of Liquid Boiling Temp. of Liquid Azeotropes
A-B interactions are more favorable than A-A and A-B interactions are less favorable than A-A and B-B interactions (GE < 0), thus decreasing V.P. of B-B interactions (GE > 0), thus increasing V.P. of mixture and increasing B.P. In effect the favorable mixture and decreasing B.P. Unfavorable A-B A-B interactions stabilize the liquid. interactions destabilize the liquid. Deviations from Raoult's Law
Raoult's Law is purely statistical - chemical behavior is found in deviations from statistical behavior. E ideal Deviations discussed in terms of excess thermodynamic functions: X = DmixX – DmixX
Regular Solution Theory
E E Excess Enthalpy S =0; H = nRTxAxB, where depends on energy of A-B interactions, relative to A-A interactions
< 0: mixing is exothermic, i.e., A-B interactions are more favorable than A-A or B-B interactions, e.g., the mixture forms H-bonds
> 0: mixing is endothermic, i.e., A-B interactions are less favorable than A-A or B-B interactions, e.g., H-bonds are disrupted; extreme case: immiscibility
Thus, ∆Gmix becomes: DGmix nRT( x A ln x A x B ln x B x A x B )
The maximum concentration of ethanol that can be obtained by fractional distillation of an ethanol-water mixture is a low-boiling azeotrope that is 95.63% ethanol by mass and boils at 78.2oC, which is lower than the boiling points of either ethanol (78.4oC) or water (100oC). Distillation of Immiscible Liquids Liquid-Liquid Phase Diagrams
nAlA
nBlB Liquid-Liquid Phase Diagrams
A mixture of 50 g (0.59 mol) of n-hexane and 50 g (0.41 mol) of nitrobenzene at 290K occurs in the T two-phase region of the phase diagram. One of the uc phases (the purple line) is hexane-rich, and the other (the blue line) is rich in nitrobenzene. The amounts of the two phases are given by the lever rule applied to the endpoints of the tie line at 290K: nll n 0.83 0.41 0.42 0.35 0.41 0.83 H N 7 n l nNH l 0.41 0.35 0.06 Thus, there is ~7 times more of the hexane-rich phase than of the nitrobenzene-rich phase.
If a sample with xN = 0.41 were heated to 292K or higher, the two phases would become a single phase. Liquid-Liquid Phase Diagrams Liquid-Liquid Phase Diagrams Solid-Liquid Phase Diagrams Simplest type of binary solid-liquid phase diagram is one in which both components are completely miscible in both the liquid and solid phases. This type of system is called isomorphous.
Example is copper-nickel, for which the phase diagram has only a very narrow two-phase (Liquid + Solid) region. Copper and nickel are completely miscible in all proportions and form true alloys, not merely physical admixtures.
The line above which the mixture is a liquid is called the liquidus line. At temperatures just below this line, crystals of a solid solution begin forming.
The line below which solidification is complete is called the solididus line. At temperatures below this line, only a solid solution exists. Solid-Liquid Phase Diagrams
For cooling along the isopleth beginning at a1,
a1→a2: System enters two-phase region “Liquid + B”. Pure solid B begins to come out of solution, and remaining liquid becomes richer in A.
a2→a3: More solid B forms. Relative amounts of solid and liquid (approximately equal at this point) given by Lever Rule along tie line.
Composition of Liquid phase given by b3.
a3→a4: At a4, there is less liquid than at a3, and its composition is given by e2. At the temperature of e2 and below, the remaining liquid solidifies to give a two-phase system containing an admixture of (almost) pure A and pure B.
The isopleth at e2 corresponds to a eutectic composition, i.e., the mixture of A and B with the lowest M.P. Solutions to the right of e2 deposit pure B as they cool, while those to the left deposit pure A. Solid-Liquid Phase Diagrams Solid-Liquid Phase Diagram of Tin-Lead Electron Micrographs of Metal Mixtures
Scale Line = 250μ Scale Line = 125μ Scale Line = 25μ Solid-Liquid Phase Diagram of NaCl-H2O Melting Behaviors of Mixtures fp lowering fp lowering of A by B of B by A • Eutectic - minimum melting point; two curves intersect • Solid solutions Liquid fp of A fp of B atomic-level random mixture composition variable melting Solid A+ Solid B+ point is not sharp. T Liquid Liquid
• Compound formation Eutectic Convention: A vertical line on a two-component phase- Solid A + Solid B diagram denotes the formation of a compound. XB
• Incongruent Melting Points - some compounds do not Liquid fp of B melt into a liquid of the same composition, but Solid B+ Liquid fp of A decompose into another solid. Solid A+ Solid C+ T Liquid Liquid Solid B+ Solid C Eutectic Compound Solid A + Solid C Formation
A C B XB Solid-Liquid Phase Diagrams
962oC
232oC Solid-Liquid Phase Diagrams Zn-Mg Phase Diagram
What is the melting temperature of the
eutectic between Mg and MgZn2?
A. 410 B. 590 C. 347 D. 650 Na-K Phase Diagram