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NAME:
Section Number: 10
CHEMISTRY 443, Fall, 2014 (14F)
Examination 2, November 5, 2014
Answer each question in the space provided; use back of page if extra space is needed. Answer questions so the grader can
READILY understand your work; only work on the exam sheet will be considered. Clearly indicate your answer and all
indications of your logic in arriving at your answer. Please answer the question asked and refrain from providing irrelevant
comments or information. Write answers, where appropriate, with reasonable numbers of significant figures. You may use only
the "Student Handbook," a calculator, and a straight edge.
DO NOT WRITE
THIS SPACE
IN
p. 1_______/25
p. 2_______/15
p. 3_______/20
p. 4_______/20
p. 5_______/20
=============
p. 6 _______/5
(Extra credit)
=============
TOTAL PTS
/100
1 Problem 1 (25 Points) Lattice Models, again………… ! …. ☺
Condensed phases (liquid, solids) have historically been treated as lattice models. In this case,
consider a bulk liquid as a lattice site model. There are N particles, and the particles interact with a
nearest-neighbor pair-wise interaction that leads to an energy contribution of ‘w’ to the total internal
energy, U, of the system from a pair of particles interacting. In this lattice model, the number of
nearest neighbors for any particle in the bulk (neglect the edges) is ‘z’. For an N-particle system, the
total internal energy from interactions is U interaction = 12 Nzw . The per-particle interaction contribution to
U
U is thus interaction
= 12 zw . It is important to note that this type of interaction implicitly builds in a
N
reference state where the particles do not interact (i.e. ‘large’ separation, or ‘dilute’ conditions); in
such a reference state, Uinteraction = 0 (much like in the case of an ideal gas). In the lattice, dynamic
effects are taken into account€since the lattice is an ‘average’ representation of the positions of
particles.
All lattice sites are filled, and the particles are indistinguishable.
€
A. (5 Points). Using the definition of the Helmholtz Free Energy, please provide an appropriate
equation for this thermodynamic quantity for this lattice model liquid based on the given information
and in terms of the variables presented in the above discussion and associated figure.
A(T,V,N) ≡ U − TS = U interaction − TS = U interaction − T ( k B ln(W ))
(
(
= U interaction − T k B ln
N!
(N!)(0!)
)) = U
interaction
− T ( k B ln(1)) = U interaction
A(T,V,N) = 12 Nzw
€
B. (5 Points). Using the definition of the chemical potential, please provide an appropriate equation
for this thermodynamic quantity.
µliquid = (
)
∂A
∂N T ,V
=
(
∂ ( 12 Nzw)
∂N
)
T ,V
= 12 zw
2 C (5 Points). Using the results from Parts A and B and your knowledge of the chemical potential of a
pure ideal gas, determine a relation between the vapor pressure of the liquid and the interaction
strength, ‘w’. You are free to define the reference state ideal gas chemical potential to be 0.
Since this question asks about vapor pressure, we think of vapor - liquid equilibrium.
At conditions of vapor - liquid equilibrium, we know that vapor and liquid chemical
potentials are equal.
µ liquid = µ vapor
We have computed the liquid chemical potential (for lattice model
liquid) in Part B. For ideal gas, we know the behavior of chemical potential as a function of
pressure :
µ ideal,vapor = µ ref + RT ln
( )
p
p ref
Equating chemical potentials gives :
1
2
zw = µ ref + RT ln
( )
p
p ref
Feeling free to set the reference state ideal gas chemical potential to 0 :
⎛ p ⎞
zw = RT ln⎜ ref ⎟
⎝ p ⎠
Rearranging :
1
2
p= p
ref
zw
2RT
e
€
3 D1. (5 Points). The vapor pressure of water is 22 mmHg at T = 300K and 760 mmHg at T = 373K.
Estimate the vaporization enthalpy using appropriate analytics.
This is a Clausius-Clapeyron analysis. Assume vaporization enthalpy is constant over the
temperature and pressure ranges probed, and liquid is transferred to an ideal vapor phase:
ln
( )
p2
p1
=
−Δh vap ⎛ 1 1 ⎞
⎜ − ⎟
R ⎝ T2 T1 ⎠
Δh vap = −R ln
(
⎛ 1 1 ⎞ −1
⎜ − ⎟
⎝ T2 T1 ⎠
( )
p2
p1
= − 8.314 J mol
−1
K
−1
)
−1
⎛ 760mmHg ⎞⎛ 1
1 ⎞
ln⎜
−
⎟
⎟⎜
⎝ 22mmHg ⎠⎝ 373K 300K ⎠
Δh vap = 45.14kJ mol −1
€
D2. (5 Points). What is an estimate of the interaction strength ‘w’ considering water to be modeled as
a lattice fluid? Consider z = 4 for a lattice model of water. Refer to your analyses of Parts A through E
for inspiration.
S1: In this approach, we compare the differential forms of the vapor pressure expressions we have
derived and have at our disposal:
From Clausius - Clapeyron Analysis
⎛ Δh vap ⎞⎛ 1 ⎞
d(ln P) = ⎜
⎟⎜ 2 ⎟dT
⎝ R ⎠⎝ T ⎠
From Analysis of Part 1C :
⎛ −zw ⎞⎛ 1 ⎞
d(ln P) = ⎜
⎟⎜ ⎟dT
⎝ 2R ⎠⎝ T 2 ⎠
Comparing the expressions, we have :
⎛ −zw ⎞ ⎛ Δh vap ⎞
⎜
⎟ = ⎜
⎟
⎝ 2R ⎠ ⎝ R ⎠
⎛ −2Δh vap ⎞
w = ⎜
⎟
z
⎝
⎠
⎛ −2(45.14kJ mol −1 ) ⎞
−1
−1
= ⎜
⎟ = −22.57kJ mol = −5.4kcal mol
4
⎝
⎠
€
4 S2. If we think of vaporization as a molecule moving from the condensed phase (liquid) to the ideal
vapor (where there are no interactions between particles), then the vaporization enthalpy
corresponds to the loss of interaction per particle on a mole basis. This is simply the interaction
contribution of a single particle to the potential energy of the liquid, and so:
⎛ −zw ⎞
⎟
2 ⎠
(Δhvap ) = U final −U init = 0 −U init = ⎜⎝
⎛ −2Δh vap ⎞
w = ⎜
⎟
z
⎝
⎠
⎛ −2(45.14kJ mol −1 ) ⎞
−1
−1
= ⎜
⎟ = −22.57kJ mol = −5.4kcal mol
4
⎝
⎠
€
5 Problem 2 (15 Points) Multiple Choice? Why yes……......
For the following, match the statement in the left-hand side column with the
most appropriate answer(s) from the right-hand side column.
I. Under the Ehrenfest classification scheme, a first-order phase transition is
one in which the Gibbs Free Energy changes continuously (smoothly) and
properties related to first derivatives of G change discontinuously. The pure
substance liquid-vapor phase transition is first-order, true or false?
__________B______
II. How many intensive degrees of freedom are available for a system
comprised of a binary liquid solution in equilibrium with its vapor (both species
present in both phases)? ___P_____.
A. ∞ (diverges)
B, true
C. false
D. greater than 0
E. equal to 0
III. Negative absolute temperatures arise under particular conditions, such as:
_____________N,Q___________
F. 1
G.
∑ν µ
j
j
=0
j
IV. What is the theoretical value of Cp(T) at the liquid-vapor phase transition
temperature? _______A______
V. The Legendre Transform of U(S,V,N) to a function of T,V,N results in what
thermodynamic potential? ______S____________
VI. The maximum non-expansion work related to a transformation (under
conditions of equilibrium along the process) of a chemical system between two
equilibrium thermodynamic states is equal to ___T_____. Consider the process
to occur with external temperature and pressure constraints (values are
constant).
VII. The Gibbs Free Energy of mixing of two ideal gases, A and B, is identically
zero (due to ideality of the vapor phase species). True or False? _C________
VIII. For a binary mixture of ideal gases at constant temperature and pressure,
what mole fraction ratio would maximize the entropy of the system?
________F________
IX. What constraint arises when there is a reversible (equilibrium) chemical
reaction? ______G______.
X. In a two-state system (i.e., 2 energy levels, ‘high’ and ‘low’) with 10 noninteracting particles what value would represent the ratio of occupancies of the
two states (or equivalently, what value would represent the ratio of ‘high’ to
‘low’ labeled particles?) _____F_______
6 H. entropy
I. isenthalpic
J. 0 J/K
K. 3
L. 4
M. internal energy
N. relatively few
quantum states
available/accessible
at a particular
temperature
O. T dQ
P. 2
Q. population
inversion of
quantum states
R. q is isentropic
S. Helmholtz Free
Energy
T. Gibbs Free
Energy
Problem 3 (20 Points) Put me to sleep already………..
A common anesthetic drug molecule is halothane (2-bromo-2-chloro-1,1,1-trifluoroethane). Its mode
of action is presumed to involve partitioning from water (state A) into lipid bilayer membranes (state
B). The values of the equilibrium constant representing this reversible partitioning, determined at two
different pressures, are the following:
P (atm)
ln K
T (Kelvin)
0
7.84
300
280
7.6
300
What is the corresponding change in molar volume of the bilayer/water/drug system as a result of the
pressure change from p1 = 0 atm to p2 = 280 atm?
Solution:
ΔGrxn = −RT ln K
At constant T (data given at T = 300K = constant)
d ln K =
−d(ΔGrxn )T −1
=
(ΔVrxn dP − ΔSrxn dT )
RT
RT
−1
(ΔVrxn dP )
RT
⎛∂ ln K ⎞
= −RT ⎜
⎟
⎝ ∂P ⎠T
d ln K =
ΔVrxn
Assume change in V is constant; use a constant slope approximation
from data given :
⎛ Δ ln K ⎞
ΔVrxn ≈ −(8.314 JmolK )(300K )⎜
⎟
⎝ ΔP ⎠T
⎛ (7.6 − 7.84) ⎞
= −(0.0821 L atm mol −1 K −1 )(300K )⎜
⎟
⎝ (280atm − 0atm) ⎠T
= 0.021L / mol
= 21cm 3 / mol
€
7 Problem 4 (20 Points) What are Legendre Transforms Good For Anyway.......
Here we will consider the utility of Legendre Transforms in determining useful thermodynamic
potentials that reach extrema under conditions of chemical equilibrium with constant pressure,
temperature, and chemical potential. That is, in order to use, as a control ‘knob’, a species
chemical potential, we need to determine what thermodynamic potential depends on the chemical
potential of interest. This is not an irrelevant question to ask, since biochemical reactions often occur
under conditions of constant pH, where the concentration of proton is controlled (specified). Knowing
that chemical potentials, in appropriate limits can be expressed in terms of concentrations, constant
pH scenario is a case of constant proton chemical potential. Let’s consider a simple equilibrium
(reversible) reaction:
A+B ⇔C
A. (1 Points) Because of chemical equilibrium, what constraint arises involving species chemical
potentials (recall the discussion of ideal gas reaction equilibrium, though in this exercise, we are
considering a general situation). €
∑ν µ
j
j
=0
j
B. (1 Points) Without taking into account your constraint, write down the total differential of the
extensive Gibbs Free Energy (knowing that Gibbs Free Energy is a function of T, P, and the amounts
of species, A, B, C). Be careful and provide the complete expression. Use the nomenclature:
ni=moles of species ‘i’ and ‘µi’ is chemical potential of species ‘i’.
dG(T,P,{n j }) = VdP − SdT + µ A dnA + µ B dnB + µ C dnC
C. (5 Points) Using your constraint from Part A, eliminate the chemical potential of species C from
your results of Part B. You can make the substitution:
€
n ,A = n A + nC ; dn ,A = dn A + dnC ; n ,B = n B + nC ; dn ,B = dn B + dnC .
These substitutions are another way to say that A is ‘distributed’ between free A and ‘bound’ A that
has transformed to species C; similarly for species B.
€
Chemical reaction equilibrium constraint :
µ A + µ B = µC
Thus :
dG(T,P,{n j }) = VdP − SdT + µ A dnA + µ B dnB + µ C dnC
= VdP − SdT + µ A dnA + µ B dnB + (µ A + µ B )dnC
= VdP − SdT + µ A (dnA + dnC ) + µ B (dnB + dnC )
= VdP − SdT + µ A dnA' + µ B dnB'
D. (3 Points) Based on your results of Part C, what are the variables that G really depends on?
Excluding temperature and pressure, there are 2 (two).
€
8 dG = VdP − SdT + µ A dnA' + µ B dnB'
Thus :
dnA' and dnB'
€
E. (7 Points) Now, we want to control the chemical potential of species B in our reaction----that is, we
want to specify the value of µB. Provide a Legendre transform of the Gibbs Free Energy (the
thermodynamic potential you have been working with so far in Parts A- D) to another thermodynamic
potential, G’, that would leave you with µB as one of the independent variables. Show that indeed the
new potential you generated is in part dependent on µB.
⎛ ∂G ⎞ '
G = G − ⎜ ' ⎟nB
⎝∂nB ⎠
'
new potential defined by Legendre transform
To show that the new potential is indeed dependent on µB:
⎛ ∂G ⎞ '
G = G − ⎜ ' ⎟nB = G − µnB' new potential defined by Legendre transform
⎝∂nB ⎠
'
€
dG'= dG − d(µnB' ) = VdP − SdT + µ A dnA' + µ B dnB' - µ B dnB' − nB' dµ B
dG'= VdP − SdT + µ A dnA' − nB' dµ B
The last expression shows that G' is dependent on µ B
F. (3 Points) At constant temperature, pressure, and µB, what does the total differential of G’ become
in terms of µA and dna’? That is, what is ( dG')T ,P,µ ? What is the integrated form of G’ and what
B
€
variables does G’ depend on?
dG'= µ A dnA'
Integrated form :
€
G'= µ A nA'
€
9 Problem 5 (20 Points) It’s bubbalicious……..
For the binary system acetonitrile(1)/nitromethane(2), liquid-vapor equilibrium is adequately
represented through Raoult’s expression. Antoine equations for saturation pressures as a function of
temperature (in Celsius) are:
ln ( P1sat ) = 14.2724 −
2945.47
2972.64
; ln ( P2sat ) = 14.2043−
t + 224.0
t + 209.0
What is the bubble temperature at P = 70 kPa and initial composition of 0.6 mole fraction of
acetonitrile at initial temperature t = 90 Celsius.
This is best done via iterating over the total pressure. The liquid phase composition at the bubble
temperature is x1=0.6 and x2=0.4. We plot in the last column the relative error in our predicted total
Pressure. The last temperature gives a fairly low relative error, and we state that value as our
solution.
P total = x1 P1sat + x 2 P2sat
relative error =
total
Ppredicted
P total
P
t
(x1)(P1sat)
(x2)(P2sat)
70
100
106.7636578
39.16623773
70
90
79.92563735
28.39092264
70
70
70
70
70
80
70
75
75.5
76
58.70520086
42.22316204
49.92411448
50.75194804
51.59068036
20.12690971
13.92082071
16.79308198
17.10485805
17.42129796
70
70
70
76.5
76.4
76.42
52.44041316
52.26958166
52.30371247
17.7424528
17.67784218
17.69074907
€
10 diff/P
1.084712794
0.547379428
0.126173008
0.197943104
0.046897193
0.030617056
0.014114595
0.002612371
0.000751088
7.91208E-05
Problem 6. Extra Credit (5 Points) Time to jam……….
In this figure, the y-axis represents the vapor pressure of the solution of an ethanol-water. Straight
lines are the vapor phase partial pressures based on Raoult’s equation. Treating the vapor as ideal
and still considering the liquid solution as non-ideal, what thermodynamic quantity related to the
species in the liquid is also being plotted on the y-axis. Please show your work in order to receive
credit.
Solution:
On the y-axis, we are plotting partial pressures of the species in the ideal gas vapor mixture. If we
consider the fugacities of the species in vapor and liquid, we come to the following conclusion:
fˆi vapor = fˆ liquid
fˆi vapor = φˆivapor pi = (1)yi P total
fugacity coefficiet = 1 for ideal vapor
Thus
fˆ liquid = pi
The y-axis plots the liquid phase fugacity of species ‘i’.
€
11 Potentially Useful Information
Stirling’s Approximation:
ln(N!) = (N ln N) − N
N →∞
NA R = kB = Boltzmann Constant
€
Number of ways to place N indistinguishable objects into M bins:
W=
M!
N!(M − N )!
S = kB ln (W) (isolated system, statistical mechanical form of entropy for lattice model)
µ i = fi
dµ = RTdp
fˆi vapor = φˆivapor pi = yi P total
ref
µ (T,P{x}) = µ still
+ RT ln(γci )
dµ = RTdp
ref
du = PRdT + µ still
+∇•
fˆ vapor = fˆ liquid (at mixture vapor - liquid equilibrium)
i
ref
µ (T,P{x}) = µ still
+ RT ln(φci )
€
12 NAME:
Section Number: 10
CHEMISTRY 443, Fall, 2014 (14F)
Examination 1, October 1, 2014
Answer each question in the space provided; use back of page if extra space is needed. Answer questions so the grader can
READILY understand your work; only work on the exam sheet will be considered. Clearly indicate your answer and all
indications of your logic in arriving at your answer. Please answer the question asked and refrain from providing irrelevant
comments or information. Write answers, where appropriate, with reasonable numbers of significant figures. You may use only
the "Student Handbook," a calculator, and a straight edge.
DO NOT WRITE
THIS SPACE
IN
p. 1_______/25
p. 2_______/15
p. 3_______/20
p. 4_______/20
p. 5_______/20
=============
p. 6 _______/5
(Extra credit)
=============
TOTAL PTS
/100
13 
1/--pages
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