EFN307G: Varmafræði og Inngangur að Safneðlisfræði 2013

EFN307G: Varmafræði og Inngangur að Safneðlisfræði
2013
Lokapróf, 5. desember, 9:00-12:00
Leyfð hjálpargögn eru reiknivél og tvær A4 blaðsíður af jöfnum sem hver nemandi hefur
skrifað sjálfur (‘cheat sheet’). Prófið samanstendur af 5 spurningum sem eru mislangar
en hafa sama vægi og gilda samtals 100 punkta.
Problem 1: (20 pts)
Consider the expansion of an ideal gas from a pressure pi to a pressure pf .
(a) Make a sketch of p vs. V if the gas is monatomic and the process is carried out in
a reversible way with the system (i) immersed in a heat bath, and (ii) isolated from the
environment. The two curves should be on the same graph and marked clearly.
(b) Sketch the entropy, σ, vs. V for the two processes (i) and (ii) and mark the curves
clearly.
(c) Give an expression for the change in entropy of the gas if the expansion is irreversible
with a rapid change in external pressure from pi to pf and the system is immersed in
a heat bath so that the final temperature eventually becomes the same as the initial
temperature. Explain your reasoning.
(d) Give an expression for the change in volume in the expansion if the gas consists of
atoms with two internal states that differ in energy by . Also, add curves to the graph
in part (a) for such a gas and mark clearly.
(e) Give an expression for the change in entropy in the expansion if the gas consists of
atoms with two internal states spaced apart by energy . Also, add curves to the graph
in part (b) for such a gas and mark clearly.
Problem 2: (20 pts)
Consider a water solution that is in thermodynamic equilibrium with the atmosphere
at 1 bar pressure and room temperature.
(a) Explain how the chemical potential of oxygen in the solution is related to the chemical
potential of oxygen in the atmosphere and give an expression in terms of the partial
pressure of O2 in the air (which can be assumed to be an ideal gas).
(b) When the partial pressure of O2 is changed slightly while maintaining constant partial
pressure of other components in the atmosphere, the chemical potential changes and the
number of O2 molecules dissolved in the solution, NO2 , changes. Write symbolically the
partial derivative expressing the change in the number of O2 molecules in the solution in
this process. Make sure you specify the variables that are kept fixed.
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(c) The O2 molecules can adsorb on the surface of a metal electrode that is dipped into
the solution. Let M be the number of sites for O2 molecules on the surface and be the
binding energy of each molecule to the surface with respect to gas phase. Assume the
adsorbed molecules do not interact with each other. Write an expression for the grand
partition function for this system and simplify as much as you can. (It can be convenient
to introduce here λ = exp(µ/τ ) and make use of the binomial expansion).
(d) Give an expression for the average number of O2 molecules that will be bound to the
surface of the electrode at a given temperature as a function of the partial pressure of
oxygen, and demonstrate that your expression gives the right limit as the partial pressure
of oxygen becomes very large.
Problem 3: (20 pts)
Consider the enthalpy, H, a state function defined as
H = U + pV
(a) Derive an expression for the total differential of H using the combined first and
second law of thermodynamics for reversible processes in a closed system and state what
the natural variables of H are (simplify as much as possible).
(b) Give a mathematical expression for the total differential of H in terms of a linear
combination of the partial derivatives with respect to the natural variables and compare
with your result in part (a) to evaluate each of the partial derivatives.
(c) Use a Maxwell relationship for H to obtain an expression for the change in volume
when the entropy changes at constant pressure.
(d) Obtain an expression for the change in H as temperature is changed at constant
pressure.
(e) The enthalpy of van der Waals gas is to first order in the parameters a and b
H(N, τ, p) ≈ 25 N τ + N bp − 2N ap/τ. Use this and the results from part (d) to derive
an expression for the constant pressure heat capacity of van der Waals gas.
Problem 4: (20 pts)
Consider the formation of a critical nucleus of a crystal from a melt. Assume the
shape of the nucleus is cubic. Let the interfacial energy per surface area be denoted as γ
and neglect the free energy increase associated with edges and corners of the cube. The
pressure is maintained constant.
(a) Give an expression for the change in the free energy of the system when a cubic
nucleus of width L forms.
(b) Sketch the change in free energy as a function of L.
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(c) Give an expression for the size, Lcr , of the critical nucleus.
(d) Comment on the feasibility of forming a cubic nucleus as compared with a spherical
one.
Problem 5: (20 pts)
Consider an ideal gas of atoms confined to a trap that can be described by a parabolic
potential well in three dimensions with a minimum located at (x, y, z) = (0, 0, 0). Along
the x-direction is an escape channel which has a parabolic shape in the y- and z-directions
with the same curvature as the well. The equation for the potential energy surface is
1 2 2
k (x + y 2 + z 2 ),
x ≤ x0 ;
2 0
(1)
Vext (x, y, z) =
x > x0 .
Vb − 21 kb2 (x − xb )2 + 12 k02 y 2 + 12 k02 z 2 ,
where k0 and kb are curvatures with kb > k0 , Vb is the height of the energy barrier located
at xb , and x0 is the cross-over point between the parabola for the well and the parabola
for the escape channel along the x-direction. The number of gas atoms in the trap is N ,
the temperature τ and the mass of each atom m.
(a) Use harmonic transition state theory to estimate the rate constant for the escape
of an atom and simplify the expression as much as possible. Assume the atoms can be
described as classical particles.
(b) Use your results in (a) to estimate the rate of effusion of the gas from the trap.
(c) Give an expression for the time an atom spends on average in the trap.
(d) Consider now the effect of quantum mechanical zero point energy and revise your
estimate from part (a).
(e) Revise your estimate in part (a) using full transition state theory rather than the
harmonic approximation and explain your choice of the transition state dividing surface.
The expression for the rate constant can be written in terms of definitive integrals where
the integration limits are clearly specified.
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