3.020 Lecture 17 1 Solution modeling • Ideal gas mixtures −→ ideal solution model • Non-ideal solution models – Activity : activity coeﬃcient • Dilute solution model • Regular solution models 2 Ideal gas mixtures • model as isothermal expansion of each component Z Z Pi RT Pi dµi = Vi dPi −→ Δµi = dPi V = dP ∗ ∗ = RT ln ( ) P P T P dµi : This is Δµi for isothermal expansion T dµi : This is also Δµi for the mixing process T • Gibbs free energy of this mixing process 1 ΔGmix = X Xi Δµi i = X = X i Xi RT ln (Pi /P ) | {z } Dalton Xi RT ln Xi i - note that ΔGmix < 0 d2 ΔGmix >0 dX22 everywhere =⇒ mixing is always spontaneous 3 The ideal solution model Δµi = RT ln Xi • motivated by ideal gases mixing • approximates a broader class of real-world systems ∂Δµi ∂T ∂Δµi ΔV i = ∂P ΔS i = P,nk = −R ln Xi > 0 process driven by entropy increase =0 no volume of mixing =⇒ T,nk ﬁnal volume is sum of pure components ΔH i = Δµi − T ΔS i = 0 ΔU i = ΔH i − P ΔV i = 0 no interactions between molecules, no bonds made/broken , no change in enthalpy or energy Q. What if molecules interact? What if ideal model doesn’t apply ? A. Capture deviation from ideal model (more bookeeping !!) 2 Let Δµi = RT ln ai = RT ln γi Xi ai = activity of component i = γi Xi γi = activity coeﬃcient of component i ΔGmix = X Xi Δµi = X = X i Xi RT ln γi Xi i |i X Xi RT ln Xi + {z ideal case } |i Xi RT ln γi {z } derivation from ideal • Non-ideal behavior captured by γi 6= 1 4 Dilute solution model Assume : 1. Each solute molecule is surrounded by solvent, and solute-solve interactions are negligible 2. Each solvent molecule is, on average, surrounded by other solvent, and therefore “acts” like a pure substance. Assumption 1 −→ Henry’s law of the solute γ20 X2 | {z } lim a2 = X2 →0 Henry’s constant Assumption 2 −→ Raoult’s law of the solvent lim a1 = X1 X1 →0 Can be derived from Henry via G-D integration. See 8.2.3 and Pset 5 3 • variation of activity with composition Could look like this (DeHoﬀ Fig. 8.4) 5 Regular solution models Assume : 1. Entropy of mixing is captured by ideal model, ΔSmix,ideal will show later that this is just “conﬁguration entropy ” 2. Intermolecular interactions are captured by non-zero ΔHmix 6= 0 ΔGmix = ΔHmix − T ΔSmix,ideal X Xi ln Xi = ΔHmix + RT i • the Simple Regular Model is the simplest case of regular solution models for binary systems ΔHmix = a0 X1 X2 a0 > 0 : endothermic mixing; ΔGmix a0 < 0 : exothermic mixing X = a0 X1 X2 + RT Xi ln Xi i 4

Lecture 17: Solution Models - Ideal, Dilute, and Regular

Related Documents

Recommended Documents

Description

Related Documents

Almost There!

Free up your schedule!

Take 5 seconds to unlock