Molar Heat Capacity (Cv) for Saturated and Compressed Liquid and Vapor Nitrogen from 65 to 300 K at Pressures to 35 MPa

Molar heat capacities at constant volume (Cv,) for nitrogen have been measured with an automated adiabatic calorimeter. The temperatures ranged from 65 to 300 K, while pressures were as high as 35 MPa. Calorimetric data were obtained for a total of 276 state conditions on 14 isochores. Extensive results which were obtained in the saturated liquid region (Cv(2) and Cσ) demonstrate the internal consistency of the Cv (ρ,T) data and also show satisfactory agreement with published heat capacity data. The overall uncertainty of the Cv values ranges from 2% in the vapor to 0.5% in the liquid.


Introduction
Accurate measurements of thermodynamic properties, including heat capacity, are needed to establish behavior of higher order temperature derivatives of an equation of state P(p, T). In par-ticular, the heat capacity at constant volume {Cv) is related to P(j^T) by: An experimental heat capacity is the apphed heat (Q) corrected for the heat applied to the empty calorimeter (Qo) per unit temperature rise (AT) per mole (N) of substance. In terms of the observed measurements, the heat capacity is given by.
The applied heat is the product of the time-averaged power and the elapsed time of heating. Measured power is the product of instantaneous current and potential applied to the 100 fl heater wound on the surface of the calorimeter bomb. During a heat measurement a series of five power measurements with an accuracy of 0.01% were made at 100 s intervals. Time was determined with a microcomputer clock to a resolution of 10"* s. Elapsed time was computed with an accuracy of 0.001%. The heat (Qo) applied to the empty calorimeter has been determined by several series of calorimetric experiments on a thoroughly cleaned and evacuated sphere. These results include those of Roder [6] from 86 to 322 K and of Mayrath [7] from 91 to 340 K in addition to new data from 29 to 99 K, as presented in Table 1. The combined Qo data sets were fitted to the expression, 12 QoiT)='ZCi(T2'-Ti') by applying a chord-fitting method to AT values ranging from 0.5 to 20 K. Details will follow in a later section. Temperatures were measured with an automated circuit consisting of a 25 H encapsulated platinum resistance thermometer calibrated on the IPTS-68 by the NIST Temperature and Pressure Division, a 10 fl standard resistor calibrated by the NIST Electricity Division, a stable (±2 ppm) electronic current source, and a bank of ultralow thermal emf (<lxl0""' V/K) relays multiplexing a precise nanovoltmeter. Potential measurements were made with the thermometer current flowing in both forward and reverse directions. An average thermometer resistance was calculated in order to avoid errors from spurious emfs. It is thought that the absolute temperatures derived this way are accurate within 0.03 K and precise to ±0.002 K. During this work we reproduced the generally accepted triple point temperature of N2 to less than 0.002 K as a further check on the validity of this claim. Temperature rises (AT) were established within 0.004 K by linear extrapolation of the preheating and the post-heating temperature drift data to the midpoint time of the heat cycle.
Pressure was measured with an oscillating quartz crystal pressure transducer whose signal was fed to a precise timer/counter. This instrument had a range of 70 MPa and was calibrated with a piston gauge. The experimental uncertainty of the measured pressure is estimated to be ±0.01% of full scale at pressures above 3 MPa, or ±(0.03-0.05)% of the pressure at lower pressures. Finally, the number (N) of moles of substance in the calorimeter is the product of the calorimeter volume (Fbomb) and the molar density (p) derived from the equation of state [8] which has an uncertainty of ±0.1%. The calorimeter volume was obtained from a previous calibration [9] as a function of temperature and pressure, and is accurate to ±0.1%. The value of A'^ derived in this way is believed to have an uncertainty of ± 0.2%. If a weighing method was used to evaluate N, the error would drop to ±0.01%. Other details will follow in a later section.
The spherical bomb depicted in Fig. 1 is constructed of Type 316 stainless steel with a wall thickness of 0.15 cm and an internal volume of 72.739 cm^ at 100 K. To prepare for an isochoric experiment, N2 was charged at a pressure of 10 MPa and at a suitable bomb temperature until the target density was obtained. Then the sample was cooled to near 63 K with liquid Ne refrigerant, or to near 80 K when liquid N2 was used. Each run commenced in the vapor + liquid region. The heater power was set to obtain about a 4 K temperature rise during each experiment. Apparatus control was then turned over to the microcomputer. A Fortran program was responsible for control of the cell heater. The guard and shield heaters followed the rise of the cell temperature using a specially tuned proportional-integral-derivative algorithm [10]. The program recorded, at periodic intervals, the bomb temperature, the cell pressure, and the voltage and current applied to the cell heater. Another Fortran program calculated heat capacity using the raw data as input. The raw data were not processed when the initial (Ti) and final (Tz)

Results
A significant adjustment must be applied to the raw heat capacity data for the energy required to heat the empty calorimeter from the initial (Ti) to the final temperature (72). For this work, QJQ ranged from 0.89 to 0.27. Since the published Qo data had a lower limit of 86 K, experiments were conducted to extend the data to temperatures as low as 29 K. The results are shown in Table 1. An examination of the empty calorimeter's heat capacity (Co) revealed that it is s-shaped when plotted against temperature. Further, Co has a sharp curvature below 100 K. Combined, these properties make a high quality fit to raw Co data difficult. In the face of these difficulties, efforts to deflne a Co(T) function were made by previous workers [6,9]. For this work, however, I fitted the data to the integral heat (Qo) function, Eq. (3), which is monotonic with no inflection. Values of Co can then be recovered from the derivative with temperature, Co = dQo/dr. Table 1 presents the raw data (j2o, T, AT), j2o values calculated from the best fit to Eq. (3), and Co from an earlier study [11]. The coefficients of Eq. (3) are presented in Table 2. Calculated Co values establish that the new experimental measurements of Co are both smooth and consistent with previous measurements to less than 0.19% at temperatures from 90 to 100 K, the region of overlap. This is depicted graphically in Fig.  3, which also shows that an extrapolation of our earlier calibration [11] would have led to serious errors at temperatures below 80 K, The nitrogen sample used for this study is of very high purity. An analysis was furnished by the  vendor. The impurities present in the research grade sample are 0.2 ppm CO2, 0.2 ppm total hydrocarbons, 1 ppm O2, and 1 ppm H2O. In addition, we performed our own analysis using gas chromatography-mass spectroscopy and confirmed these results. The raw and reduced data for each run are presented in Table 3 for two-phase states, and in Table 4 for single-phase vapor and liquid states. Sufficient raw data are presented in Tables 3 and 4 to allow rechecking these computations or to reprocess the raw data using other equations for any adjustments to the experimental data. Data for the number of moles (N) in the calorimeter are provided in both Tables 3 and 4. These data identify and tie together the two-phase and single-phase portions of each isochoric run. Table 3 presents values of the two-phase heat capacity at constant volume (Cv^^^) and the saturated liquid heat capacity (C") at the midpoint temperature (T) of each heating interval. Values of the saturated liquid heat capacity C" are obtained by adjusting C^^^ measurements with the thermodynamic relation, where pt and P" are the density and pressure of the saturated liquid and p is the bulk density of the sample residing in the bomb. The derivative quan-tities were calculated using the formulation of Jacobsen et al. [8].
Corrections to the experimental heat capacity calculated using Eq. (2) for vaporization of sample are given by CAH=8JVCAH,A^-> Ar-' (5) where 8JVc is the number of moles vaporized during a heating interval and A//v is the molar heat of vaporization calculated using the equation of state [8]. Thus, Eq. (5) corrects for the heat which drives a portion of the sample into the capillary by evaporation during a heat capacity experiment in the two-phase region. It is at most equal to 0.06% of Cv^^\ In Table 3 the column labeled difference refers to calculations for C, made with the equation of state in Ref. [8]. This equation of state correctly predicts the values within ±2%. Corrections for PV work on the bomb are given by where k = 1000 J • MPa"' • dm"^, the pressure rise is AP =P2-Pi, and the volume change per mole is AVm=p~2^-p~i^. The derivative has been calculated with the equation of state [8]. The PVv/ork correction is important only for single-phase samples and varies between 0.26 and 3.8% of the value of C The largest such corrections occur for the highest density isochores. Table 3. Experimental two-phase heat capacities  •looca-c.cicVa.
= d2o/dr. While we have observed that Cv*^^ values are a function of both T and p, C" values are a function of T only. Hence, Ca data provide us with a valuable check of the accuracy of our measurements by direct comparison with published data. Figure 4 shows the behavior of C" from near the N2 triple point to near the critical point temperature where it rises sharply. Also shown in Fig. 4 are results of Weber [2] and Giauque and Clayton [3], whose data have uncertainties of ±0.5% and ±1%, respectively. In order to intercompare the data sets, our data were fitted to the expression. The coefficients of Eq. (7) are given in Table 5.
Deviations of the C" data of Refs. [2] and [3] from this expression were calculated also. The data of Refs. [2] and [3] were the most accurate available. This work overlaps the temperature range of both previous studies. The deviations of all the C" measurements from Eq. (7) are shown in Fig. 5. We may conclude from Fig. 5 that the data of Refs. [2] and [3] are consistent with this work within ±1% with 95% of these data within ±0.2%.
It is also important to examine the internal consistency of our data. Perhaps the most interesting test of the internal consistency of the data derives from the relation c,<^Vr = -dV/dr^+F" d'/'jdr^ (8) due to Yang and Yang [12], where ju. is the chemical potential and Vm is the molar volume. This thermodynamic relation implies that when plotted on isotherms, Cj-^^IT should be linear versus molar volume. To simplify this test, the measured Cv^^' data in Table 3 were fitted to the expression, and new values were computed at integral temperatures from 65 to 125 K. Selected Cv^^^ isotherms are shown in Fig. 6. We have observed that C"^^' varies linearly with Fm within the experimental precision (±0.15%) of the data.   Further, we have obtained values of d^PJdT^ at integral temperatures, given in Table 6. Also shown in Table 6 are experimental values from Weber [2] and calculated values of this derivative which are from published vapor pressure equations [8,13]. The agreement of this work with published values is better than ±3 x 10"^ MPa • K'l Values of the molar heat capacity at constant volume are depicted in Fig. 7. Shown in this plot are single-phase Cv isochores at each of the 14 different filling densities of this work. As expected, C increases with density up to the critical density (approx. 11.21 mol • dm"^), where it has a maximum value. Then at densities between the critical and twice the critical, Cv decreases to a local minimum value. These data are found in Table 4. Also given in Table 4 is a column labeled "diff." which gives the percent difference of this work from the equation of state in Ref. [8]. The authors of the equation of state estimate an accuracy of ± 2% for their calculated heat capacities. With only a few exceptions, these calculations are in fact within ±2% of the data. Most significantly, however, in the temperature range from 66 to 78 K, the values calculated with their equation fall 1 to 5% below the results of this study. Undoubtedly, accuracies would be improved by a new fit of the equation of state which includes this data.
At highly compressed liquid densities greater than twice critical, Cv shows a rising trend which is indicative of hindered rotation of N2 molecules. A broad generalization can be made for low molecular mass gases with regard to the existence of a minimum liquid Cv at 2.0 pc. If we examine a plot of reduced residual heat capacity {Cv -Cv)IR at saturated liquid states versus reduced density pipc, shown in Fig. 8, we find identical behavior for Ar [14], O2 [14], and N2, A single parabola represents data for these three gases within experimental error. As shown by Fig. 8, the vertex of this parabola is found at 2.0 pc. I have not found a satisfactory   Reduced Density, p/pc 3.0 Figure 8. Reduced residual heat capacity evaluated at saturation plotted against reduced density; Ar (O) Ref. [14]; O2 (A) Ref. [14]; N2 (<0) this work.
explanation of this phenomenon based on firmly grounded theory. Further study is expected to lead to new insight and understanding of the behavior of liquid heat capacities.

Analysis of Errors
Uncertainty in G arises from several sources. Primarily, the accuracy of this method is limited by how accurately we can measure the temperature rise. The platinum resistance thermometer has been calibrated on the IPTS-68 by NIST, with an uncertainty of ±0.002 K due to the calibration. Other factors, including gradients on the bomb, radiation to the exposed head of the thermometer, and time-dependendent drift of the ice point resistance lead to an overall uncertainty of at= ±0.03 K for the absolute temperature measurement. Uncertainty estimates of the relative temperature, however, are derived quite differently. The temperatures assigned to the beginning (ri) and to the end (T2) of a heating interval are determined by extrapolation of a linear drift (approximately -0.0005 K min"') to the midpoint time of the interval. This procedure leads to an uncertainty of ± 0.002 K for Ti and Tz, and consequently ±0.004 K for the temperature rise, AT = Ta-Ti. For a typical experimental value of AT of 4 K, this corresponds to an uncertainty of ±0.1%. The energy applied to the calorimeter is the integral of the product of the applied potential and current from the initial to the final heating time; its uncertainty is ±0.01%. The energy applied to the empty calorimeter has been measured in repeated experiments and fitted to a function of temperature; the estimated uncertainty is ±0.02%. However, the adjustment is considerably larger for vapor than for liquid. For low density vapor the ratio Qo/Q is as large as 0.89, while for the highest density liquid it is as low as 0.27 . This leads to considerably larger (approximately 10 times) uncertainty propagated to the heat capacity measurements for vapor states. The number of moles of each sample was determined within ± 0.2%. A correction for PF work on the bomb leads to an additional ±0.02% uncertainty. For pressures, the uncertainty due to the piston gauge calibration ( ± 0.05% max.) is added to the cross term [(ot)(dP/ dT)p] to yield an overall maximum probable uncertainty which varies from ±0.06 to ±0.8%, increasing steadily with the slope of the P(pi T) isochore to a maximum at the hi^est density and lowest pressure of the study. However, the pressure uncertainty does not appreciably contribute to the overall uncertainty for molar heat capacity. By combining the various sources of experimental uncertainty, I estimate the maximum uncertainty in Cv which ranges from ±2.0% for vapor to ±0.5% for liquid.