General Physical Chemistry I

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1 General Physical Chemistry I Lecture 13 leksey Kocherzhenko pril 7, 2015"

2 Last time "

3 Phase diagrams" Maps showing the thermodynamically stable phases for each and," phase regions in a phase diagram are separated by phase boundaries:! p T Solid and liquid phases in equilibrium,"g (l) m = G (s) m Liquid and gas phases in equilibrium,"g (g) m = G (l) m Points:" Solid and gas phases in equilibrium," G (g) m = G (s) m gas phase stable" B gas and liquid in equilibrium" C liquid phase stable" D liquid and solid in equilibrium" E solid phase stable"

4 Characteristic points" Normal vs. anomalous liquid! Highest temperature at which a liquid can exist" Lowest pressure at which a liquid can exist" (for a normal liquid, also the lowest temperature at which liquid can exist)" Equilibrium between phases is dynamic: there are as many molecules leaving a phase per second as there are molecules coming back to it!

Gibbs free energy variation at"t = const Ø For perfect gasses: "pv m = RT Variation of the Gibbs free energy with pressure for a perfect gas:" Z p f Z p f = p i V m (p)dp (p) is not a straight line"

5 Gibbs free energy variation at"t = const Ø For perfect gasses: "pv m = RT Variation of the Gibbs free energy with pressure for a perfect gas:" Z p f Z p f = p i V m (p)dp (p) is not a straight line" G (g) m G (s) m (p) = RT G (l) and" are straight lines " p i m (p) dp p ) V m = RT p G (g) m = G (g) m (p f ) G (g) m (p i ) = RT (ln p f ln p i ) or:" G (g) m Standard molar" Gibbs free energy" = = RT ln p f p i (p) =G m + RT ln p p 1 bar" The stable phase of a substance is determined by both the temperature and the pressure"

6 Partial molar properties"

7 Partial molar volume " Partial molar property the contribution per mole that a substance makes to an overall property of a mixture " Total volume of a mixture of several substances:" V = X i i V i (x i ) Partial molar volumes as functions of mole fractions for a C 2 H 5 OH/H 2 O mixture" Partial molar volumes" Mole fractions: "x i = i P i i Ideal solution:"v m = X x i V i i Independent of mole fraction" Real solution: partial molar volumes depend on the mole fractions for each substance, due to different molecular packing, as compared to pure substance"

8 Negative partial molar volume " The partial molar volume of substance i is the change in the total volume of the mixture that occurs when a small amount of substance i is added, while all other parameters of the mixture are held constant" For a mixture of two substances, and B:" T,p, B V = V + B V B If adding changes the molecular packing in the mixture so as to reduce the total volume of the mixture, then is negative" V

9 Chemical potential " Chemical potential = partial molar Gibbs free energy:!µ i = The total Gibbs free energy:"g = X i i i T,p, j6=i Ø The chemical potential is a measure of the ability of a substance to bring about chemical or physical change in a mixture" Ø The chemical potential depends on the conditions of a mixture (pressure, temperature, composition)" Let us study the change in the total Gibbs free energy of two substances when they are mixed " (starting with the case of two perfect gasses)"

10 Mixtures of perfect gasses"

11 Gibbs free energy of mixing: Perfect gasses " In lecture 12 (slide 10) we have found the molar Gibbs free energy for a perfect gas as a function of pressure:" G (g) m Standard molar" Gibbs free energy" (p) =G m + RT ln p p 1 bar" For partial molar properties in a mixture:" µ (g) i (p) =µ i + RT ln p i p Standard chemical potential" Partial pressure" The chemical potential increases with pressure and, at a given pressure, with temperature"

12 Gibbs free energy of mixing: Perfect gasses " ll gasses mix spontaneously, implying that the Gibbs free energy for this process is " negative" Initial state" Final state" Gibbs free energy for final state:" G final = µ final + B µ final B = = µ + RT ln pfinal Gibbs free energy for initial state:" G initial = µ initial + B µ initial B = = µ + RT ln pinitial p p + B µ B + RT ln pinitial B + B µ B + RT ln pfinal B p B p B + The initial and final temperature is the same: there is no change in the kinetic energy of molecules, which is the total energy of molecules for perfect gas"

13 Gibbs free energy of mixing: Perfect gasses " G initial = µ + RT ln pinitial G final = µ + RT ln pfinal Initial state" Final state" p + B µ B + RT ln pfinal B p B Change in the Gibbs free energy:" G = G final G initial = = RT ln pfinal ln pinitial + B RT ln pfinal B p p p B = RT ln pfinal p initial ln pinitial B p B + B ln pfinal B p initial B Using the perfect gas law, it can be shown that if" p initial = p initial = p, then the total pressure in" p B + B µ B + RT ln pinitial B the final state is also "p = p final + p final B p B (see, e.g., problem 2 of test 1 or the problem from recitation 3)" =

14 Gibbs free energy of mixing: Perfect gasses " We found that:" If p initial G = RT ln pfinal p initial = p initial B = p and" p = p final ) G = RT ln x final + B ln pfinal B p initial B + p final B = x final + B ln x final B p + x final B p < 0 apple 1 apple 1 apple 0 apple 0 = x = x B = + B ) G = RT x ln x final + x B ln x final B Initial state" Final state" à for two (perfect) gasses, the Gibbs free energy of mixing is always negative! à this is always a spontaneous process "

15 Entropy of mixing: Perfect gasses " We found that:" G = RT x ln x final + x B ln x final B We also know that:" = 0, since internal energy doesn t change and no work is being done" ) S = R x ln x final G = H T S No entropy change for mixing of two volumes of the same gas" = T S Constant temperature" + x B ln x final B Entropy of mixing (for two perfect gasses)" (s you may have found in HW4)" Initial state" Final state"

16 Mixtures of liquids"

17 Raoult s law " Raoult s law: p J = x J p indicates property of pure substance" J," the partial vapor pressure of a substance in a liquid mixture is proportional to its mole fraction in the mixture and the vapor pressure of the pure substance" Ø Solutions that obey Raoult s law are called ideal solutions! Ø Most reliable for mixtures of substances with molecules of similar shape held together by intermolecular interactions of similar type and strength! Real mixtures show deviations from Raoult s law"

Chemical potentials for mixtures of liquids " Ø liquid in a closed vessel is in dynamic equilibrium with its vapor" There is as much liquid evaporating per s as there is vapor condensing" Ø Since the

18 Chemical potentials for mixtures of liquids " Ø liquid in a closed vessel is in dynamic equilibrium with its vapor" There is as much liquid evaporating per s as there is vapor condensing" Ø Since the system is in equilibrium, the difference in the Gibbs free energy between the two phases is:" G l/g = G l G g =0 ) µ (l) = µ (g) = µ (g) + RT ln p p Ø For a component in a mixture, we can apply Raoult s law:" p J = x J p J ) µ (l) J = µ (g) J + RT ln x Jp J p J = µ (g) J Independent of the composition of the mixture" + RT ln p J p {z J } = µ J +RT ln x J Chemical potential of pure liquid"

19 Chemical potentials for mixtures of liquids " We found that" µ (l) J = µ J + RT ln x J The mole fraction" x J < 1 ) ln x J < 0 ) The chemical potential of a solvent is lower in a solution than when it is pure" ln x J =0( x J =1

20 Chemical potentials for mixtures of liquids " Interactions between unlike molecules weaker than between like molecules à vapor pressure greater than for ideal solution" Easier to vaporize mixture than individual substances" Interactions between unlike molecules stronger than between like molecules à vapor pressure smaller than for ideal solution" Harder to vaporize mixture than individual substances"

General Physical Chemistry I

General Physical Chemistry I Lecture 14 Aleksey Kocherzhenko April 9, 2015" Last time " Chemical potential " Partial molar property the contribution per mole that a substance makes to an overall property