Ideal and Regular Solutions

• Ideal solutions are a special case of regular solutions
• Both ideal and regular solutions share the same entropy of mixing, denoted as ∆mS
• The key difference between ideal and regular solutions lies in their enthalpy of mixing, ∆mH
• For ideal solutions, ∆mH is zero
• Regular solution models attempt to estimate a non-zero ∆mH realistically
• The Gibbs free energy of a system is represented as G = Gunmixed + ∆mG, where ∆mG is the change in Gibbs free energy upon mixing
• The change in Gibbs free energy is expressed as: ∆mG = ∆mH - T∆mS

Ideal Solutions

• Defines ideal solutions by the chemical potentials of all components across the concentration range
• The chemical potential: µi = µ?i (T, P) + RT ln xi
• µ?i is the chemical potential of the pure liquid (xi = 1) at temperature T and pressure P
• An ideal solution's behavior originates from its zero enthalpy of mixing (∆mH = 0)
• Absence of mixing enthalpy (∆mH = 0) in ideal solutions does not mean the absence of interactions between particles. Rather, it means all interactions are identical
• A lattice model of a binary mixture consisting of N1 molecules of type 1 and N2 molecules of type 2 with total N = N1 + N2 molecules is used to express the entropy of mixing
• Both molecule types occupy one lattice site and have the same interactions.
• Boltzmann's equation for entropy: S = kB ln Ω
• The entropy of mixing: ∆mS = Smixed − Sunmixed = kB ln Ω(N1, N2) − kB ln Ω(N1, 0)Ω(0, N2)
• The number of ways to randomly distribute N1 molecules of type 1 and N2 molecules of type 2 over N = N1 + N2 lattice sites is given by Ω(N1, N2)
• The product Ω(N1, 0)Ω(0, N2) gives the number of ways to arrange molecules before mixing
• Only one way to place N1 molecules on N1 sites and N2 on N2 sites, thus ln Ω(N1, 0)Ω(0, N2) = 0
• Simplified mixing entropy due to indistinguishability of particles of the same type is: ∆mS = kB ln Ω(N1, N2) = kB ln (N! / (N1!N2!))
• In a non-ideal solution, account for different interactions between different types of molecules
• By applying Stirling's approximation, ln N! ≈ N ln N − N, we obtain the mixing entropy: ∆mS = −kB [N1 ln x1 + N2 ln x2] = −R [n1 ln x1 + n2 ln x2]
• The number of moles is represented as ni = Ni /NA and the mole fraction of component i as xi = ni /n
• Equivalent mixing entropy: ∆mS = −nR [x1 ln x1 + x2 ln x2]

Ideal Solution Equations

• Gibbs free energy of a binary ideal mixture can then be written as G = Gunmixed + RT [n1 ln x1 + n2 ln x2]
• Molecular redistribution during mixing accounts for the logarithmic term in the chemical potential
• A process of solution of a solid includes heat change (enthalpy), followed by entropy change in an ideal solution
• The "ideal" solubility depends on temperature because of the change in enthalpy
• Condition for chemical equilibrium: µ2(s) = µ2(aq)
• Expression for the ideal solution model: µ∗2 (s) = µ∗2 (l) + RT ln x2 ⇒ ln x2 = (µ∗2 (s) − µ∗2 (l)) / RT
• Equations lead to: d ln x2 / dT =Afus H / RT²
• Slope enables the calculation of the enthalpy of fusion

Raoult's Law

• This states that the partial vapour pressure Pᵢ of every component i varies linearly with its mole fraction xᵢ in solution
• Vapour pressure is the product of component pressure and mole fraction: Pᵢ = Pᵢ*xᵢ
• This law follows from applying the condition of chemical equilibrium

Non-Ideal Solutions

• The partial and total vapor pressures of most solutions strongly deviate from Raoult's law, not ideal solutions
• These deviations can be both positive and negative
• Acetone/carbon disulphide shows a strong positive deviation from Raoult's law
• Acetone/chloroform are standard examples of showing a strong negative deviation
• The standard example carbonyl group is hydrogen bonded in weak chloroform

Regular Solution Theory

• Hildebrand introduced regular solutions in 1929
• Non-ideality is caused by a non-zero enthalpy of mixing, while the entropy of mixing is "ideal"
• Lattice model: N1 molecules of component 1 and N2 molecules of component 2, which each can occupy exactly one lattice site
• Volume change is nonexistent when the two components mix, which implies that ∆mH = ∆mU at constant pressure and that is why ∆mG = ∆mA
• Enthalpy of mixing formula: ∆mH = 1/2 z (N1 + N2) [2w12 − w11 − w12] x1x2
• The interaction parameter w: w = NA z [w12 − (w11 + w22)/2]
• Then we can write the enthalpy of mixing for a regular solution as ∆mH = (n1 + n2)wx1x2

Gibbs and Activity Co-efficients

• The Gibbs free energy for mixing: ∆mG = ∆mH − T∆mS = (n1 + n2)wx1x2 + RT (n1 ln x1 + n2 ln x2)
• We can compute the chemical potential formula: μ1 = μ?1 + RT ln x1 + wx²₂