NAME: _________________________
MSE 308: Thermodynamics of Materials
Department of Materials Science & Engineering
Boise State University
Spring 2005
Practice Midterm Exam 1
March, 2005
Problem
1.
2.
3.
4.
5.
6.
7.
8
Grand
Total:
Total Points
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1
Points Obtained
NAME: _________________________
1. Write down the mathematical equations for the 1st and 2nd laws of
thermodynamics. Describe each law with one sentence each. (5 points)
2. Write down the equation describing the combination of the 1st and 2nd laws of
thermodynamics (otherwise known as the Fundamental Equation of
Thermodynamics - FEOT). Is this equation an exact or inexact differential? (4
points)
3. Write down the functionality, i.e., Z = Z(x,y), of the FEOT. (2 points)
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2
4. The notion of a "reversible" process is a fiction in the real world. What makes this
concept, which at first glance would appear to be only of academic interest, so
useful in applying thermodynamics to real-world "reversible" processes? Limit
your answer to no more than 3 succinct sentences. (3 points)
3
5. Derive a relationship that describes the dependence of Gibb's Free Energy upon
temperature and volume.
6. Using your answer from developing a relationship that describes the dependence
of Gibb's Free Energy upon temperature and volume, derive an expression if you
consider the system you are dealing with is an ideal gas with n not equal to one.
4
7. Using a graph of volume and temperature to assist you, explain with words and
equations (do not solve them) how you would use your answer in problem 6 to
find the Gibb's Free Energy change from some initial state to some final state. Be
succinct in your explanation. (7 points)
5
8. Derive the differential equation for the change in volume when the entropy of n
moles of an unknown gas increases to twice its original value and the Gibb's free
energy decrease by half of its initial value. Circle all answers including the
coefficients you have derived and your derived general equation. (13 points)
6
9. Use the differential you obtained in the previous problem to derive the differential
3
assuming the gas is an ideal gas. Remember that Cv for an ideal gas is nR .
2
Simplify your answer as much as possible. Circle your answer. (7 points). (20
points)
7
10. Draw a plot showing the process path you would take to find ∆V. Label each
process path. (? points)
8
11. Using your plot and process paths, derive ∆V (i.e., perform the integrations). State
each process path for each integration. (? points)
9
12. The relationship between Cv and Cp given is by:
TV α 2
CV = CP −
β
Using this relationship, derive the relationship between Cv and Cp for an ideal gas
for which n is not equal to one.
10
13. The rather mundane material Borium (Bo) has the following properties:
a. Cp = a + b T
b. α = thermal expansion coefficient
c. β = compressibility
d. V = molar volume
e. Bo is solid over all temperatures of interest to us here.
Express the pressure required to keep the volume of one mole of Bo constant
when the temperature in increased from 298K to T, in terms of T and the materials
properties.
Assumptions: α and β are very small numbers.
11
14. Without using toolbox 3, derive the following state variables as a function of T &
P:
a. volume
b. entropy
15. For the following reaction, answer the following questions using the information
provided to assist you:
Al + O2 = Al2O3
Cp
[J/(mol K)]
a
b x 103
c x 10-5
Temp. Range
[K]
Al(s)
Al(l)
Al(g)
O2(g)
Al2O3(s)
Al2O3(l)
20.67
31.76
20.77
29.96
106.6
192.46
12.38
0
0
4.18
17.78
9.52×10-11
0
0
0
-1.67
-28.53
5.6×10-3
298-937(Tm)
934-2793(Tb)
2793-4000
298-3000
298-2325(Tm)
2325-4000
∆H k298K
∆S k298K
→2
∆H 1Xform
[kJ/mol]
-1675.7
-1620.57
[J/(mol K)]
28.30
39.55
164.6
205.1
50.9
67.24
[kJ/mol]
10.7
330.0
107.5
unknown
a. Balance the reaction equation. (3 points)
b. Draw the ∆Srxn diagram. Label all important aspects of the diagram. (5 pts)
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c. Write the symbolic equation (no numbers) down for
∑S
products
. Integrals should
∑S
reactants
. Integrals should
have temperature intervals. (6 pts)
d. Write the symbolic equation (no numbers) down for
have temperature intervals. (6 pts)
13
16. Write down the general expression for ∆S rxn
17. Describe how you would determine ∆Grxn and write down the general equation
for ∆Grxn .
14
18. Using the Clapeyron equation, derive the Clausius-Clapeyron equation. State all
assumptions that you are making.
19. Draw a P vs T phase diagram given the following
α → β → δ → γ → L → ν with α being the lowest temperature phase.
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system:
20. For the system, α → β → δ → γ → L → ν , the following are the vapor pressures
for the condensed-vapor phase equilibria:
A
ln pα = α + Bα ln T + Cα
T
A
ln pβ = β + Bβ ln T + Cβ
T
A
ln pδ = δ + Bδ ln T + Cδ
T
Aγ
ln pγ =
+ Bγ ln T + Cγ
T
A
ln pL = L + BL ln T + CL
T
Using these equations, do the following:
a. Describe a method and derive equations to find the triple point pressure
and temperature for the δ − γ −ν triple point.
b. Describe a method and derive equations to find the boiling point pressure
and temperature for the δ − γ −ν triple point.
16
c. Describe a method and derive equations to find the ∆H β →δ .
d. Describe a method and derive equations to find ∆C pβ →δ .
17
21. On April 15th of last year, you performed experiments to determine the heat of
vaporization of the element Taxinium. From your measurements, you have
acquired the data tabulated in table 2. From this data, calculate the heat of
vaporization (∆Hvap). Assume the ∆Hvap is not a function of temperature and thus
has Arrhenius behavior. Provide a rough sketch and label all important aspects
including the slope. Show all work. (10 points):
Table 2: Vaporization data for Taxinium.
Temperature (K)
Pressure (atm)
600
2.73x10-7
1000
2.05x10-2
1300
1
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22. Using the definition of S and Ω, where:
S = kb ln Ω and
n!
,
Ω= r
Π i = 0 ni !
use Sterling’s approximation, given by:
ln x ! ≈ x ln x − x ,
to derive:
r
n
S ≈ − k B ∑ ni ln i .
n
i =0
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