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Chapter 5: the Thermodynamic Description of Mixtures

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Chapter 5: the Thermodynamic Description of Mixtures 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 • Colligative properties 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.
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