Measurements of the Specific Heats, Cσ, and Cv of Dense Gaseous and Liquid Ethane*

The specific heats of saturated liquid ethane, Cσ, have been measured at 106 temperatures in the temperature range 93 to 301 K. The specific heats at constant volume, Cv have been measured at 19 densities ranging from 0.2 to 3.1 times the critical density, at temperatures between 91 and 330 K, with pressures to 33 MPa, at 200 PVT states in all. The uncertainty of most of the measurements is estimated to be less than 2.0 percent. As the critical point is approached the uncertainty rises to about 5.0 percent. The measurements were performed to provide input data for accurate calculations of the thermodynamic properties for ethane. They are believed to be the most comprehensive specific heat measurements available for the liquid and vapor states of ethane.


Introduction
For the calculation of the thermodynamic properties of a fluid, properties such as internal eneray, enthalpy, entropy, and velocity of sound, especially at temperatures less than critical, one n eeds either the latent heat of vaporization or a s pecific heat along a path traversing the temperatures of interest.
Reat capacity measurements are much easier to make than latent heat measurements, and they are not restricted to the liquid-vapo r curve but can be made at temperatures and densities in the single phase fluid region as well.
For ethane, the specific heat of the saturated liquid, Gu, was measured from 93 to 301 K, and specific heats at constant volume, Gv, were measured on 19 isochores with den sities ranging from 1.5 moljl to 21 mol /l with temperatures from 91 to 330 K and pressures up to 33 MPa [1]. 1 In a forthcoming publication Goodwin [2] uses the present results together with extensive PVT data to construct a complete thermodynamic network for ethane from the triple point to 600 K with pressures up to 70 MPa.

Experimental Method
The basics of the specific h eat experiment are deceptively simple. The heat capacity of a sample of fluid is determined in principle as follows. A sample holder is filled with a known amount of 'This work was carried out at the Nalional Bureau of Standards under the sponsorship of the Ameri can Gas Associatio n. sample, N, and is placed in an adiabatic environment. If we now apply a carefully measured amount of heat, Q, to the sample holder, then the temper ature of sample and hold er will rise to a new value, from an initial temperature , TI , to a final temperature, Tz, the change in temperature b eing I1T. To obtain the h eat capacity of the sample we must account for the heat absorbed by th e container. This is accomplish ed by conducting a second experiment with the sample ho lder empty to find the heat capacity of the empty co ntainer, Go. With the desired specific heat of the sample can be calculated from Thus, the parameters we must measure in the experiment are Go, Q, I1T, and N.

Apparatus and Procedures
The apparatus used is a constant volume adiabatic calorimeter fully described by Goodwin [3]. The essential features are a spherical sample holder, a filling capillary, a heating/ cooling interceptor. guard ring, an adiabatic shield, and a platinum res1s ta~ce thermometer mounted on the sample holder. Calonmeter and cryostat are shown in figure 1. The refrigerants used were liquid nitrogen, ice, and cold water. The instrument has been used to measure the specific heats, C. and C" of hydrogen [4,5], oxygen [6,7], fluorine [8,9], and methane [10]. Measurements of the heat of fusion and of the solid· solid transition in ethane with this apparatus have been reported elsewhere [11]. Minor modifications to the system have been described by Goodwin and Weber [6]. The major experimental parameters are Q, ;;".T, and N. These are measured as follows. We obtain the calorimetric heating rate from nearly simultaneous readings of the potential and current applied to the calorimeter heater. The time of the heating interval is measured by an electronic counter triggered by the potential across the calorimeter heater. Temperatures are measured with the ulatinum resistance thermometer. The thermometer ~as calibrated by the NBS Temperature Section. Temperatures are on the IPTS-68 scale. The temperature of the adiabatic shield and guard ring are controlled to the sample temperature with difference thermocouples and automatic power regulation. The amount of sample is determined from an observed temperature T and pressure P in the single-phase domain, from the bomb volume at this T, P, and from the fluid density derived from an equation of state [2].
The ethane used in these experiments was commercially available research grade with minimum purity certified by the supplier at 99.98 percent. This purity was verified by chromatographic analysis.
The procedures used for measurement of the empty calorimeter, for loading of the sample, and for the specific heat measurements are the same as those used previously [4][5][6][7][8][9][10] except for filling the sample holder at low densities, and in the sequence of measurements. Differences in the filling of the calorimeter arise because the critical temperature of ethane is above room temperature . The ethane supply tank is normally at room temperature, about 296 K and the corresponding supply pressure, vapor pressure, is about 4 MPa. Fillings with liquid densities down to 12 mol/l are determined by selecting the temperature of the calorimeter, as before. However, different techniques had to be used to achieve densities below 12 mol /I. One was to raise the filling pressure by placing the ethane supply tank in a hot water bath, up to 40 °e. The other was to fill the sample holder around 12 mol/I, heat it to a temperature above critical, and then bleed it in small increments down to the desired density.
The sequence of measurements adopted was to conduct the C. and Cv measurements with a single filling rather than with different fillings as was the practice before. In this scheme the sample holder is filled to a known density in the single phase region and is then cooled to a temperature where both liquid and vapor are present in the calorimeter. Heating intervals are applied to the two phase sample, the data reduction yields values of Cu.
During these measurements both liquid and vapor densities are changing, gradually filling the sample holder. From that point on the data reduction is carried out to yield values of CV• The heating interval in which the sample holder contains both two phase and single phase fluid is called the "breakthrough" point. A sharp change in rate of temperature rise can be seen on a recorder trace of calorimeter temperature, and both guard ring and shield heaters show a slight "bump" on the recorder traces of the differential thermocouples corresponding to a change in power requirement.

. Ca lculations and Adju stments
The data reduction applicable to this experiment has been desr,ribed in detail by Goodwin and Weber [6,7]. However, for ethane the separate programs of Cu and CI' were combined and a phasefinder developed that would pinpoint the "breakthrough" point of each filling. The phasefincler, the filling conditions, and the PVT conditions at which each point was measured are based on the equation of state by Goodwin [2].
One of the primary experimental parameters is the total amount of sample in the system, N. The pressure and temperature at filling are measured, the corresponding density is calculated from the PVT surface, and N is evaluated from a knowledge of the calorimeter volume, and the various ancillary volumes such as capillary, connector, and valve volumes. ' As mentioned before, the critical point of ethane, at 305.33 K, is above room temperature. A number of fillings and experimental measurements were made between 305 and 330 K. For these runs the portion of the sample in the capillary is not negligible, and ha s to be accounted for accurately. All of the "nuisance" vol urnes were revised, in particular the valve volum e which, nominally at room temperature, was larger than previously estimated . To partially all eviate the problem the valve was thermostated at 40 D C , and a variable valve temperature was includ ed in the data reduction routines. The amoLint in the capillary is determined by assumin g a temperature distribution along the capillary. This distribution was changed to accommodate a variabl e temperature at th e valve end.
Several oth er corrections made in the programs are reviewed briefly. The calorimeter volum e depends both on temperature, thermal expansion , and on pressure [4 , 6]. Since the sampl e holder is a thin stainless steel sphere it stretches as the pressure increases. Thus, in a 0, measurem ent work is done by th e sample du e to the increase in sam ple volume. This correction developed by Walker [12] ranges from 0.5 to 5 percent of the resulting Ov value.
However, it can be mad e accurately. The density for each 0" mea surement is calculated from the filling den sity after correctin g for sample holder expansion and the amount compressed into the filling capillary [7]. In the case of a 0, measurement the effects of the latent heat of vaporization and heat absorbed by t he vapor must be subtracted [4,6,8]. This type of correction has been derived by I-loge [13].
It is worthwhile to m ention that of the three state variables, pressure, temperature, and density, only temperature is measured during the meas uremen t of a specific hea t point. The amoun t of smnple in the ca lorimeter is used to es tablish th e density and pressure at the mean temperature of the experiment. Whi le the total amount of sample remains constant, the distribution between calorim eter and capillary changes from poin t to point because the c aIOl'im eter volume changes with temp erature and pressure. Thus, whil e the results for O. are corrected to be a tru e 0", the measurements of fi. given run are made at slightly changin g mean densities .
Curvature adjustments have been made for the 0, values a,t temperatures above ] 0l.5 K . Adjustments to the experimental gross heat capacityliquid and vapor-ran ge from 0.002 J /mol-K to 0.366 J /mol-K, or up to 0.16 perce nt of the total va lue of 0 , . Curvature adjustments for the values of 0 , were not significant, and were, therefore, omitted .

Heat Capacity of the Empty Calorimeter
E arl y estimates reveal ed that und er the best of circumstan ces 50 percent of the applied heat is required for the calorimeter; for the very low densitie,-; at th e highest temperatures up to 93 percent of the heating goes to raise the temperature of the calorimeter. Since the critical temperature of ethane is 305.33 K it appeared desirable to make at least some of the 0 , measurements at temperatures above critical. An upper limit of 338 K is imposed by the fact that the platinum resistance thermometer is mounted with Wood's metal, which melts at 65 DC. Measurements on the other fluid s had been carried out to only 300 K, th erefore, an extension of the measurements on th e empty calorimeter were indicated.
R emeasuring th e heat capacity of the calorimeter 0 0 provided an opportunity to conduct additional checks of th e sys tem with regard to systematic errors, and to see if th e precision of the m eas uremen ts could be improved. The m eas urements of t he hea t capacity of th e empty calorimeter included large and small tJ.T's from 8 to 0.5 K ; large and small appl ied powers, from l.0 to 0.23 W; runs with deliberate temperature offse ts in both gua,rd ring and shield temperatures, 3 K (100 p.V) hot and cold ; as well as different coolants in th e refrigerant tank, runs 2,3,4, with ice and run s 5, 6, 7 with liquid ni t rogen . The results of th ese meas urements, some 92 points, are shown in ta,bl e 1. Th e a,pplied tempera t ure differences are sma ll enough so that a curv a ture corr ection is not required . Intercomparison of t he da ta, is achieved by using the functional re presentation developed by Goodwin and Weber [6].  Points 208 throu gh 304 are includ ed in table 1 to show the most extreme v ari ation in tJ.T. Th ey were not used to obtain t he coefficients, 0 1, because durin g th ese run s one of the d. c. amplifiers had a, large bias which was not corrected un til the start of run 4. The analyti cal curve represen ts th e heat capacities of th e empty calorimeter with an imprecision of 0.07 percen t. To the level of 0.1 percen t in Co there are no discernibl e sys tematic errors that can be related to th e size of th e tJ. T , the size or ra te of the applied h eatin g, th e temperat ure gradient of the capillary, or to temper ature elTors in shield or guard ring systems. The agreement of th e present values with those measured by Goodwin and Weber [6] is well within th e imprecision of th e separate measurements. In th e tempera t ure range 87 to 320 K the uncertainty in the quantity (Q-OotJ.T) /tJ.T will range from 0.04 to 0.08 Jj K due to the uncertainties in 0 0 alone.

. Results
The results to be presented include values for the specific heat of saturated liquid ethane, values f<;)1' the specific heat of single phase ethane, b?t~ III compressed liquid and in gaseous states, and a lllmted set of measurements on methane, made for purposes of comparison. As mentioned above, both 0, and 0 , measurements were made during a single fillin~.   sure, obtained by compu tation from laboratory observations, are in effect direct m easurem ents. The volume of the calorim eter is computed , the density is obtained from the equ ation of state [2].
The total number of moles, N, includes the amount in capillary and valve wi th th e upper stem temperature equ al to the indicated v alve temperature . Breakthrough density and temperature define the point on the saturation boundary applicable to the run in question. The values are calculated from the equation of state, the loading conditions and the vapor pressure by considering th e variation of calorimeter volume with temperature and press ure.

.1. Performance Tests: The Specific Heats
Co and Ct· of Methane Prior to m aking measurements on eth ane we made a limited se t of m easurements on m ethane. Th e purpose was to ch eck on the operation of the instrum ent by comparison to the values previously m easured by Younglove [10]. Several values of C. and 29 values of C, were m easured at three different filling densities and at widely differing temperatures. R esults and comp arisons are shown in table 3. Two conclusions can b e dru,wn from the m easurements on m eth an e. One is r u,ther surprisin g, n amely that the values of the specific h eats calcu lated from the r aw da ta will differ , if slightly different PVT surfaces are used in the d ata reduction process. The other is expected, namely that the valu es of the specific h eats depend directly on the values measured for the heat capacity of the empty cu,lorimeter.
To caleul ate th e pres en t resul ts, which u,re shown in column 5 of table 3, we used Goodwin's most recent formul ation of the PVT surface of methane [14]. Since Younglove used a differen t, earlier formulati on [15], a second calcul a tion of our resul ts usin g the earli er PVT surface is shown in column 6 of table 3. On e of the most important difi'erences between th ese two PVT surfaces is the assignm en t of critical density, 10 .0 m ol/ I for reference [14], and 10.15 m ol /I for r eferen ce [15]. The in tercomparison of th e two calc ul ations is given in col umn 7. Clearl y, both O. and Cv arc sensitive to the PVT surface use d in the data r ed uction process. Furthermore, the differences vary from poin t to point on th e PVT surfa ce. R ecompu tin g all of You nglove's results with th e two differen t PVT surfaces leads to maximum differences of .08 percent in both C. and C v.
Accordin gly, th e m ost consisten t way to compare th e present results wi th those of Youn glove [10] is to use t he same PVT surfa ce. This comparison involves columns 6 an d 10 of tabl e 3, with differen ces given in column 11. Th e disagreement b etween the two exp erim ents run s from -1 to +1 percent. Several explan ations were consid ered , only one of which is di sp layed in table 3. Th ere exists a consistent offset, 0.4 percen t , between the Co m easured in the course of thi s experimen t and that m easured by Youn glove [10]. In column 8 we h ave calcula ted our present results on m eth an e using Younglove' s [10] values for 00 and Goodwin's earlier formulation of the PVT surface [15]. Column 9 shows the departure of the valu es in column 8 from those in column 6. Finally, a comparison of col umns 9 and 11 suggests if not quantitatively, th en at leas t quali tatively, that ind eed th e difference in the values of the Oo's is the expl a nation for th e differen ce between the present m easurements and those of Youn glove [10].

Ethane
The specifi c heat of saturated liquid eth ane w as m easured for 106 temper atu res. The lowest was 93.7 K , th e triple point is 90. 348 K , th e hi gh est temperature was 301.5 K, t he critical point is 305 .33 K. Values of O. along with the experim ental conditions, experimental p ar ameters, and the various correction t erm s are given in table 4. A plo t of C. is sh own in figure 3. Th e IneaS Ul'em en ts for C. were mad e with /j,T between 3 and 5 k elvin , and with loadin gs su ch th at at eac h tem perature C. is defined by at least t wo diffel'en t fillings (see column 11, table 4). C urvature corrections were necessary only Itt temper atures above 101 K . The results for O. are repre ented with an analytical eq uation as follows: where Tc is th e cri t ical temperature, 305 .33 K and values of the coeffi cients fire given in th e h eadin g of tltble 4. Val ues cal cuI ated from eq . (4) and differen ces between experim ental and cal cul ated values expressed in percen t are also given in table 4. The standard deviation of th e entire fit is 0.3 J /m ol-K . Fo]' temperatures b elow 260 K th e imprecision in the experiment is ± 0.1 p er cent, not mu ch l arger th an th a t experienced for measllrem en ts of the empty calorimeter. Consid ering all so urces the est~ mated un cer tltin ty in the measured value of O. I S about 0.5 p ercent generally, in creasin g to ab out 5 percent within a few kelvin of t h e cri tical point.
Comparison with th e earlier m easurements of Wiebe et a1. [16], and Witt and K emp [17] is made using eq (4) for interpolation. Differences in C. are shown in table 5. They are negligible u,t low temperatures but increase grad ually to 5 p ercent at the highest t emperatures of com p arison. The explanation of the differences lies in the different PVT surfaces used to evaluate the experimental data and correction terms, in particular the rather large difference in assignment of the critical density, 6.80 mol /l this experiment and 6.99 mol/l for the other authors. The specific heats at constant volume were measured at 19 densities ranging from 0.2 to 3. 1 times the critical density, at t emperatures between 91 and 330 K , and with pressures to 33 MPa. A shown in figure 2, a given density is limited eith er by the maximum allowable sys t em pressure, about "' "   [14) Co: This paper This paper Ref. [10) Inter polated from ref. [10) PVT: Ref. [14) ref. [15) Ref. [ The specific h eat, C" of singleph ase fluid methane PVT conditions based on R esults this paper calculated with Co a nd PVT sources below reference [14] Interpolated from r ef. [   35 MPa, which in turn leads to a maximum pressure of 33 MPa at the mean temperature of the experiment, or by the upper lim~t in temperatUl'e, 330 K.
Values of Cv along with the experimental conditions, the major experimental parameters, and the correction term are given in table 6. The temperature and density dependence of Cv is illustrated in figure  4. For a wide range of d£nsities to either side of the critical density the specifi c heat increases sharply as the coexistence enevelope is approached. At liquid densities far removed from critical the temperature dependence is relatively weak.
Representation of the specific heats has b een achieved by Goodwin [2] who correl ates the available PVT data, the specific h eats of the ideal gas, the specific heat of the saturated Jiquid from this experiment, and selected values of Cv from runs 1, 8, and 9. Values of Cv ca.1culated from his equation of state and differences between ex perimental and calculated values expressed in percent are given in table 6. A scan of column 7 in table 6 reveals that the agreement between experimenta.! fmd calculated values is excell ent over much of the PVT surface, i. e., a nominal 2 percent or less. It is only in the region near the critical point where the experimental h eat capacities increase drastically that the representation is not able to match the experimental surface of C v , departures reach values of up to 18 percent. The agreement for the very lowest density, run 19, is particul urly gratifying since for this run the experimental uncertaint} in Cv is about 2 percent, whereas the calculation of Cv from ideal gas and a very small PVT contribution should have very little error attached to it.
Experimental measurements of specific heats have been made by other authors [18,]9], however these measurements are m easurements of the specific h eat at constan t pressure, Cpo The values can be compared only indirectly to the presen t measure-

Discussion
It is readily apparent that accurate values of Co are essential if we wish to obtain accurate values of either Cu or Cv' A chan ge of 0.1 percent in Co, for example, will r esult in a change of 1 percent in the values of Cv calculated for run 19. The temperature incremen t, t::. T, is evaluated at the middle of the heating interval by extrapolating the temperature ?rift ~'ates just before heating and after an equilibratmg tIme has elapsed. Since the drift is linear the statistics of the extrapol ation can be used to estimate an uncertainty in the t::.T. For the first 14 points of Co the average slope un certainty was 0.19 X lO-3 K /min , since th e average elapsed time to the center of th e measurement interv al is about 20 min th e average uncertainty in t::.T turns out to be ± 0.004 K .
This in turn implies that if we seek 0.1 percen t precision in the specific h eats the t::.T must be 4 K or larger. The choice of t::.T, as shown in tables 4 and 6, was based on this consideration and on the idea that there ought to be at least 5 points per experimental run. The imprecision in the temperature time data is attributed to the exact setting or resetting of the platinum thermometer current rather than potentiometer inaccuracy. Potentiometer inaccuracy was actually reduced from values given by Goodwin and Weber [6] to a maximum of 0.003 K by consideration of a potentiometer calibration. Heat leak to and from the sample is estimated to be less than 0.1 percent by considering the difference in drift rates before heating and after equilibration has been reached. Shield temperatures lag at the start of the heating interval by about 0.02 K. They lag again at the end of the heating interval after the power is turned off. The two lags compensate to produce a nearly adiabatic environment during the entire heating interval. Deliberate changes of temperature along the capillary were introduced to see if the applied heat, and therefore the resul ts could be changed. In run 14 points 1401-1413 were obtained with liquid nitrogen in the refrigerant tank. These points were duplicated for 1414-1425 using cold water as coolant. The results, as shown in tables 4 and 6, are virtually identical. However, when we applied deliber ate heating to the capillary, actually quite drastic heating 100 ma to a 140 n heater, the results changed. Points 508-517 differ from those obtained in a duplicate run 520-527 without heating the capillary by 1 percent at the lowest temperature. The difference disappears entirely at the highest temperature of the run. However, rather than changing the applied heat, heating the capillary apparently changes the distribution of sampl e between calorimeter and stem.
The same problem, distribution of sample between calorimeter and stem, is thought to give rise to the curvature of the runs. Runs 18, 4, 5, 6, and 7 show a definite curvature as to two phase boundary is approached, see figure 4, if compared to the values calculated by Goodwin [2]. The curvature seems to abate at pressures above the critical pressure, or at a point wh ere mass change between calorimeter and stem has stabilized. It is possible that a heat of vaporization correction to Q should be included for the Cv calculation as long as sample is being transfen'ed from calorimeter to stem. It should be noted that the correction term, column 13 of table 6, is irregular for the first few points of the runs in question.
Sample distribution is also thought to explain the departure of point 208 from the rest of run 2. The possibiity exists that for run 208 the capillary was frozen, because if point 208 is recalculated with zero stem volume the value of Cv is increased by about 2.5 percent.
We had hoped to employ the breakthrough points to resolve the problem of sampl e distribution. Experimental breakthrough temperatures agree well with calculated values; densities and total sample agree to a point where we are confident that the calorimeter volume has not changed. However, to calculate Cu and Cv values from a breakthrough point requires that we know the time of breakthrough exactly in order to proportion the applied h eat Q. There is simply too mu ch Jag in the res ponse of the recorders to permit an accurate determination of th e breakthrough time.
The imprecision in the experim ent depend s primarily on the imprecision of Co and on the amount of sampl e since in mo st cases th e t:>T is around 4 K.
For liquid den sities th e imprecision from point to point alon g an isochore is about 0.1 percent with occasional differences as large as 0. 3 percent. For densities less than criti cal the imprecision, the variation of Cv point to point from a smooth curve, increases to about 1 percent. The inaccuracy or uncertainty of the present measurements is estimated from th e comp a rison to th e experim ents of others and from th e comparison to values cal cula ted from th e PVT correla tion. We consid er th e excellent agreement betwee n the present resul ts and th e experiments of oth ers [10 , 16, 17, 18], in p articul ar th e agreement with expe rim ental v alues of Cp , and we consid er the resul ts of delib era tely introdu cin g changes in th e prese nt ex perim ent. It is diffi cult to see how systematic errors larger th an about 2 percent for liquid densities or larger tilaH 5 percent for va por den sities clo se to critical co uld remain undetec ted in the present experim en t.