Stanley J. Gill and Earl M. West University o f Colorado Boulder
The Indirect Determination of Heat Capacity, C,, of a Liquid
TIlc iudirecl mr.tlrod of heat capacity determination illustrates scvcral hasic thermodynalnic concepts. The method is based ou a thennodynamic ~ l e which , shows that the combination of t,wo partial derivative measurcmmts pcnnits the determination of a third.' The use of a Maxwell relat,ion pcrmit,s the substitution of an experiment~ally accessible partial derivative for an experimentally difficult derivative. In this way thc heat capacity of a liquid ran hc dctermined without making n y direct heit metlsurcmeuts. Burlew2 has used the rncthod in prcrisc determinations of liauids.
dG
=
+ (%)*d~
(%)nd~l ar
(5)
aud idcn(,ifyS and T' in eqn. (4) as -S
=
($1
(6)
:md Ir
=
(::)
(7)
%-r,.
For a statc fuuclior~such as C sccoud derivatives iuvolving T aud p arc equal:
Principle of Method
A brief derivatiou of lhc ucressary ttherulodynamie equation for the method may he giver1 by considering entropy, S, of a purc liquid as a funct,ionof tcmpcraturc, T , and pressure, p. The differer~t,ialof thc nitropy for variations of tcmpcrature ant1 pressure is
If we hold the eutropy of the system coustanl, theu eqn. (1)becomes
where we recognize the couditior~of constaut, S, upon the differentials dT and dp. Equation (2) states that the combination of two part,inular experiment,^, described by the partial differentialsi n this equation, must equal the result of a third experiment. One of the partial differential terms in eqn. (2) is related to the heat capacity at coustant pressure, C, by
The partial derivative ( b S / b ~expresses )~ the resull of an experiment whcrc entropy is added to the system and a change in pressure is made to occur such that the temperature is held constant. This type of an expcriment is difficult to execute. The result of this diicult experiment is equivalent to a particularly simple and familiar experiment. The relation is given by one of Maxwell's relations, derived from the differential for the Gibhs free energy, G, dG
=
-SdT
+ Vdp
(4)
where V is the volume of the system. Since G is a function of T and p, 'LEWIS, C.N., A N D RAND.^., hl., "Thermody1~1~mic~," McGrm-Hill Publishing Co.,New York, 1923,p. 1Ri. z B ~ ~J.~S.,~J . w Am., Chem. Soc., 62,681, 6!10, 6% (1!l40).
and Lhcrcfom usiug cqns. (6) and (7) :
This equaliou states the equality of two particular experimental results. Thc experiment described by ( b V / b T ) , involvcs making a measurement of the increase of volume with increase of temperature at a fixed prcssore, an cxpcriment easily performed. From eqn. (2) we can now substitute the results of cqns. (9) and (4) and find:
A nieasurernent of ( d p / b T ) , ( b V / a T ) , and 1' enables determination of C, witahout ever making a direct measurement of heat. The partial derivative ( b p / b T ) , requires measuring the temperature increase upon an incrcasc of pressure while the system is held a t a fixed valuc of entropy. The latter condition can be accomplished provided the change in pressure and its attending change in tempcraturc is conducted in a reversible manncr which is also adiabatic. These conditions can be closely realized by suddenly imposing a pressure change andmeasuring the increase in temper* turc within a time interval beforc any heat exchange can occur with the surroundings. The Experiment
The apparatus (Fig. l), consists of a simple mercury column pressurizihg system, thc sample container with thermocouple, and associated electronic equipment. Temperature changes, at pressures of 1-2 atm Hg applied to a liquid such as benzene, are 0.02°C0.050C with a resulting thermocouple emf output of 1-3 fiv. These small emf's require a low noise and high gain amplifier syslem for accurate measurement on a convenient read-out device. These requirements have been met by ulilizing a Beclcman Model 14 DC breaker Volume 43, Number 10, October 1966
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557
amplifier for signal amplification and a n L and N Speedomax H strip chart recorder with 1 mv sensitivity for readout purposes. The amplifier htw an input impedance of 20 ohms and up to 80 DB gain with extremely low intemal noise. I t also supplies bucking voltages for balancing out residual voltages and micro-
FIGURE 2 0 : ,
2b:
FIGURE
TheImOmeter
r
#14 Rubber Stopper
UPPER RESERVOIR
Hg COLUMN
RESERVOIR &SS'Y IN BATH
AMPLIFIER
RECORDER
Figure 1. General Scheme of Equipment for Indirect Heot Capacity Meowremant.
volt test signals for calibrating external readout devices. The amplifier input cable shield is grounded internally in the amplifier and, ideally, should not be grounded elsewhere. However, we shield the thermocouple leads to the feed-through from the reservoir assembly, which is immersed in the water bath, thus creating a path of approximately 10 I< ohm to eart,hground. This resistance is relatively high, however, and its effect is not evident. Ground currents are further reduced by operating the amplifier and recorder ac inputs above ground. To reduce thermal emf's, we use a special solder having thermal properties approaching that of copper t o attach the copper thermocouple leads to the ampliier input cable and the thermocouple to its mounting posts. The pressure system consists of two meter sticks mounted end to end on a vertical support with a mercury reservoir att,arhed to a sliding device. Teflon tubing of 2 mm od is attached to the bottom of the mercury container and the other end slipped over a short section of stainless st,eeltubing cemented into the capillary tube on the bottom of the test liquid reservoir. A4ultiple windings of small gauge wire wrapped tightly around this portion of tubing provide a tight seal against mercury leakage. The test liquid may he subjected to a maximum of 2000 mm Hg of pressure by positioning the mercury reservoir on the meter sticks. The glass assembly (Fig. 2a & b ) , consist,^ of a resenroir containing the test liquid, a reservior containing mineral oil and a Teflon plug with attached thcrmocouple. The thermocouple plug connects the two reservoirs and provides a leak proof seal at applicable pressures. The junctions of t,he differential thermocouple are constructed of 3fi gauge constantan wire and 38 gauge copper wire fused together by discharging a 10,000 MFD. capacitor, charged to 28 v, through the twisted junctions. Exact wire size is not critical except where heat 558
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Journal o f Chemical Education
6 2 m m Teflon Tvbinq ESERVOIR
ASSEMBLY
LOWER RESERVOIR Figure 2. Detailed Description of Solution Cell, Thermocouple, and Prerwriring Connections.
conduction away from the junction becomes a ~ o i n of t concern. The thermocouple is calibrated by varying the temperature of one junction from 20'-30°C, while holding the other iunction at 25 =tO.Ol°C. and measurine the emf on thk recorder. The DC amplifier test s&als are used as a voltage source for calibrating the full scale deflection of the recorder. The assembly is enclosed in a 3-in. diam copper cylinder, which is then filled with water, reducing the effect of bath temperature variations transmitted to the
Figure 3.
Simple Pycnometer.
junctions to a negligible amount after temperature equilibrium. These gradients are reduced further by using mineral oil, which has a relatively slow thermal time constant, in the upper reservoir and by completely immersing the cylinder in the water bath. About 1.5 min are required for thermal equilibrium. Measurement of the thermal expansion coefficient is taken at O.Ol°C and is accomplished by use 15, 25, and 35'C of the simple apparatus shown in Figure 3. Actually any pycnometer would suffice.
*
Results a n d Discussion
A photograph of a typical recorder trace of thermocouple voltage for application of two different pressures to a sample of benzene at 25°C is shown in Figure 4.
removed after a short interval. This serves to indicate that the process is essentially reversible. I n Table 1 the data for benzene is used to calculate the experimental part,ial derivative (AT/AP)~for the two pressure applications shown in Figure 4. The table Table 1.
Benzene Toluene
Determinations of (ATIAp), for Benzene and Toluene a t 25'C.
1.01 2.02 1.01 2.02
0.0244 0.0484 0.0218 0.0439
2.41 2.40 2.16 2.17
X X X X
lo-'
10P 10-'
2.398 X
10Y
2.189 X 10-'
10F
also contains data for toluene and gives Burlew's determinations for these two materials. The conversion of atmospheres to bars is given by 1.013 bar/atm. Table 2 illustrates a typical set of experiments using the pycnometer shown in Figure 3. Toble 2.
Thermal Expansion of Benzene and Toluene
Material
Sample weight w
expansion (AV cc)
Temp. ("C )
Toluene
13.4.59 g
0.164 0.168
15 25 35
Figure 4. Recorder Trace d Thermocouple Voltoge for Two Application3 of Pressure t o o Sample of Benzene at 2SDC.
The thermocouple calibration factor for this experiment was measured as 0.0248"C/pvolt. The trace shows that the thermocouple responds quickly to the temperature change upon application of pressure and returns to the normal baseline when the pressure is
L ! ! w
AT
1 . 2 3 x IOP
The values given by Burlew for the thermal expansion and 1.252 X 1 0 F coefficient a t 25'C are 1.395 X cc/g-deg for benzene and toluene. The combination of the appropriate values of Tables 1 and 2 give heat capacities for benzene and tcluene as 0.41 and 0.39 cal/g-deg as compared to Burlew's values of 0.4182 and 0.4074 cal/g-deg. The agreement is within our experimental error. Student results are comparable to these determinations. Acknowledgment
We wish to acknowledge the assistance of the National Science Icoundation Grant GE-5109 in the purchase of instruments used for this experiment.
Examination Committee Plans Test for One Semester Physical Chemistry A recent survey made under the auspices of the Exmination Committee of the Division of Chemical Education obtained data. from abont 750 colleges. Of these 207 had a one semester physical chemisiry course, 26 contemplnted introducing such a course, while the rest (69Y0) had none. An overwhelming majority indicated that the students enrolled were premedical and life science majors. A moderate amount of calculus (prerequisite) was used in 29% of the courses, 23% used a limited amount (no prerequisit,e calculus), and the remainder indicated that c a l c u l ~ ~ s was not used. Comments by respondents indicated that. a definite trend toward increasing the amount of caleulw exists. Persons wishing more detailed analysis of the results and those wishing to cooperate in the construction of the examinat,ion should write to Professor Alfrcd P. Mills, 1Jniversity of Miami, Cornl Gahles, Flo~idaSS194.
Volume 43, Number 10, October 1966
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