Chapter 18: Solutions - Materials Science

1 Notes on the Thermodynamics of Solids Morris, Jr.; Fall, 2008 Page 396 Chapter 1 Chapter 188: Solutions : Solutions Chapter 18: Ideal A solution of ideal The ideal The vapor pressure of a component in an ideal Thermodynamics of Real Excess thermodynamic The activity and the activity Dilute The chemical potential in a dilute The fundamental equation of a dilute Generalization to "semi-dilute" Dilute Solutions : Equilibrium of two Solutions with the same Influence of a dilute solute on two-phase The vapor pressure of a dilute solution: Raoult's Osmotic Dilute Solutions with the same solute: Henry's Solute-solvent equilibria between dilute The interference of solutes in a dilute ternary Behavior near a Critical The Simple The thermodynamics of the simple Decomposition and ordering in the simple The Phase Diagram of a Binary Special points in the T-x Binary phase Solid solution Low-temperature behavior of a solid Phase diagrams with eutectic or peritectic Structural transformations in the solid Systems that form Mutation lines in binary phase Miscibility gap in the INTRODUCTION This Chapter treats multicomponent fluid phases ( Solutions ) that are sufficiently simple that their thermodynamic behavior can be studied in some detail.

2 We are, of course, particularly interested in solid Solutions , which obey the thermodynamics of fluids if the stress is hydrostatic. The theory developed in this Chapter includes that part of the Notes on the Thermodynamics of Solids Morris, Jr.; Fall, 2008 Page 397 thermodynamics of solid Solutions that is independent of the details of the atom arrangements. We begin with a discussion of the one completely solvable case: a solution of ideal gases. We shall then use classical thermodynamics to generalize this result to the simplest model of a real solution, the ideal solution. While almost no real Solutions (except the ideal gases) are ideal Solutions over the whole range of composition, we shall discover that every solution behaves very nearly like an ideal solution in the limit of small solute concentration (the dilute solution). This result, and its logical extension to higher solute concentrations through an expansion of the free energy in powers of the concentration, is the basis for most of the useful theory of Solutions , and we shall explore its applications in some detail.

3 We shall then consider models that attempt a global representation of the fundamental equa-tion of a solution. Finally, we shall discuss the phase diagrams that represent the equilib-rium of binary Solutions and offer a simple classification of the more important of these. IDEAL Solutions A solution of ideal gases The fundamental equation of a mixture of ideal gases was developed as an exercise earlier in the course. The classical derivation is based on Dalton's Law, Pk = NkTV where Pk is the partial pressure of the kth specie, and the caloric equation of state E = NCT = 32 NT where N is the total mole number. From eq. the energy can be written as the sum of the energies of the individual species: E = k Ek = k 32 NkT Given the additivity of the energy and the pressure, S = ET + PVT - i iNi = k Sk where Sk is the entropy the kth specie would have if it filled the volume V, at temperature, T, alone.

4 It follows that a component in a solution of ideal gases has the entropy it would have if it were present alone. Notes on the Thermodynamics of Solids Morris, Jr.; Fall, 2008 Page 398 The entropy of a one-component ideal gas can be found by integrating the thermal and caloric equations of state, and is S(E,V,N) = 32 Nln EN + Nln VN + Ns The entropic form of the fundamental equation of a mixture of ideal gases is, therefore, S(E,V,{N}) = N s(e,v,{x}) = k 32 Nkln EkNk + Nkln VNk + Nks k = 32 Nln EN + Nln VN - N k xkln(xk) + N k xks k Comparing eqs. and , the major difference between the entropy functions of the one-component and n-component ideal gas is the inclusion of the extra term Smix = - N k xkln(xk) on the right-hand side of This term is called the entropy of mixing, and plays a cen-tral role in the thermodynamics of Solutions . Note that the entropy of mixing is a straight-forward consequence of Dalton's Law; it does not enter because the species interact, but because they do not interact.

5 Its appearance is a consequence of writing the entropy as a function of the global content of the mixture (E,N) rather than the contents of the individual species (Ek, Nk). Essentially, it results from the fact that the various species in the mixture fill the same space at the same time. The Helmholtz free energy can be found by expressing S as a function of T,V and {N}, and can be written F(T,V,{N}) = N f(T,v,{x}) = V Fv(T,{n}) = - VT 3n2ln(T) + n - k nkln(nk) + k nkf k = - NT 32ln(T) + ln(v) + 1 - k xkln(xk) + k xkf k Notes on the Thermodynamics of Solids Morris, Jr.; Fall, 2008 Page 399 The advantage of using the Helmholtz free energy is that it is easily computed from statistical thermodynamics, which lets us evaluate the unknown constants, f k . The result is f k = 32 ln mk2 N0 2 where mk is the atomic mass of the kth specie, N0 is Avogadro's number, = h/2 , h is Planck's constant, and T is measured as RK, where R is the molar gas constant and K is the temperature in degrees Kelvin.

6 [I leave it as an exercise to confirm that, with this result, the quantity in braces on the right-hand side of is dimensionless.] The form of the fundamental equation that is most commonly used in the study of Solutions is the Gibbs free energy. To find it we compute the chemical potentials of the components as a function of T, P, and {x}. From equation , k = F Nk = - 3T2 ln(T) + T ln NkV - Tf k = T ln(Pk) - 5T2 ln(T) - Tf k Equation shows that the chemical potential of a component in a solution of ideal gases is the chemical potential it would have if it were present by itself. Since Pk = xkP, where xk is the atom fraction of the kth component, k = T ln(P) - 5T2 ln(T) - Tf k + T ln(xk) = 0k(T,P) + T ln(xk) where 0k(T,P) is the chemical potential of component k at temperature T and pressure P. The Gibbs free energy per mole is, then, g(T,P,{x}) = k xk 0k(T,P) + T k xk ln(xk) where the first term is the Gibbs free energy per mole when the components are separated at constant P and T, and the second term (which is negative) is the free energy of mixing.

7 Note that the free energy of the mixing comes from the entropy of mixing; it does not rep-resent any physical interaction between the species, but is due to the fact that they do not interact, and can, hence, fill the same space at the same time. Notes on the Thermodynamics of Solids Morris, Jr.; Fall, 2008 Page 400 The ideal solution The mixture of ideal gases is one example of an ideal solution, which is a solution of atoms of species that have no preferential interaction with one another. The ideal solu-tion is the simplest solvable model of a real solution. It is useful to define an ideal solution by its Gibbs free energy: an ideal solution is a solution that obeys a fundamental equation of the form , where 0k(T,P) is the chemical potential of a pure sample of the kth component at temperature T and pressure P. Equation shows that the change in free energy on forming an ideal solution from the pure components at (T,P) is always negative.

8 Hence species that form ideal Solutions are miscible in any proportions. The entropy of mixing of an ideal solution is the same as that of a mixture of ideal gases (eq. ). The chemical potential of the kth component of an ideal solution is given by equation The molar enthalpy of an ideal solution is h = g + Ts = k xk 0k(T,P) + Ts0k(T,P) = k xkh0k(T,P) where s0k(T,P) is the molar entropy of the kth component in its pure state at (T,P). Equation shows that the heat of mixing, h, is zero; the enthalpy of an ideal solution is just the weighted average of the enthalpies of its pure components, so no heat evolves when the solution forms. It follows as a corollary that the miscibility of the species that form an ideal solution is due to the entropy of mixing. The relative chemical potential of the kth component in an ideal solution is k(T,P,{x}) = g xk = 0k - 0n + T ln xkxn = 0k(T,P) + T ln xkxn where n designates the reference component, or solvent, and 0k(T,P) is the relative chemi-cal potential of the pure solute.

9 Note that k depends on the concentrations of the other solutes only through their effect on xn, the mole fraction of the solvent. For a binary solu-tion, Notes on the Thermodynamics of Solids Morris, Jr.; Fall, 2008 Page 401 k(T,P,x) = 0k(T,P) + T ln x1 - x where x is the mole fraction of solute. The vapor pressure of a component in an ideal solution One of the properties of an ideal solution that can be easily calculated is the equilib-rium vapor pressures of its components. Let the ideal solution be a condensed phase in equilibrium with a vapor that is a mixture of ideal gases (this is almost always a good ap-proximation since the total vapor pressure of a condensed phase is usually very small). The condition of chemical equilibrium requires that the chemical potential of every component be the same in the solution and the vapor. From equation the chemical potential of the kth component in the vapor phase is k = k(T) + T ln(Pk) where k(T) is a function of the temperature only and Pk is the partial pressure of the kth component.

10 The chemical potential in the solution is, from equation k = 0k(T,P) + T ln(xk) where 0k(T,P) is the chemical potential of the pure component in the condensed state at (T,P), and is a weak function of P when P is small. Equating the chemical potentials yields Pk = xk exp 0k - kT ~ xkP0k(T) where P0k(T) is the vapor pressure of the kth component in its pure form. It follows that the vapor pressure of a component in an ideal solution depends only on its own atom frac-tion. Conversely, the concentration of a component in an ideal solution in equilibrium with a dilute vapor depends only on its partial pressure in the gas. THERMODYNAMICS OF REAL Solutions Excess thermodynamic quantities There are two model systems that provide good reference states for the thermody-namics of a real solution in the sense that their properties are well-defined and relatively simple to measure experimentally.