3.012 PS 5 Issued: 10.15.04 Due: 10.22.04 3.012 Fall 2004 2. Formation of an ideal solution is always thermodynamically favorable. Consider the mixing process described schematically below for an isolated two-component system. a. Using the ideal solution model of chemical potentials, show that at constant temperature and pressure, the Gibbs free energy change for taking an initially pure quantity of a component A and mixing it with a quantity of a second pure component B is always favorable (mixing will occur spontaneously) if the two components form an ideal solution. b. Show that the entropy change on mixing two components to form an ideal solution is always positive or zero. (Hint: How can I directly get the entropy change in the process from the Gibbs free energy change calculated in part (a)? Or in other words, how is S related to G?) (a) The chemical potential of each component in an A-B ideal solution is: µA = µA,o + RT ln X A µB = µB,o + RT ln X B To determine whether the mixing process will occur spontaneously, we need to calculate the Gibbs € if it is less than zero: free energy change and determine € 3.012 PS 5 1 of 2 10/27/04 ΔGmixing = ΔGsystem = G final − Ginitial < 0? The initial free energy of the system, for the unmixed components is simply: € Ginitial = n A µA,o + n B µB ,o The free energy of the final state, where A and B have mixed to form an ideal solution is: € = n A (µA,o + RT ln X A ) + n B (µB,o + RT ln X B ) G final Combining these equations, the free energy change for the mixing process is: ΔGmixing = G final − Ginitial = RT ( n A ln X A + n B ln X B ) € Because XA and XB are related, we can simplify this expression to: ΔGmixing = RT ( n A ln X A + n B ln(1− X A )) € For a spontaneous process, ΔG must be less than zero: € n A ln X A + n B ln(1− X A ) < 0 Now, nA and nB are always positive numbers. The value of XA can range from 0-1. We see that for any value of XA, the prescribed quantity will be negative in total, meaning that these liquids will mix in any ratio spontaneously- mixing € is always favored. (b) The entropy is always related to the Gibbs free energy by the partial derivative:  ∂G  −S =    ∂T  P ,n …we know this from the complete differential expression for G: € C  ∂G   ∂G  dG = −SdT + VdP + ∑ µi dn i =   dT +   dP + ∑  ∂T  P,n  ∂P T ,n i i=1  ∂G  dn i    ∂n i T ,P ,n j ≠i Applying the partial derivative to our ΔGmixing, we obtain the entropy change on mixing: €  ∂ΔGmixing  ΔSmixing = −  = −[ R( n A ln X A + n B ln X B )] = −R[ n A ln X A + n B ln(1− X A )]  ∂T  P ,n The natural log terms in this expression are always less than or equal to zero- implying that the entropy on mixing for the ideal solution is always positive (or zero). € 3.012 PS 5 2 of 2 10/27/04