Exam Molecular Thermodynamics (CH 3141) 06-11-2013 Write your name and student number on each paper sheet that you hand in!

Write the answers to problems 1 to 3 and to 4 on separate sheets, as they w i l l be coiTected by different people.

Symbols that are not defined here have the same meaning as in the lectures and/or Sandler's book and/or the formulary. Define symbols that you use, that were not defined in Sandler's book or in the lectures, or that may be ambiguous!

Wi th this exam you may use the 'FoiTnulary Molecular Thermodynamics'.

Wi th each question: read carefully, and think before you start scribbling!!!

Success!

1

Say o f the statements below i f they are true or false. Give a brief explanation (no explanation means no points)!

a) For a one-component system. -

b) To specify a microstate requires a small number o f variables, to specify a macrostate requires a large number o f variables.

c) A l l systems within a canonical ensemble are in the same macrostate.

d) The probability that a system in a Grandcanonical ensemble has N molecules is

e) A canonical ensemble contains many micro-canonical ensembles.

f) I n a system wi th pairwise additive interactions, changes in fi-ee energy wi th respect to an arbitrary order parameter are given by

()A 7 = 47iNp\g(r,AX(rydr

g) Monte Carlo simulations can be used directly to compute the thermal conductivity o f a f lu id .

h) In Monte Carlo simulations, ensemble averages should only be updated after accepted trial moves.

i) The Debye length increases wi th increasing electrolyte concentration,

j ) The total charge o f the double layer around an ion is zero.

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T

O Composition 1

In a temperature vs. composition phase diagram, the so-called binodal curve ( = coexistence curve) provides the compositions o f coexisting phases. Compositions in-between the binodal pomts at a certain temperature are not thermodynamically stable. Systems wi th such an overall composition w i l l phase separate into two coexisting phases. In the case of polymer solutions these two phases w i l l be a solvent-rich phase wi th a relatively low polymer concentration, and a polymer-rich phase wi th a relatively low solvent concentration.

a) What is larger (according to the Flory-Huggins theory and in reality), the volume fraction o f polymer in the solvent-rich phase or the volume fraction o f solvent in the polymer-rich phase? (NB. The above illustration is not for a polymer solution, so you caimot deduce the answer from the figure.)

b) Explain this by considering the transfer o f a polymer molecule f r o m a phase that mostly consists o f the same polymer to a phase that mostly consists o f solvent, and the transfer of a solvent molecule f rom a phase that mostly consists o f solvent to a phase that mostly consists o f polymer.

c) Give the expression for the Helmholtz energy o f mixing according to the Flory-Huggins theory.

d) Derive f rom the expression for the Helmholtz energy an expression for the chemical potential o f polymer as a ftinction o f the volume fraction o f polymer.

e) Indicate how at a given temperature (in the Flory-Huggins theoiy this is at a given value o f the Flory-Huggins parameter) the composition o f coexisting phases can be calculated f r o m the expressions o f the chemical potentials. (you do not have to perform the calculation).

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f ) The spinodal cui-ve (see figure) separates compositions that are thermodynamically metastable from those that are thermodynamically unstable. Metastable compositions are those between the binodal and the spinodal curves, unstable compositions are those between the two branches o f the spinodal.

The spinodal is where d^^jdij)^^ = 0 . Derive an expression for the spinodal according to the Flory-Huggins theory. Write the result in terms of as a function o f (f) .

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Keto-enol tautomei ism o f acetone.

k e t o enol

o

H

Acetone is an example of a molecule that can exist in two forms: the 'keto tautomer' and the 'enol tautomer' (the keto form is most abundant). These two forms are in a dynamic equilibrium in which they are interconveited all the time. Obviously, the internal single-molecule partition flinctions are different for the two forms. Call the internal partition fiinction o f the keto form q. , and o f the enol fo rm q. , .

a) Give an expression for the internal partition function q.^^ o f acetone in terms o f q.^^^^

and ? i , , , .

We consider a dilute vapour o f acetone, which may be considered as an ideal gas. b) Give the equilibrium constant for the tautomerisation reaction in terms o f g.^ ^ and

q.^^^^. You may use the general expression as given in the 'Formulary'.

p c) Give the expression, in terms o f q,^^^ ^ and q.^^^ ^, for the equilibrium constant Kp= ,

where P^ and P^ are the partial pressiu'es o f the enol and the keto tautomers.

Now consider the system as an ideal-gas mixture o f A^ ,^ 'keto mlecules' and 'enol molecules'. d) Write down the expression for the canonical partition function o f the mixture.

e) Deduce f r o m this expression for the partition f imction the expression for the Helmholtz energy o f the mixture.

f ) Explain in words, how the equilibrium values o f A^ ,^ and A^ ^ can be deduced f rom this expression for the Helmholtz energy. (You do not yet have to do the derivation)

g) Peiform the derivation described under ( f ) , and demonstrate that the result is consistent wi th what was found under (b).

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N B . write tlie answer to problem 4 on a separate sheet!

4 Consider a system of W hard spheres inside a volume V at temperature T.

(a) Show that the configurational integral at low densities can be expressed as Z{N, V,T)^V(V- c)(F - 2c)...(F - (TV - l )c) in which c is a constant.

(b) What is the physical meaning of the constant c ? Explain in 10 words maximum.

(c) Which approximations are made in (a]? Explain why these approximations are only valid at low densities. Briefly explain your answer.

(d) Explain that c=4ua^/3 in which a is the diameter of a hard sphere.

(e) Show that at low densities, lnZ(N,V,T) ^ NinV IV

pV [ f ] Use the equation of (e] to derive an equation for the compressibility Z = in

NkJ terms of the number density p=N/V

(g) The second virial coefficient for a system with pair interactions equals

B = 27r][\ - exp[-^(/..(/-)])/-V/-0

in which uij is the pair interaction potential. Show that the expression derived in (f) is in agreement with the value of B of this system.

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