Thermodynamics of
Multicomponent Systems
Laboratory Manual
Materials 3C04
Materials Science and Engineering
McMaster University
2014–2015
Materials 3C04
List of Experiments
1.
Heat Capacity Ratio For Gases: An experiment similar to the Joule-Thomson
experiment. Measurement of the heat capacity ratio from pressure measurements.
2.
Determination of Heat Capacity and Entropy Changes for a variety of metals. A
simple experiment to determine the heat capacity of various metals.
3.
Binary Phase Diagrams.
Metallographic analysis will be used to identify
compositions in the Al-Cu phase diagram.
4.
Single Electrode Potentials. The single electrode potentials for a variety of different
metals are measured versus a saturated calomel electrode.
Requirements for Laboratory Experiments
1.
SAFETY GLASSES ARE MANDATORY. PLEASE WEAR THEM AT ALL
TIMES DURING THE EXPERIMENT.
2.
Read and familiarize yourself with the lab before the experimentation period begins.
All the experiments require less than three hours to perform.
3.
The reports should be clear concise and free from grammatical and spelling errors.
The reports should not contain long quotes or paraphrases from textbooks or this
manual.
4.
The lab reports should be typed, printed and stapled.
5.
The report document should include headings of Objective, Experimental Method,
(including one paragraph on how you understand the theory, in your own words),
Results and Calculations, Discussion, Conclusion and any Appendix.
Any
references made should be credited. The written reports must follow this style.
6.
The experimental method should explain clearly what was done in the experiment.
7.
The results and discussion section of the report should show all calculations and
detailed calculations for important aspects of the experiment. Any graphs should be
produced in a professional manner such that they are easily read. Considerations for
errors and sources of error should be taken into account.
8.
The discussion should analyze the results and report on any of the questions
introduced in the laboratory manual.
9.
Conclusions should include your final remarks on what was important in the
experiment.
10.
During the lab, your attentiveness and method of performing the experiment will
also be graded, and added to your final grades.
Safety
At the end of each experiment there is a section on safety. This section notes
hazardous materials used in the experiment and precautions that should be taken to avoid
injury or illness. Care should be taken when handling glassware. Compressed gas cylinders
are dangerous and care should be taken when using them. Use your best judgment and care
when in the laboratory.
Your safety in these laboratories is of primary importance. If you feel unsafe tell
your laboratory demonstrator and the course instructor.
Experiment #1
Heat Capacity Ratios For Gases
Introduction
This experiment investigates the heat capacity ratio for several gases by using
a) sudden, irreversible expansion of a gas followed by
b) reversible heating.
The heat capacity ratio cp/cv can be calculated by:
1.
allowing a gas to quickly and irreversibly expand from P1 to P2 under adiabatic
conditions.
2.
allowing the gas to return to its initial temperature slowly.
From thermodynamics we know that for an adiabatic irreversible expansion
ΔU = Qirr + Wirr = QirrPΔV and Q = 0 for adiabatic conditions. For a monatomic ideal gas,
U is equal to 3/2 nRT. This indicates that for a change in internal energy resulting from a
sudden change in volume under adiabatic conditions the temperature must also change.
Using the first law of thermodynamics in partial molar form:
du = qrev + wrev = 0Pdv
(1)
Also for adiabatic conditions and an ideal gas:
du = cv dT
(2)
P = RT/v
(3)
Equating (1) and (2) and substituting (3):
cv dT = RT/v dv
Assuming cv is independent of temperature and integrating (3) between state 1 and state 2:
cv ln (T2/T1) = R ln (v2/v1)
(4)
For an ideal gas:
P2v2/T2 = P1v1/T1
or T2/T1 = P2v2/P1V1
(5)
Substituting (5) into (4)
cv ln (P2v2/P1V1 ) = R ln (v2/v1)
or
ln (P2/P1) + ln (v2/v1) = R/cv ln (v2/v1)
ln (P2/P1) =  (1+R/cv) ln (v2/v1)
Also:
(6)
cpcv = R
(7)
Substituting (7) into (6)
ln (P2/P1) = (cp/cv) ln (v2/v1)
ln (P2/P1) = (γ) ln (v2/v1)
(8)
where γ is the heat capacity ratio.
A slow heating after the initial expansion mimics a reversible heating of a gas. For a
system at constant volume after heating to ambient temperature (state 3):
v2 = v3
(9)
T3 = T1
(10)
P1v1/T1 = P3v3/T3
(11)
P1v1/T1 = P3v2/T1
(12)
v2/v1 = P1/P3
(13)
and
For an ideal gas
Substituting (10) into (11)
or
Substituting (13) into (8)
ln (P2/P1) = (γ) ln (P1/P3) = (γ) ln (P3/P1)
γ = (ln P1ln P2) / (ln P1ln P3)
(14)
This expression for the heat capacity ratio is solely in terms of three pressures in a system
which undergoes an adiabatic expansion and a reversible heating.
Experimental Method
This experiment is conducted in a carboy as shown in the figure where A is the gas
inlet for gases lighter than air and B is the gas inlet for gases heavier than air. The
manometer used should be filled with a suitable liquid such as water (to avoid use of
mercury). To convert to mm Hg, multiply the manometer reading by the ratio of the
densities (1.00 g/cm3 for water/13.55 g/cm3 for mercury). The readings should be added to
the current corrected barometric pressure.
The corrected barometric pressure (CBP) is given by:
CBP = (1/13.6) x Barometer reading (in cm) + 76 cm of Hg.
A : Lighter than air
Clamp
To manometer
Clamp
Clamp
B: Heavier than air
1) FLUSHING
1) To fill the carboy with gas, clamp the line to the manometer and open clamps A
and B.
2) Allow the gas to fill from line A or B depending on the density of the gas in the
carboy prior to filling.
3) Allow 10 minutes of flow at 6 l/mn to flush the carboy of the old gas and fill it
with the new gas.
2) FILLING WITH GAS
1) After the 10 minutes, reduce the gas flow into the carboy by partially closing off
the inlet line (A or B).
2) Carefully open the clamp to the manometer to avoid blowing all the liquid out of
the manometer.
3) Close the outlet clamp (A or B).
4) When the manometer reads about 60 cm of water close the inlet clamp (A or B).
5) Allow 10 to 15 minutes for the manometer reading to stabilize (i.e. the gases
must reach room temperature). Take a reading of the pressure, convert it to mm
Hg
and
add
it
to
the
corrected
barometric
P1 = 1/13.55  barometric pressure + 76 cm of Hg.
pressure
to
get
3) ADIABATIC PROCESS
1) Remove the rubber stopper in the top of the carboy and replace it as quickly as
possible (a sudden irreversible adiabatic expansion), make sure the rubber
stopper is tight. This allows for an expansion such that the pressure in the carboy
will be the current corrected barometric pressure or P2.
2) Again allow 10 to 15 minutes for the pressure to stabilize and take a reading
(reversible heating) P3.
The experiment should be repeated three times for three different gases: nitrogen,
carbon dioxide, and argon.
Discussion
Calculate the heat capacity ratio for each run using the equations given earlier and
compare the results with values in the literature. Discuss the validity of the thermodynamic
assumptions made and note any problems in the experimental method that could lead to
errors.
Safety Precautions
Mercury should not be used as the liquid in the manometer due to the possibility of
liquid being blown out of the manometer. Extreme care should be taken when filling the
carboy and opening and closing valves to avoid high pressures in the carboy.
Experiment #2
Determination of Heat Capacity and Entropy Changes
Introduction
This experiment attempts to determine the constant pressure heat capacity of a metal
and uses this value to determine entropy changes in the system.
If a warm metal is placed into liquid nitrogen at its boiling temperature, liquid
nitrogen will boil off until the metal reaches the temperature of the liquid nitrogen. If the
system is well insulated, the amount of nitrogen boiled off will be dependent upon the
temperature change in the metal. Thermodynamically the total enthalpy of the system will
be constant. That is to say the enthalpy change from nitrogen boiling will equal the enthalpy
change in the metal.
ΔHnitrogen + ΔHmetal = 0
(1)
The enthalpy change of the nitrogen boiling is equal to the number of moles evaporated
times the molar enthalpy of evaporation:
H nitrogen  n H v  ( WN 2 / M N 2 ) H v
(2)
w = weight of nitrogen boiled off
M = molar mass of nitrogen
The enthalpy change in the metal is a function of the heat capacity and is given by:
ΔHmetal = n cp dT
if cp is independent of T
ΔHmetal = (Wmetal/Mmetal) cp (TfinalTinitial)
(3)
Substituting (2) and (3) into (1)
[(W/M) ΔHv]nitrogen = [(W/M) cp (TfinalTinitial)]metal
1/cp = (Wmetal / Wnitrogen) (Mnitrogen/Mmetal) (TfinalTinitial) / ΔHv
The heat capacity of any metal can be calculated knowing its molar mass and the amount of
material present by placing it in liquid nitrogen and measuring the amount of nitrogen boiled
off.
In this experiment it is also possible to measure the entropy change in the system.
Entropy cannot be measured directly and as a result it must be calculated from measurements
of volume, mass, pressure and temperature. The system in the case of this experiment is
assumed to be the liquid nitrogen, the metal, and the evaporated nitrogen gas. This leads to
five different entropy changes in the system:
ΔS1 is the entropy of evaporation of the liquid nitrogen at its boiling point,
ΔS2 is for the cooling of the lead sample,
ΔS3 is the entropy of warming the nitrogen gas to room temperature,
ΔS4 is the entropy from the heat flow from the surroundings,
ΔSnet is the total entropy change in the system.
ΔS1 is given by:
ΔS1 = 1/ Tboiling  dq = qp / Tboiling = n ΔHv / Tboiling
ΔS2 is the entropy change in the metal and is an isobaric process given by:
S2  n metal  (c p metal / T )dT
Assuming that cp is constant:
S 2  n metal c p metal ln(Tfinal / Tinitial )
ΔS3 is an isobaric heating of the nitrogen gas:
S3  n N 2
 (c p
N2
/T )dT
where
c p N  30.6285 + 0.00477T J/mol-K
2
ΔS4 is the entropy associated with the heat supplied by the surroundings to the nitrogen gas.
The heat supplied to the nitrogen gas is equal to the heat absorbed by the gas but opposite in
sign.
S4   n N 2 /Tfinal  (c p N )dT
2
again where
c p N  30.6285 + 0.00477T J/mol-K
2
The total or net change in the entropy of the system is:
ΔSnet = ΔS1 + ΔS2 + ΔS3 + ΔS4
Tinitial = Tboiling = 77.32K and Tfinal = room temperature
Experimental Method
The vessel for this experiment is composed of two styrofoam cups glued together
such that their rims meet all the way around with the bottom cut out of one of the cups.
1) The first styrofoam cup is filled 2/3 full of liquid nitrogen.
2) Place the filled vessel on a balance and measure the decrease in mass of liquid
nitrogen at 30 second intervals over a 5 minute period. This establishes the rate of nitrogen
loss due to heating by the surroundings.
3) Measure the mass of the metal.
4) Using the set of wood splints slowly lower the metal into the liquid nitrogen and
record time t1 and total weight w1.
5) Wait until the liquid nitrogen ceases boiling violently, record time t2 and total
weight w2. (To be sure of getting an accurate result, wait for a sufficiently long
period of time to get a large value of t2). During this period (t2 - t1), the nitrogen
loss (w1w2) is caused by the absorption of heat from the surroundings and heat
from the metal.
6) Continue taking mass readings for 10 minutes at 30 second intervals.
7) Take measurements for Pb, Cu, Ni, and Al.
Styrofoam cups
Liquid Nitrogen
Discussion
Plot the data points before the metal is put into the nitrogen and the data points after
liquid nitrogen ceased boiling violently, calculate the best fit slopes for both these lines and
average the values, this will give us a more accurate rate of nitrogen evaporation due to
heating by the surroundings. Using this rate and the weight loss (w1w2), as well as the
amount of time (t2t1), calculate the amount of nitrogen boiled off solely due to the addition
of the metal and calculate Cp.
Compare the Cp values to those found in the literature. Calculate the various entropy
changes and discuss the results.
Discuss the assumptions made in the derivations in the introduction. What entropy
term was left out of the net entropy. Are there any sources of error in the experimental
method?
Safety Precautions
Liquid nitrogen can cause frostbite if not handled with care. Be sure not to touch the cold
metals or liquid nitrogen directly. The styrofoam cups should be handled with gloves.
Experiment #3
Binary Phase Diagrams
Introduction
The bismuth-tin phase diagram is an example of a simple eutectic system. These
elements exhibit complete liquid solubility and limited solid solubility. Our objective is to
determine the bismuth-tin phase diagram. The aluminum-copper phase diagram is a more
complex system of eutectoids, peritectics and intermetallic compounds. The microstructure
of the alloys formed in this system depends on the size and distribution of phases. The
second objective of this lab is to correlate the microstructure with the phase diagram.
Definition of Terms
1) Liquidus line: The line on the phase diagram, above which the liquid phase is
stable.
2) Solidus line: The line on the phase diagram, below which the solid phase is
stable.
3) Solvus line: The line on the phase diagram separating a solid solution from a
two-phase region.
4) Phase: On the phase diagram, a phase is defined as a region of varying
temperature, pressure or composition, but having a common structure. A phase
may be a pure element, a liquid or a solid solution, a stoichiometric or a nonstoichiometric compound.
5) Component: A component in a phase diagram is an end-member of fixed
composition but a variable structure. Any phase transformation of a component
compound must be congruent.
6) Degrees of freedom: The number of variables (pressure, temperature and
composition) which can be changed independently without changing the state of
the phase or phases in equilibrium.
The Al-Cu System
The Al-Cu system provides a good example of the correlation between phase
equilibria and the thermal history and mechanical properties of an alloy. It is the shape of
the solvus boundary which enables the 2000-series Al-Cu alloys to be heat treatable (Figure
2). The heat treatment is a sequence of “solution treat, quench, age”. The strategy of
introducing small particles of another phase is known as age hardening or precipitation
hardening. During the ageing treatment, the strength and hardness increase up to a critical
point and further ageing causes softening. The highest strength is always associated with
precipitates of very small size and spacing.
Figure 2: Binary Alloy Phase Diagram
Experimental Method
For this experiment, a series of Cu-Al alloys have been prepared by melting in an
induction furnace, casting into a mold and cooling. The samples differ in composition and
heat treatment. Some have been quenched to “freeze in” the high temperature equilibrium
structure and others have been aged to give a structure part way between quenched and fully
equilibrated.
treatments:
The alloys are characterized by the following compositions and heat
Al - 4.5% Cu (wt%)
etchant: Keller’s reagent
a)
annealed: heated at 540C for 2 hours, quenched in water
b)
properly aged: annealed as above, heated at 190C for 8 hours
c)
over-aged: annealed as above, heated at 190C for 10 days
Al - 32.5% Cu eutectic
d)
etchant: Keller’s reagent
annealed: heated at 500 for 2 days, quenched in water
Al - 88% Cu eutectoid
etchant: 5 g ferric chloride, 2 ml HCl, 93 ml ethanol
e)
β phase: heated at 635C for 2 days, quenched in water
f)
α+γ mixture: heated at 535C for 2 days, quenched in water
Keller’s reagent: 1.0 ml HF, 1.5 ml HCl, 2.5 ml HNO3, 95 ml H2O
The TA will distribute one mounted sample to each student to polish, etch and
photograph (or sketch). The micrographs will be used by the whole lab group for analysis.
You will also measure the hardness of each sample.
Discussion
Examine all the micrographs produced by your lab group and identify samples a-f
from their microstructures. Explain the features of the microstructure in terms of the heat
treatment for each composition. Try to identify the phases in the micrograph. Use the lever
rule to determine the amount of each phase in 2-phase samples. Account for the relative
hardness of samples a-d based on their microstructure. What range of composition of Al-Cu
alloys is effectively heat-treatable by age-hardening? The solubility of Cu at the eutectic
point is 5.62 wt%.
Safety
When etching the samples, use protective gloves and exercise care. Note: The use
of HF requires documented training. Your TA will etch all samples requiring HF.
References
1.
Avner, Sidney H., Introduction to Physical Metallurgy, McGraw-Hill, New York,
1974 (chapter on phase diagrams).
2.
Callister, William D., Materials Science and Engineering, An Introduction, 6th Edition,
John Wiley and Sons, New York, 2003 (section on heat treatment).
3.
ASM Handbook, Metallography, Vol. 9, 1991.
Experiment #4
Single Electrode Potentials
Introduction
In this experiment the standard electrode potentials for several different metals are
measured. A system composed of a metal in a solution of its ions connected to a saturated
calomel electrode is used for this experiment.
It is possible to measure Gibbs energy changes very accurately and directly for a
redox reaction by allowing this reaction to occur in an electrochemical cell and measuring
the voltage generated.
In an electrochemical cell the oxidation and reduction steps occur at different
electrodes and are separated by an ionic solution. The reactions that occur at the electrodes
are known as half cell reactions. These two reactions are connected by the ionic solution
between the electrodes and a conducting wire between them. The potential developed
between the two electrodes is dependent upon:
1.
The nature of the electrodes (what the electrodes are made of)
2.
The concentration of the ionic solutions used.
3.
The manner in which the potential is measured.
A table of standard electrode potentials indicates that each metal has its own distinct
potential when measured against a hydrogen electrode. The potential of the hydrogen
electrode by convention is set to 0.0V. The magnitude and sign of the potential determines a
metal’s tendency to release electrons. A metal with a highly negative standard potential
easily loses its electrons whereas a metal with a highly positive potential holds its electrons
tightly. In the table of standard potentials which follows (Table 1) the potentials are all
written for oxidation or loss of electrons.
The table is very easy to use if the electrochemical cell in question has the standard
hydrogen electrode for one half cell. However, it is dangerous and difficult to set up the
hydrogen half-cell. Instead of using the hydrogen cell, this experiment will use a saturated
calomel electrode. The calomel half cell is a mercury-mercurous chloride half cell and is
usually denoted KClsat. | Hg2Cl2, Hg. This half cell shows some of the conventions that are
necessary for electrochemistry.
Electrode Reaction
Potential
Li = Li+ + e
3.045
K = K+ + e
2.925
Ca = Ca2+ + 2e
2.87
Na = Na+ + e
2.714
Mg = Mg2+ + 2e
2.37
Al = Al3+ + 3e
1.66
Zn = Zn2+ + 2e
.763
2+

0.44
2+

0.136
2+

0.126
3+

-0.036
+
-
0.00
-
0.153
Cu = Cu2+ + 2e
0.337
Ag = Ag+ + e
0.7991
Hg = Hg2+ + 2e
0.854
Fe = Fe + 2e
Sn = Sn + 2e
Pb = Pb + 2e
Fe = Fe + 3e
H2 = 2H + 2e
+
2+
Cu = Cu + e
Table 1: Table of Standard Potentials
The electrochemical cell is denoted by two half reactions such that the electron flow
from left to right produces a positive cell potential. In this experiment the electrochemical
cell should be denoted with the metal on the left such that all the metals lose electrons.
M | M2+ _ KClsat. | Hg2Cl2, Hg
The | in the system indicates a change in phase in the system and the _ denotes a salt bridge.
The half cell reactions in this system are:
M - 2e-  M2+
Hg2Cl2 + 2e  2 Hg + 2 Cl
The potential of this cell is calculated as the difference between the single electrode potential
on the right and that on the left.
A positive voltage for this cell indicates electron flow left to right indicating that the
reactions proceed spontaneously in the way they are written. This corresponds to a decrease
in the free energy of the system and the Nernst equation can be used:
ΔG = zFE
where z is the number of electrons being exchanged, F is the Faraday constant
(96485 coulombs) and E is the cell potential.
As noted earlier the potential of the cell can vary not only with the metals used but
the concentration of the solution. The standard cell reaction potential is achieved when
components have an activity of 1. However, it is very difficult to produce a situation where
the activity is 1 and as a result the Nernst equation is modified for the activity and recast in
terms of the potential E:
E  Eo 
RT
n Q
zF
where Eo is the standard cell potential and Q is the activity quotient:
Q = a gG a hh / a bB a cC for the equation bB + cC  gG + hH
For this experiment the total potential of the cell is given by:
E = E (ocalomel electrode) + E j  E ( M _ M 2 ) 
RT
ln a M 2+
zF
where Ej is the potential for the liquid junction and a 2M is the activity of the metal ion in
solution. For the purpose of this lab:
E(oc alomel electrode) + E j = .2444 , or E j  0
Activities may be written in the form of coefficients and molality:
ai = γimi
Where ai is the activity of component i, γi is the activity coefficient for component i and mi is
the concentration of component i in molality. The following table shows the activity
coefficients in the .01 and .1 molality range for the metals used in this lab.
Electrolyte
Activity Coefficients
0.01 m
0.1 m
Copper Sulfate
.4
.16
Lead Nitrate
.69
.37
Silver Nitrate
.90
.731
Zinc Sulfate
.387
.15
Table 2: Activity coefficients for some metal ions in solution
A potentiometer is used to measure the voltage in the cell because a voltmeter draws
a small amount of current and would lead to inaccurate results. A potentiometer balances the
voltage in the cell with an equal and opposite applied voltage. Thus there is no current flow
in the cell.
Experimental Method
Three different metals should be tested in this part of the experiment. The solutions
provided will be 1 molal stock and should be diluted to 0.1 molal by pipetting 10 ml into a
100 ml volumetric flask and filling the remainder with distilled water. The cell is set up as
shown in Figure 1. Your lab demonstrator will help familiarize you with the potentiometer
and how it is used. Record the potential being sure to note the sign of the voltage.
You should:

Read carefully the lab manual before you come to do the experiment

Have your safety glasses all the time

Clean all glassware and components with distilled water prior to beginning each
analysis

Be sure that the wires are firmly connected before you record the measured values.
Figure 1 shows the cell setup that is used for this experiment.
Figure 1: Experimental Setup
Redox reaction
Consider the red-ox reaction,
cathodic:
Mn+ + ne-
Anodic :
M
M
Mn+ + ne-
The magnitude of the potential is the same for both reactions. However, the sign of the
potential that you read in the voltmeter depends on whether you are reading the potential for
the cathodic reaction or the anodic reaction. Be sure to note which reaction you measured
when writing the report.
Power from an Electrochemical Cell
In the second part of the experiment, a commercial button cell battery is to be tested.
In order to obtain uniform temperature, the cell is kept inside a sealed bag submerged in a
bath of water with the wires connected to the positive and negative terminals.

Add ice to the bath and wait until the temperature reaches 0C

Connect the wires across a 10 k, 1000 , and 100  resistor with the button cell and
in parallel with the potentiometer.

Take readings of the potentiometer and record them.

Slowly increase the temperature of the bath using a hot plate at a rate of 2C/min
until you reach 60C, taking readings every 20C.

Plot the data as potential of the cell versus temperature. Explain what you see.
Results and Discussion
Calculate the standard potentials for each of the metals tested. Be sure to adjust for
the activity of the metallic ions. Calculate the free energy for each of the electrochemical
cells. Discuss what assumptions have been made in the derivation of the cell potential
equation. Derive the equation for potential including the activity correction term from the
Nernst equation. Explain the effect of temperature on the potential of the dry cell.
Safety
Some of the solutions used are toxic. Avoid contact with the solutions of metallic
ions. Care should be taken with all the glassware and avoid touching the lead electrode. The
calomel electrode is also highly toxic.