Download Thermodynamics and Statistical Mechanics II - Home Exercise 1

Thermodynamics and Statistical Mechanics II - Home Exercise 1
1. Gibbs free energy
For an ideal gas, the Helmholtz free energy is given by F (τ, V, N ) = N τ [ln(nλ3T ) − 1].
(a) Find the chemical potential using µ(τ, V, N ) =
∂F
∂N τ,V
Show that if the indepen-
dent variables are N , V and τ - the chemical potential depends on N and that
F 6= N µ.
(b) Calculate Gibbs’ free energy out of the Helmholtz free energy for ideal gas (Hint:
try using the equation of state to express G as a function of N , P and τ ).
∂G
(c) Find the chemical potential using µ(τ, P, N ) = ∂N
. This expression for the
τ,p
chemical potential may be obtained from the expression given in (a) using a change
of the independent variables.
(d) Show that for this example µ(τ, P, N ) = µ(τ, P ) , demonstrating the fact that if
the independent variables are N , P and τ the chemical potential is independent
of N .
Also show that G(τ.P, N ) = N µ(τ, P ).
Thermal Expansion
(a) Prove that thermal expansion is a purely unharmonic-potential related phenomenon.


∞
, for r < σ
(b) Given the potential u(~r) =
, calculate the thermal
h
i

−ε exp − r−σ
, for r > σ
ξ
expansion for r ≈ σ . You can do this by following these steps, but any other
(correct) way is also fine:
• Plot the potential. Understand it’s shape and characteristics.
• Linearize it for r ≈ σ.
• Calculate the period of bound motion.
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• Calculate r̄ ≡
1
T
RT
r(t0 )dt0 , where r(t0 ) is the solution for the equations of
0
motion. Assume v0 = 0.
• Show that r̄ ∼ E ∼ τ . Discuss how, if at all, the specifics of the potential
affect the thermal expansion phenomena.
2. Thermal Properties of Water
Water density is ρ ≈ 1 cmg 3 and molar mass is MW = 18 amu.
(a) Molecular polarizability α is defined as the tendency of a molecule to be polarized
~ Find the Molecular polarizability in
as a response to an electric field: p~ = αE.
~ assuming that d~E · E
~ τ.
finite temperature in an electric field E,
(b) Estimate the electric susceptibility of water χ. Electric susceptibility is related
to molecular polarizability α by χ =
nα
.
ε0
Compare the result to the table value
(≈ 80). Would you consider the agreement to be good?
(c) Estimate (numerically) the parameters of Lennard-Jones potential for water molecules
(The VDW attraction developed in class can be used.)
(d) As most of other substances water expands with increasing temperature (for water
this is true in fact only above 4◦ C). Estimate the volume expansion coefficient
of water defined as γ =
∆V
.
V ∆T
(Hint: express γ using ln(< r >) and use thermal
expansion developed in class )
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