Liquids and Solutions Introduction This course examines the properties of liquids and solutions at both the thermodynamic and the molecular level. The main topics are: Liquids, Ideal and Regular Solutions, Polymer Solutions, and Electrolyte Solutions. This question sheet covers each of these topics. Suggested Reading 1. Dr Bob Thomas has an excellent website with sample questions, quizzes, and tutorial pages including interactive Java applets at: http://physchem.ox.ac.uk/~rkt/teaching.html. 2. P. W. Atkins, Physical Chemistry, any edition, OUP. 3. Murrell, J. N. and Jenkins, A. D., Properties of Liquids and Solutions, 2nd ed., Wiley, 1994. 4. D. J. Walton and J. P. Lorimer, Polymers, OUP Primer 85, 2000. 5. K. A. Dill and S. Bromberg, Molecular Driving Forces, Garland, 2002. Notation and Common Pitfalls Units and accuracy As ever, give your answers to an appropriate number of significant figures, with their correct units, and label the axes of your graphs clearly. This can be particularly tricky when deducing polymer molecular masses and virial coefficients, so take extra care here. Diagrams Parts of the Liquids and Solutions course are quite descriptive - illustrate your answers to descriptive questions with clear labelled diagrams wherever possible. Notes Make your own notes for use in revision on the following topics. Don t hand them in. Ideal liquid solutions; Raoult s law; Henry s law. Non-ideal liquid solutions; excess enthalpy and Gibb s energy of mixing; regular solutions; immiscibility. The radial distribution function. Polymer solutions; the random-walk model; self-avoidance; the Flory- Huggins model of polymer solution thermodynamics; the van t Hoff equation and its derivation; the polymer virial coefficient.
Electrolyte solutions; Debye-Hückel theory and its application; implications for solubility: salting-in and the common-ion effect. Questions 1. [ls2-i] (a) What is an ideal solution? (b) Show that the Gibbs free energy of mixing and entropy of mixing of two liquids to form an ideal solution of mole fractions x A and x B is mix G = nrt (x A ln x A + x B ln x B ), mix S = nr(x A ln x A + x B ln x B ), where n = n A + n B is the total amount of solution resulting from n A of liquid A and n B moles of liquid B. (c) What mole fractions give the maximum entropy of mixing? Explain why an ideal solution is always miscible. (d) What are the main features of a regular solution? Give expressions for mix H, mix S, and mix G for a regular solution. (e) Define the parameter β that appears in the theory of regular solutions in terms of the pairwise interactions between its components, H AA, H BB and H AB. (f) For a regular solution, the first and second derivatives of mix G m with respect to mole fraction vanish at the upper critical solution temperature (UCST), T c, above which all compositions of the solution are miscible. Writing mix G m in terms of x A alone: mix G m = RT (x A ln x A + (1 x A ) ln(1 x A )) + βx A (1 x A ), evaluate the second derivative at x A = 1 2 to show that T c = β 2R. (g) The excess free energy of mixing of an equimolar mixture of acetonitrile and CCl 4 is 1190 J mol 1 at 45 C. Assuming that the mixture behaves as a regular solution, calculate the UCST. 2. [ls2-d] How is the activity of a solution defined? Show by writing a A = γ A x A, that mix G has the same form as that of a regular solution if the activity coefficients satisfy the Margules equations: RT ln γ A = βx 2 B RT ln γ B = βx 2 A. Hence show that the partial pressure of A is given by p A = x A e β(1 xa)2 /RT p A. Determine the limits of p A as x A 1 and x A 0 and sketch p A vs. x A for a solution with β > 0, commenting on the limiting behaviour in relation to Raoult s law and Henry s law.
3. [ls2-f] (a) What is the osmotic pressure, Π, of a solution? Derive the van t Hoff equation, stating your assumptions: Π = [B]RT, where [B] = n B /V is the molar concentration of the solute, B. (b) In practice, the known concentration is in terms of the mass of polymer dissolved in a known volume of solvent, c, e.g. in g dm 3. If M is the molar mass of the polymer, show that Π c = RT M. (c) For dilute polymer solutions, the deviation from ideality is such that the osmotic pressure has to be written as a virial expansion: Π c = RT (1 + Bc +...) M The table below shows values of the osmotic pressure for a solution of a given sample of poly(styrene) at two different temperatures: c /kg m 3 1.0 2.0 3.0 5.0 T = 320 K Π/ N m 2 28.1 59.2 93.3 170.5 T = 330 K Π/ N m 2 30.1 65.8 107.0 205.7 By plotting a suitable graph, determine the molar mass of the polymer and values for the second virial coefficient, B, at the two temperatures. (d) Explain why B migh vary with temperature and, assuming that B varies linearly with temperature over the range 300 to 340 K, calculate the Flory θ-temperature for poly(styrene) in this solvent. 4. [ls2-a: Q2003.Adv.Q4.solutions] (a) The chemical potential of a monovalent electrolyte, M + X, in solution satisfies µ = µ + 2RT ln γ ± (m/m ) where ln γ ± is given by the Debye-Hückel limiting law as ln γ ± 2.303A I. (i) Define the term I, the ionic strength, in this equation. (ii) Given that A is proportional to (ɛt ) 3/2, where ɛ is the relative permittivity of the solvent and T is the temperature, show that I µ µ ideal C ɛ 3 T, where C is a positive constant.
(iii) Use the formula in part (ii) to deduce the circumstances for which the Debye limiting law will be most accurate. (b) Use your understanding of the physical basis of Debye-Hückel theory to explain why µ µ ideal : (i) is negative, (ii) increases in magnitude with increasing ionic strength, (iii) decreases in magnitude with increasing solvent permittivity, and (iv) decreases in magnitude with increasing temperature. (c) EMF measurements in water and in methanol at 298 K lead to the values of the mean activity coefficient, γ ±, of HCl shown in the following table. 5. [ls2-b] water methanol m = 0.002m 0.95 0.81 m = 0.005m 0.92 0.72 m = 0.008m 0.90 0.66 (i) Show that the Debye-Hückel limiting law is valid for both solvents at these molalities. (ii) Given that the relative permittivity of water is 78.6 at 298 K, calculate the relative permittivity of methanol. (iii) The Debye length λ, which is often taken as a measure of the radius of the ionic atmosphere surrounding an ion, is proportional to ɛ/i, where ɛ and I are as above. Given that λ is approximately 10 nm in an aqueous solution of monovalent electrolyte M + X of molality 0.001m, use your result from (c)(ii) to calculate the values of λ for HCl in water and in methanol at a molality of 0.008m. Comment on the results that you obtain. The Random Flight Model is the simplest model for describing the conformations of polymer chains in a solution. A polymer is treated as consisting of N segments, each of which is characterized by a vector l k of length l. In the Random Flight Model, the orientations of these segments are random, i.e. l j l k = 0 where j and k are two different segments. The vector R connecting the two ends of the polymer is R = N l k. k=1 (a) Show that the mean square distance between the two ends can be written R 2 = R R = Nl 2 (b) Calculate the predicted value, in the Random Flight Model, of the root-mean-square end-to-end distance, R 2 1/2 for polyethylene oxide of molecular weight 10 6 u, taking the segment to be (CH 2 CH 2 O) with a molecular weight of 44 u and a length of 0.36 nm.
(c) The value of R 2 1/2 inferred from light scattering is 110 nm. Comment on the Random Flight Model in the light of this value. In the previous question you assumed that each monomer unit was a random flight segment - how many monomer units actaully comprise a random flight segment? 6. [ls2-h] (a) What is meant by the radial distribution function for a liquid or a solid? Sketch and explain the form of the radial distribution functions you would expect for the following: (i) Solid Argon, (ii) Liquid Argon, (iii) An ideal gas. (b) The position of the first peak in the radial distribution funtion, r, the first ionization energy, I, and the polarizability, α, for the liquid states of Ne, Ar, and Xe are given in the table below. r /nm I /kj mol 1 α /10 3 nm 3 Ne 0.3017 2085 0.4 Ar 0.3725 1524 1.6 Xe 0.4346 1173 4.0 The magnitude of the interaction between pairs of these atoms is given by the London formula as U = 3α 2 I 4r 6. Explain the origin of this formula and calculate U for pairs of Ne, Ar, and Xe atoms at their distance of closest approach. (c) Taking the entropy of vaporization of Xe to be vap S (Xe) = 80 J K 1 mol 1 and its boiling point to be T b (Xe) = 166 K, calculate the enthalpy of vaporization of Xe, vap H (Xe). Also calculate vap U (Xe). Estimate the co-ordination number in liquid xenon. (d) Assuming the same co-ordination numbers, estimate the boiling points of Ar and Ne. [Hint: Trouton s rule]. Answers to Selected Questions 1. (g) 286.3 K 3. (d) 307.6 K 4. (c) (ii) 32.4 (iii) λ H2O(0.008m ) = 3.536 nm, λ MeOH (0.008m ) = 2.270 nm 5. (b) 54.3 nm, (c) s = 4.1 6. (c) vap H = 13.3 kj mol 1, vap U = 11.9 kj mol 1, z = 12; (d) 27 K, 89 K