The Heat Capacity of Metals
R. A. Shigeishi Carleton University Ottawa. Ont K I S 5B6, Canada
A physical chemistry experiment
Physical chemistry experiments illustrating t h e principles of classical thermodvnamics, Quantum mechanics, spectroscopy, and kinetics & abundant i n the literature b u t there are few simple experiments associated with introductory statistical th&modynamics. O n e of these, outlined b y Shoemaker a n d Garland i n their text on physical chemistry experiments ( 1 ) is the determination of the heat caoacitv of metals over the temperature range 77-300 K. T h e present article describes what a r e felt t o b e substantial i m ~ r o v e m e n tisn t h e oriainal design with t h e result t h a t heat csparities of metals are.;otttinelv ohtained within ICuc, of literature values. T h e e x ~ e r i m e n i a t Carleton University is performed by third-year honor students a n d usually requires three laboratory periods. Detailed discussions of t h e Einstein a n d Debye theory of h e a t capacities of solids a r e available i n most texts o n statistical thermodynamics (2,3) a n d hence only a brief sketch is given here. According to the Einstein model, each atom in an ordered array of N atoms of a solid vibrates at a single frequency harmonically and independently about its lattice point in a force field created by the remaining atoms. Assuming that the solid is isotropic the vibrational motion of the N atoms is equivalent to that of 3N - 6 = 3N (for large N) independent one dimensional harmonic oscillators each with a freouencv v = (%n) , . . . l. a. ) where k is the force constant and m the mass. In addition tothc \,ih;ntionalenergy rheaold hssa latticeen. ergy denoted by Noo,? where o,, is the interaction energy 1,etwt.m a central atom and the remaining atoms. Thus the energy per vibrating atom is 6 = 3(u %) hv $012. The partition function for the system of 3N wcillatars occupying a volume V a t temperature T is given hv
+
+
where q is the one dimensional oscillator partition fundion given hy
- e-'~/kT
"=a
The energy levels of the 1D harmonic oscillator are given by r, = (U
+ $1
h"
U
= 0,l.Z..
.
I t can he shown that e-R/2T
q
=1--e-~/~
where 0 =hulk has the dimensions of temperature. Thus
and the heat capacity
co= & [ k r 2 (-)MnQ DT
- -
Limiting the number of modes to he 3N and the maximum frequency to u, gives
So"" g(") dv
= 3N
The partition function in this case is Q
= ,-Nmd2kT
J
e-hr/ZkT
-
1 e-h.lk~~(u)du
and the heat capacity C, = 3Nk [ ~ D ( u )-
N,V
]
-
While the Einstein theory is correct in predicting that C, 3R at 0 as T 0 it is only in semiquantitative room temperature and C, agreement with data. The main deficiency of the theory lies in the simplifying assumption that the atoms vibrate independently with a singlefrequency. In a real solid theatoms are strongly coupled and there is, essentially,a continuous distribution of vibration frequencies. Taking these facts into account Debye proposed a model which treats the solid as a continuum and the vibrations as elastic waves with an average velocity C and frequency distribution 12nV "2 d") = 3
I
3u where u = 0oIT = hv,/kT, 0o = Debye temperature D(u) = Dehye Function =
Q(N,V,T) = e-N"aJ"T
q=
Figure 1. Universal heat capacity curve tor the Debye and Einstein theories
," JUG
If the Debye function is calculated for various values of BD/T a plot of Cd3Nk versus BD/T may be drawn. This is a universal curve from which the Debye temperature of a metal may be found if the heat capacity at a particular temperature is known. Conversely if RD is know for a metal the heat capacity at any temperature may be found. The curve is shown in Figure 1. Experimental Apparatus The calorimeter and the ancillary vacuum line are shown in Figure 2. Since it is essential to minimize conduction heat transfer from the sample to the cell walls the pumping system must be capable of reaching 5 X 10-%rr or lower. These pressures are routinely obtained with a simple Hg diffusion pump backed by an oil rotary pump and the use of liquid nitrogen cold traps to prevent mercury and oil vapor from entering the high vacuum region of the calorimeter. The pressure is measured with a Bayard-Alpert type of ionization gauge and an inexpensive controller model 831 manufactured by Norton Vacuum Equipment a division of NRC. The metal sample along with the heating and thermocouple arrangement is shown in greater detail in Figure 3. Twometal slabs of dimension 4 X 1X s i n . sandwichinga heating tape of constantan wire and silk fiber (Saraain Thurnepen Co. in Basel, Switzerland) via four small screws ensures that most of the heat from the heating tape is transferred to the metal. To avoid electrical contact between sample and tspe a thin coating of varnish is applied to the inside face of each slab. Temperature averaging over the sample is accomplished by the use of four chromel-alumel thermocouples in parallel placed in small holes drilled into the metal a t different locations. In order to make the heater and thermocouple contacts to the sample inside the vacuum calorimeter and eliminate the use of adhesives and cements, a four pin metal through glass feedthrough (Ceramaseal Inc., New Lehanon Centre, N.Y. 12126) was glass blown Volume 56, Number 1, January 1979 / 59
onto a ground glass male joint. Two of the pins were chrome1and the other two alumel. One set of pins was used for the chromel-alumel thermocouple and the remaining set for the heating Leads. The wires connecting the latter t o the heating tape were also used t o suspend the sample in the cell. The inside wall of the glass cell was silvered, aside from a venieal strip to allow for visual inspection of the sample, in order to reduce possible radiation heat transfer from the sample. A simple electrical circuit t o heat the sample and t o measure the electrical energy input and the sample temperature is shown in Figure 4. The dummy heater whose resistance is set equal to that of the sample heater is used to set the current and voltage prior to the actual heating of the sample. The voltage across the standard resistor (-10 ohms) provides an accurate measure of the current. The dc power source is a constant voltage, constant current unit with a maximum rating of 34 V and 1.5 A (Power Mate Corp., Hackensack, N.J., Model BP 34D). The use of s digital voltmeter greatly facilitates the rapid monitoring of voltages across the various resistances and the thermocouple potential. LIQUID NITPOGEN TRAPS
HELIUM STORAGE BULB
Procedure and Calculations The resistance of the sample heater and the standard resistor and the mass of the sample are assumed to be known. After ensuring that the pressure may be reduced to the torr range a Dewar with the appropriate coolant is placed around the cell. The stopcock to the diffusion pump is closed and a small pressure of helium is admitted t o the cell from the storaee " bulb. The helium orovides raoid heat transfer from samole to bath and an eauilibrium temnerature is soon ~.~ estahli,hcd. \Vhm the sample is at ur nrnr the desired temprroturr the helium ir pumped our and the prt.-.sure again restored to rhr lo-" torr range. The temperature of the aample is then monitored for 15 min to determine any temperature drift caused by residual heat transfer to or from the bath. Meanwhile, the current and voltage scroas the dummy heater are set to provide -50 t o 100 mA when the heater resistance is -200 a. After measuring the temperature drift, power is applied to the sample for a set time during which the voltage across the sample heater and the standard resistance is periodically checked and the sample temperature measured as a function of time. When the changein sample temperature is about 5-8 K the power is switched off and the sample temperature again monitored for 15 min to measure the post heating drift. An example of the plot of thermocouple emf versus time is shown in Figure 5. The actual temperature rise due to electrical energy input is found by extrapolating the drift before and after heating to the midpoint of the heating period ( I 1. Three temoeratures a t which the heat canacitv . . can be measured are easily a c c c s ~ i b lmum ~ , temperature, 0ry.lct-acctone IYSK, and liqu~dnitrogen Y K . hleasurrment~at intermediate tcmprrarures rd ~initiilllyRI the may tw carrwd out hyhearmg the r v a ~ u ~ t sample ~
~
~~~~~
~~
.
~~~
~
~
.~
VOLTMETER
EEOTHROUGH
HELIUM
CHROMEL- ALUMEL THERMOCOUPLE
HEATER LUDS
SAMPLE HEATER
HEATER
0-1000n
-2oon SAMPLE
w Figure 2. Calorimeter cell and vacuum system
0 -
OFF
Figure 4. Electrical circuit t~ heating the sample and measuring thermocouple eml.
HE4TlNGTAPE
FOUR CHROMAL-ALUMEL THERMOCOUPLES IN PARALLEL
I
20
I
TIME rmin.,
Figure 3. Arrangement of sample heater and thermocouples. 60 I Journal of ChemicalEducation
Figm 5. Te-mure versm time plot shawing pre-heat and post-heat & I for aluminum sample initially at T = 176 K.
hath temperature 77 or 195 K)to the desired temperature and then initiating the monitoring and heating procedure. In those cases where the sample and hath temperature differsubstantially heat loss and temperature drift will he larger. The heat capacity of a metal receives contributionsnot only from lattice vibrations which have already been discussed hut also from the conduction electrons (4). The later contribution, however, is very small and only significant at temperatures well below 77K where the lattice factor is also very small. Thus the heat capacity determined in this experiment may he equated to the lattice heat capacity. Since measurements were carried out at constant pressure the relevant heat capacity is that at constant pressure. This is given by
Heat CapaCBes 01 Aluminum and Copper Temperature K
J m o W 1 K-'
Jr
C. n
o ~ e ~ ~ ~ ~
Aluminum a0 = 396 K 22.6 23.4 23.8 23.0 22.7 23.2 27.6 21.6 20.1 19.6 20.6 18.9 17.8 19.1 18.6 16.1 13.5 12.1
where V is the voltage across the sample heater, I the current, t the heating time in seconds, m the mass of the sample, and A T the temperature rise. The constant volume heat capacity C, differs from the experimentally determined C, according the equation 9TVcrz
c,=c,---
C-XP J molect K-1
B
where V is the rnolal wdume, Ru the wlume coefficient of expansion, and .1 the isothermal compresihillty 15).Smre the temperature dependence of 3a and Bare often unknown the second term is replaced by an empirical term ATCD2where A is a constant evaluated from the room temperature prape&iesof themetal. Thevalues of the various constants for aluminum and copper are available in references (5-
23.4 23.4 23.2 23.2 22.2 22.1 21.8 21.7 20.2 20.0 19.5 18.8 17.8 17.3 17.3 16.9 14.2 13.2
7).
Dlscusslon Some sample results ohtained by students for aluminum and copper are shown in the table. These are compared with experimental values ohtained by Giauque and Meads (7) and ,,C ohtained from the universal curve using the accepted literature values of OD = 398 K and 315 K for aluminum and copper, respectively (2). It can he seen that most of the student values lie within 10% of the literature and universal curve values. Students also calculated the Dehye temperature values using their heat capacity values and the universal curve. While values ohtained from heat capacity measurements a t low temperatures were within 10% of the accepted values, a t high temperature deviations in most cases ranged up to 40%. The latter is understandable in view of the flatness of the universal curve nearing room temperature. I n this range small errors r in OD. In fact if the experiin C , result in much l a ~ g e errors mentalvalue exceeds the theoretical limit of C, = 3R aDehye temperature cannot he ohtained.
Acknowledgment Theauthor wishes to thank thestudentsof Chemistry 315' for the use of their data. Mr. Karl IXedrich for thediagrams, and the referee for helpful suggestions. Literature Cited (1) Shoemaker, D. P., and Garland. C. W., "Experiments in Physical Chemistry: McGraw-Hill. New York, 1962. p. 349. (2) Hill, T. L.. "InIraduction to Statistical Thermdynamies." Addison-Wesley, Reading M-chusett*. 1960, p. 86. (3) b p p , D.,"Ststistical Meeh8nies,*Holt,Rinehart and Winston, lnc., New York. 1972. "r - "7 .. (4) Allis. W . P., and Hcrlin, M. A,, "Thermadmamies and Statistical Mechanics." MeGraa-Hill. Tomnto. 1952. p. 104. (5) Zemansky, M. W.,"HestandThemd~amics,"5th Ed.. McGraw.Hil1. NovYork. 1957. p. 327. (6) "American Institute of Physics Handbook: MeGraar-Hill, New York, 1957. p. 4 (7) Giauque, W . F.,and Mead8.P. F., J A m e r Chsm Soc.. 63,1897 (1941).
Volume 56, Number 1, January 1979 1 61
