Deviation of International Practical Temperatures from Thermodynamic Temperatures in the Temperature Range from 273.16 K to 730 K

The range over which thermodynamic temperatures have been realized by gas thermometry at the NBS has been extended to 730 K. The results are preserved by measuring the corresponding international practical temperatures. The difference between them is expressed as the following polynomial: T/K−T68/K68=−120,887.784/T682+1213.53295/T68−4.3159552+6.44075647×10−3T68−3.56638846×10−6T682which is valid in the range 273 to 730 K. The difference found and the estimated uncertainties at the three defining fixed points in the range covered are t(°C) T/K–T68/K68 Uncertainty Random (99% confidence limits) Systematic 100 −0.0252 ±0.0018 ±0.00054 231.9681 −0.0439 ±0.0022 ±0.0015 419.58 −0.0658 ±0.0028 ±0.0028


Introduction
The National Bureau of Standards' gas thermometer has been described in numerous papers [1][2][3][4][5][6][7][8][9][10]1 t hat give details of many of the impor tant parts of the equipment and of some of the special measurements required. The last two of these papers also give the results of measurements with the gas thermometer from 0 to 142°C . A comparison of these results with most earlier work elsewhere shows there are systematic differences that we ascribe to our more complete elimination of the effects of sorption. It was stated in the last paper (and has been further confirmed by more recent observations) that our gas thermometer system and the helium filling gas were clean enough that any residual effect of sorption on these measurements would probably be negligibly small. Ultimately, it will be rewarding to remeasure t hese temperatures, when the benefits of further experiences with the gas thermometer can be realized. At present, it is more productive to extend the measurements to higher temperatures, both fo r their own intrinsic interest, and as a fu r ther step leading to the gold point. This paper gives the resul ts of the initial measurements of the difference between thermodynamic temper atures and international practical temperatures, at temperatures in the range fro m 14 1 to 457 °C, and combines them with the earlier data [9,10], reappraised in the light of further information.

Equipment
The essential elements of t he gas thermometer have been described in earlier papers [9,10]. Briefl y, it is a constant volume type comprising a 432. 5 cm 3 bulb of platinum-rhodium alloy in the form of a right circul ar cylinder connected by small bore 90 per cent platinum 10 percent rhodium tubing to a valve. The gas thermometer can be connected through the valve with a diaphragm pressure transducer, so that there-after equality of the gas thermometer pressure with the pressure in the manometer can be established. The gas thermometer has a counter-pressure arrangement, a narrow annulus about 0.25 mm wide formed around the bulb by a heavy Incone1 2 600 case surrounding the bulb.
The gas thermometer has been improved by installing more thermocouples to measure the temperature distribution along the small bore tube, and by modifying the gas handling and vacuum equipment. 3 The new arrangement of the vacuum system, shown in figure 1, allowed the system to be subdivided into four parts (labeled I, II, III, and IV) plus the manometer, so that all the critical sections could be pumped by ion pumps. The subsystems were arranged in such a way as to be able to attain the best vacuum in part II, which included the gas thermometer bulb in direct communication with the residual gas analyzer (RGA).
Two discrepancies from the earlier papers [9,10] should be noted. The composition of the bulb is not all 80 percent Pt-20 percent Rh as stated in [9], nor all 88 percent Pt-12 percent Rh as stated in [10]; it is some of each. The sheet supplied for the top was found to have been 80 percent Pt-20 percent Rh, but the sheet for the sides and bottom was 88 2 The use of commercial product names is for convenience and exactness and does not imply endorsement by t he National Bureau of Standards. 3 A new oil diffusion pu mp has been installed . It is stated by the manufacturer to have a backstreaming raie of "almost zero-less than 5X IO-' mg per em' p er min," percent Pt-12 percent Rh. However, the bulb is the same Bulb III referred to previously; the effect of this difference will be considered in the Discussion. Also, the volume of 454 .8 cm 3 given for Bulb III has been re-evaluated as 432.5 cm 3 • The effect of this difference on earlier results will be tabulated in table 4, section 8.
The liquids in the stirred baths were changed from organic materials. The liquid for the bath at 0 °C was changed to water, to which potassium chromate was added as a rust inhibitor, and the liquid for the bath at temperatures from 142 to 457°C was changed to a eutectic mixture of molten potassium nitrate, sodium nitrate and lithium nitrate. Despite the lesser viscosity of the n ew fluids, no important change in the performance of the thermostat was observed.
New equipment was installed for the electrical measurements. All resistances of the platinum resistance thermometers (PRT's) were measured with improved speed and precision on an ac bridge [11]. All thermocouple emf's were measured by a digital voltmeter.
For thermal expansion measurements, a new interferometer furnace, designed for use up to the gold point, was assembled, and it was operated from -25 to 550 °C so that the thermal expansion of samples of the bulb material could be measured. Details of this equipment and the results will be given in a separate paper.  c "Pp".to, ,hown in figm, 2 w", con, tmoted f07lhe determination of thermomol ecular pressure effects. It consisted of two tubes, an 0.8 mm i.d., 1.6 mm o.d. stainless steel tub e running insid e a 9.6 mm i.d. Inconel tube, connected across a diaphragm pressure transducer. The outer tube was chosen of such a size that it could be inserted into one of the PRT wells of the gas thermometer bulb case to thermostat it in the stirred molten-salt bath. The apparatus was attached to the gas handling system, so that it could be evacuated, and thereafter gas could be introduced. The overall pressure was read with sufficient accuracy on a simple U-tube mercury manometer. The valves in the system were arranged so that when valves 2 and 4 were closed and 1 was open, the null of the diaphragm could be read, and when 1 and 3 were closed and 2 and 4 were opened, the thermomolecular pressure difference could be r ead on the diaphragm gage.

Preliminary Considerations
A stud y of the early results in the extended temper ature range showed the need for modifying To System 2 / the experimental procedures. This section presents a discussion of some of those problems and their solutions.
During vacuum bakeout we observed the ubiquity and persistence of hydrogen that is a common experience, and for which there has been a common explanation: that the hydrogen was absorbed in the metal in great quantity and diffu sed out of it slowly. Perhaps it was a happy accident that we introduced some wet helium into the co unter-pressure system at 700 °C, and to our astonishment observed a drastic rise of the hydrogen peak on the RGA scan of the gas thermom eter contents within 20 s. Inasmuch as th e helium peak did not change, the hydrogen must have diffu sed through the wall, made of 88 percent Pt-12 percent Rh metal. The diffu sion of hydrogen throu gh platinum is covered in the literature, which is ample, and is adequately summa.rized in Dllshman [12] . There is consid erable variation in the results, but a typical se t of values at 750°C is as follows: H ydrogen diffu ses through pure Pt at the rate of 0.1 3 Pa-liter per cm 2 per min per mm thickness of the metal , with a hy drogen pressure of 10 5 Pa on one sid e and a vacuum on the other (the " permeation rate"). Platinum immersed in hydrogen at a pressure of 10 5 Pa absorbs 0.34 Pa-liters of the gas per cm 2 per mm thickness.
Both the permeation rate and the solubility for hydrogen vary as p V ', so their relative relation s~ip remains the same at lower concentrations. The lllescapable conclusion is that hydrogen initially dissolved in the platinum will quickly be exhausted , and conversely thfLt persistent concentrations of hydrogen observed eluring vacuum bakeout must be mfLintained by diffu sion through the platinum . Th e same range of valu es of diffusion and permefLtion can b e expected for platinum-rhodium alloys. The concentration of free hydrogen in the atmosphere is very small ; the only possible sources are water or other hydrogen bearin g materials which decompose to release hy drogen. Th erefore, we surmised that the problem mi gh t be resolved by con trol of ~he ambient atmosph er e. Th e furna ce was flu shed w~th argon during the b akeo ut, und a slow flow m amtained thereafter. Previously, t he partial pressur e of hydrogen in the gas thermometer bulb ha d never been lower than 70 j.(Pa, but it declined to 20 j.(Pa at 700°C when the protective atmosphere was provided.
For the present temperature range up to 457 °9, we found it necessary to change our procedures m three other ways which will be described in the following paragraphs : Modification oj procedure to avoid creep oj the gas thermometer b11lb.
Above 327°C, the gas thermometer bulb is subject to creep (non-elastic deformation under small FIGURE 2. AppaTatus fOT measw'ing th ennornoleculm' pTeSS1l1·c . stress). It was discovered from the early results of measurements carried out in the same manner as for the 0-142 DC range. The manometer equipment by its nature permits the accurate realization of only one pressure a day, with one or more experimentally determined values of temperature of the gas thermometer necessary to balance that pressure. Overnight, between measurements, we have always pumped out those portions of the system immediately adjacent to the gas thermometer bulb (parts II and IV) in order to maintain the purity of the system . (The pumpout of section II is also desirable for one of the "integrity checks" described b elow.) The typical pressure in the gas thermometer bulb at 457 DC was 10 5 Pa; it was expected that it would deform elastically when the counter pressure annulus (in part IV) was evacuated. However, it was found that gas thermometer temperatures determined by measurements made in descending order agreed with those determined by measurements made in ascending order only up to 327 DC. Above 327 DC, the gas thermometer temperatures differed more as the temperature (and, to some extent, time) increased. We also observed some hysteresis of the gas thermometer bulb after the counter-pressure was restored. It was necessary, therefore, in all subsequent measurements to keep the pressure inside and outside the bulb nearly equal until the bulb temperature was below 300 DC. Because it was more convenient to equalize the pressure by reducing it in the counterpressure system, all values reported in the paper were derived from runs starting at the highest temperature and proceeding by consecutive steps of about 33 DC to the lower limit.

Modification of Temperature lVfeasurement of the Dead Space.
The effect of the volume in the connecting tube of the gas thermometer, the "dead space," expressed in kelvins, is about six times as large at the zinc point as at the steam point. The uncertainties of the dead space effect can best be evaluated by examining the combined equation for the deadspace correction, Md S ' This is where the summation is over the length of the connecting tube in mm increments, TI is the fiducial or reference temperature, Tz is the measuring temperature, PI and P z are the corresponding pressures, Vo is the volume of the bulb at TJ, Vki and Tki are the corresponding values of volume and temperature at the kth position for state i. The element of volume varies between the two states only because of thermal expansion, i.e., Vki= Vk(l +i3kiotki), where i3kt is the coefficient of thermal expansion depending upon the position and the state, and otki is the difference in temperature of the kth position in the ith state from the calibrating temperature of 23 DC. The determination of the volume is thought to be suf-ficiently accurate [7]; the measurements of the Tt .,. on the other hand required improvement.
We note from the equation that for any region of the dead-space actually at the gas thermometer temperature in both states, the net contribution is zero. The accuracy is increased, then, by maintaining the level of the liquid in the bath as high as possible. Unfortunately, the molten salt, because of low viscosity and low surface tension, could not be confined as readily as the earlier bath fluid. Even at the highest temperature the bath had to be less completely filled or· the salt would overflow, and it contracted so much over the range of temperature that the level was appreciably reduced at the lowest temperatures.
Previously the temperatures for the dead space calculations for the steam point were derived on the following assumptions: (a) That the lowest junction is at the temperature of the bath. (b) That the emf's of the Pt-lO percent Rh/Pt thermocouples installed for dead space measurement follow the same temperature dependence as was determined for annealed thermocouples of the same spools of wire. (c) That the portion of the dead space above the uppermost junction was at the temperature of that junction. (d) That the values of the temperature can be adequately interpolated from least squares fits of the observed temperatures to polynomials chosen according to the complexity of the temperature profile. The consequence of these assumptions is that one may measure all emf's as differences from the bottom one and then the temperatures can be derived from the calibrations by calcul ating the equivalent emf of each couple with an ice junction.
Probably the steam point dead space calculation was adequately accurate because its contribution in kelvins was comparatively small (0.034 Ie), the height of the bath liquid was satisfactory, the gradients were not severe, and the actual thermocouple emf's were reasonably close to standard table values. However, in this present temperature range, all these factors are less ideally fulfilled. The value of the dead space correction becomes large, partly because the bath liquid level had to be lower; the gradients, of course, were substantial, and as the temperature became high, the deviation of the actual emf's versus temperature from the table values became important. It is certain moreover that the bottom thermocouple was not adequately immersed. All these factors required that a better measurement and calculation be employed.
Instead of referring the temperatures to the bottom thermocouple, if the temperature of the reference junctions of the dead space thermocouples were established, the error in the dead space calculation could be reduced. This is true because the temperature is then more accurately known than before at the lower temperatures, where the relative effect of a given volume of the dead space is larger; the same discrepancy in emf produces a smaller error in kelvins if it affects the higher temperatures where the thermoelectric power becomes twice as large as at room temperature; and the accuracy of the result then no longer relies upon the bottom couple being at bath temperature. The details of this change will be discussed in section 6.

Modijication in the handling oj the platinum resistance thermometers
We found the platinum resistance thermometers to be much more sensitive to shock in the upper range of the temperature investigated. The vibration of the stirred liquid baths caused a significant increase of the resistances of the PRT's above 200 °0 (not accountable by the effects reported by Berry [13], so that the procedures were changed to minimize the time during which the thermometers were kept in the bath. The measurements of the PRT resistances were made at the end of the time they were in the bath, therefore we used the values of the triple points that were found after each gas thermometer run to calculate their resistance ratios. With such precautions, the triple point values did not increase greatly, being subj ect to a change of <40 J.l.Q from work hardening during a measurement period, and being partially restored to their original values by annealing at 425 and 457 °0 whenever those temperatures were measured.

Procedure
The gas thermometer was prepared for operation by vacuum bakeout at 750 °0. After an initial pumpdown to a system pressure of 0.01 Pa by the oil diffusion pump, it was connected with the RGA and isolated from all other parts of the system. It was then pumped only by the ion pump of the RGA, a procedure that gives both the highest concentration in the RGA for the detection of sorb able species, and very clean pumping for the gas thermometer. The entire bakeout furnace was isolated and a protective atmosphere of argon was introduced into it in order to exclude ambient air and prevent the diffusion of water vapor to the hot zone. The pumping was continued until the partial pressure of all contaminating gases was less than 0.1 mPa in the gas thermometer.
The measurements for this paper were made over a period of 15 months during which time the gas thermometer was loaded 10 times and measured at 123 different temperatures. Of the first 51 temperatures measured only 9 ale free of creep effects, and of the remainder, the last 43 have the best measurements of the temperatures for the dead space.
The gas thermometer was loaded while being thermostated at the highest temperatures to be measured. The initial measurements were commenced on the following day. At the beginning of a measurement, parts I, II, and III of the vacuum system were isolated but each was being evacuated, part I by its diffusion pump, part II by both ion pumps and part III by its ion pump. The procedure for the measurement of a gas thermometer temperature was as follows: (1) Before loading, or at other times before introducing helium as a pressure transmitting gas, the diaphragm section was isolated by valves 1 and 2, and the rate of pressure buildup was determined over a period of a half hour or more. 4 The rate varied between the limits of 2.5 to 5 mPa/min. With a total volume of about 100 mm 3 , the relative amount of gas accumulating in the diaphragm compared with the amollnt of gas in the gas thermometer (for a pressure of 10 5 Pa) varied between 6 x 10-12 min-I and 1.2 x 1O-11 min-I. The measurement is an "integrity check" to ascertain that the diaphragm is neither contaminated nor has a leak from the outside, and further that there is no significant leakage across the seat of valve 1. The same rate was observed whether the gas thermometer was evacuated or filled, hence it indicated no significant leakage occurred across the valve seat, and probably represented mostly, or totally, degassing.
(2) At the same time, the heater for the Ti-CuO purification trap was turned on. Within 1 hI', the trap reached its operating temperature (700°C), as confirmed by measurement with a Pt-10 percent Rh/Pt thermocouple referred to ice.
(3) Two integrity checks of the ac bridge and triple point cell were made. First, the ratio of the standard 100 Q resistor to a 100 Q PRT was read to assure proper thermostating of the standard.
Then the ratio of the resistance of a standard PRT immersed in a triple point cell to the standarclresistance was determined on the ac bridge. This thermometer was treated with the utmost care, so that the ratio was reproducible within less than 1 1 part in 10' over a period of 11 month, a fact that tended to confirm that the system was fun ctioning properly. Over longer periods, ·the observed small changes in the ratio could have occurred because of drift in the resistances of either the standard resistance or the thermometer.
(4) The ratios of the resistances of our 3 measurin g PET's to the standard resistor were determined at the temperature of the triple point of water. (5) The MacLeod gage, monitoring the vacuum in the upper cell of the manometer was read (typically 0.15 mPa or less) (an integrity check).
(6) The ion pump currents were recorded (an integrity check). The RGA ion pump, open to part II, could be expected to have a current of 0.5 J.la, equivalent to 0.2 J.lPa pressure.
(7) The Dewar around the molecular sieve (#13) trap was filled with liquid nitrogen, and when the temperature of the Ti-CuO trap had reached 700°C, part I of the vacuum system was isolated and helium was slowly passed through the traps into the lines.
(S) After isolating the ion pumps, helium was admitted to sections 2 and 4, until the pressure as indicated on the U-tube manometer was higher than the pressure in the manometer lines. Invariably, the gas flow was controlled to proceed in the direc-• The subsequent discussion assumes the gas thermometer was already loaded. tion from I to II and IV so that no backflow of helium would occur from other parts of the gas thermometer. Finally the manometer gas line was opened and the system pressure adjusted. When it was nearly at the pressure needed, the motor of the valve allowing communication of mercury between the upper cell and the lower cells of the manometer was turned on. The length of time it was necessary to open the valve before flow commenced was recorded (an integrity check). Final adjustments of the amount of gas in the system were made, the mercury valve was opened fully and the level of mercury in the cells was checked. The mercury level was remarkably stable over periods of weeks (within about 2 Mm), a fact that depends not only upon a stable temperature but also upon a stable contact angle with the wall. After any needed adjustment of mercury level, the gas pressure was controlled by "thermal injection", governed by feedback from the capacitance bridge to maintain the pressure at the manometer setting. (9) After reading the diaphragm zero, we closed the by-pass valve and opened the valve that connected the gas thermometer valve with the diaphragm. The pressure in the gas thermometer was made equal to the pressure in the manometer by adjusting the temperature of the liquid thermostat. When the pressures were about equal, the gas thermometer valve was closed, the by-pass valve opened, and the null of the diaphragm reread. The valves were then reset to reconnect the diaphragm to the gas thermometer so that any small difference between the pressure in the gas thermometer and the pressure in the manometer system could be observed and recorded. Then the gas thermometer was again isolated and the null reread . (11) Another set of null-pressure-pressure-null measurements was made with the diaphragm.
(12) Data also included were the time, the values of the gage blocks used in the manometer, and the regulator bridge and control settings. (13) The resistances of the platinum resistance thermometers were measured at the temperature of the triple point of water.

Equations and Calculations
The calculations for the gas thermometer temperatures can be made "symmetric", in that the equations can be devised so that the same calculation is performed for each measurement including that of the reference state. The details are given in [9]; the essentials are repeated here. We calculate a quantity designated Z that is an "augmented pressure", expressed in cm of Hg at 20°C, where the thermal expansion, the pressure head and the dead space effects are accounted for. It is (2) where i refers to the ith measurement, hot is the height of the gage blocks used in the manometer, and

Wt=7rt+J3tt+7rtJ3tt
In the order of the terms in eq (3), the significance of the symbols is as follows: 7r t is the pressure head and is given by where M is the molecular weight of the gas, g is the acceleration due to gravity, R is the molar gas constant=8.3137X 10 6 cm 3 Pa mol-1 K-l, T kt is the thermodynamic temperature for the kth element of length and ith measurement, and lk is the increment of length. The volume thermal expansion coefficient of the bulb, 13, is derived from + ( +t) 3 dt 1 t::.l 1 J3t t= 1 ao ,an ao r; T as determined from samples of material of the bulb of length lo at 0 °C. The term 7r ,J3t is negligible. The remaind er of eq (3) is the deadspace term. T t is the thermodynamic temperature of the bath, B t is the second virial coefficient of the thermometric gas at T t , P t is the pressure of the ith measurement, R is the molar gas constant, Vo is the volume of the gas thermometer bulb at 0 °C, Vk is the volume of the kth element of the connecting tube at 23°C, J3kt is the thermal expansion coefficient of the volume of the tube at temperature tkt , in the kth position during the ith measurement, and otkt =tkt -23 °C. This same temperature tkt, is expressed as an absolute thermodynamic temperature for Tkt in the denominator. The quantity Zt is also given by the expression (4) where with no the number of moles of gas, Po the density of mercury at 20°C, and other quantities as defined before. oZt is a term for thermomolecular pressure, and any variation of the pressure head in the manometer line from the level of the gas thermometer in the colder thermostat. We have calculated an "approximate gas thermometer temperature", T' GT, from the equation Then the gas thermometer temperature, T GT, was • Scale comparisonswill be given as pure numbers. All temperature scale quantities are therefore divided by their corresponding unit. derived by correcting T' GT for the effects of thermomolecular pressures and differing pressure heads, so that (5a) We can .substitute the expression for Z jO and ZiO from 4a mto 5a, and then because Po, g, and Vo are constants, and no is maintained constant, we get (6) with additional, insignificant second and higher order terms. Thus the gas thermometer temperature is equal to the thermodynamic temperature plus a term for the effects of gas imperfection. Substantially (PdPj)(TdTi)=(I+ktj)=Const. , for t j constant, (7) so. ~hat T GT/KGT = (Tj+ aPj)/K. Thus by de ter-mIlling T GT/KGT at a constant pressure ratio but ?iffering pressure ranges, T j/K may be found as the mtercept o.f the straight line of TGT/KGT versus Ph and the dIfference of the second virial coefficients can b e found from the slope.
The calcul ation is made by comp uter in a series of programs: (1) The international temperatures of the gas thermometer and the manometer reference station are calculated by a program in Fortran IV named B~IDGE. This program is given in appendix I, WIth a file of the calIbration constants A and B of the PET's (TCAL75). A calibration on the ac bridge at the zinc, tin and triple points was made in Jun e 1.974, and. the .last gas thermometer measurements m cluded m thIS paper were made in September of tl~e same year. (Th.e exact tim e does not appear to be Important. Startmg before the measurements were commenced, and extending to a time one year afterward, 4 sets of the thermometer calibrations have been shown to give a reprod ucibili ty of values at the temperatures of the fixed points within a standard deviation of 0.45 mIL) (2) The e~ect of .the pressure head is calculated by a. pro~ram III B.aslc named PYE. This prograJ~l is g~ven ~n append~x 2, and the subprogram LSFITI is given m appendiX 3. In this program, the the treatment of the temperature distribution is made as described in section 6. The in ternational temperatures, t68, were determined from the emf's of the thermoco~lpl es for the. locations shown in figure 3. A corr~ctlOn was applIed to t68 to give the thermo-dynamIC temperature, ttll, and this in turn was converted to a value of reciprocal absoln te temperature, 7  that the good thermal contact provided by the vacuum sea] between the tube and the top of th e header, H, brought th e temperature of the tube substantially into agreement with the temperature of the header at that point. Between the valve and the top of the header (33 mm), the val ues of 7 were interpolated line' al'ly, and simil arl y between the top of the header and the position of the thermocouple Al (19 mm). The length of the tube above the top of the header varied by an amount D2, which reflected the differential expansion between the conn ecting tube of the gas thermometer and its lnconel suspension , (Direc t measurements confirmed the calculated value of D2 within experimental err6r.) The thermocouples Al .. . , AI2, and B1 on the other hand terminated close to t he connecting tube but were independently suspended, being installed in a harness that was self-supporting and through which the tube could move. The expansion of the thermocouple harness was included in the therm al expansion term for the tube.
In order to interpolate the reciprocal temperature between the thermocouple lo cations, a quadratic equ ation, 7= B(1)+B(2) I+B (3) ]2, was fitted by least squ ares to the first four pairs of values of 7 versus position, 1, for Al . . . A4 (76 mm). A polynomial giving r as was fitted by least squares to the 9 pairs of reciprocal values of temperature versus position for A4 . . . A12 (203 mm). In the interval from A12 to B1 (25 mm) r was linearly interpolated. If the difference in emf between B1 and the calculated value for the temperature of the bath (the level of which was near or not more than 20 mm below B1) was sufficiently lar~e (>0.1 mV), the difference of the corresponding recIprocal absolute temperatures was linearly inter-pola~ed over the next 136 units of length (126 mm), and If <0.1 mV, over 82 units of length (76 mm). The · balance of the length of the connecting tube, and an additional length up to the center of the gas thermometer bulb, for a total length count of 688 (639 mm) was included in the calculation. The unit of length was 0.929 mm, which arose as a chart coordinate in measuring the tube diameter. The quantity 'lrt IS calculated as: as given in eq (3a), with the added definition that Ci ki is the linear expansion coefficient of the kth element for the ith measurement, and fltkt =tkt -23 . The output of the program gives the notebook referen?e number (8<p is the notebook and page, 81 the aSSIgned run number), the value of 'lrt in ppm, and the displacement D2 in mm.
3. The calculation of the deadspace effect depends upon the evaluation of "L,kVk t/ Tkt and involves exactly the same temperature calculations. They are associated with values of the volumes over the lengths from k-1 to k. The calculation is performed by computer with a program in basic named DEDSPS, given in appendix 4. A file of chart readings for the diameter of the tube is a part of this program, that also required the entry of D2 from the PYE calculation. The thermal expansion of the tube and of the thermocouple locations is combined in the term (l+i3kifltk i ). The output or the program gives the notebook and run reference numbers and the value of "L,kVki/Tki, designated V*TAD.
4. The value of Z is calculated by a program in Fortran IV named SUMMA, which is given in appendix 5. The calculation of the pressure is derived from eq (27) of [5], and is expressed in cm of mercury corrected to 20 °e. The output gives the notebook and run reference numbers and the value of Z in cm.

Determination of Parameters
Numerous calibrations and special measurements are involved in the gas thermometer calculations. Some of them have been mentioned in earlier sections, but they will be summarized here.

The Gas Thermometer Pressure
The value of pressure used in the gas thermometer equation is the sum of the pressure calculated by eq (27) of [5] plus the difference measured at the diaphragm. The height of the mercury column was determined by the length of the calibrated gage blocks. The density of the mercury was adjusted for its variation in temperature from 20 °e. We used the values of the thermal expansion of mercury published by Beattie et al. [14] . There is an offsetting effect of expansion of the gage blocks; the value in SUMMA reflects the difference between the volume expansion of mercury and the linear expansion of chromium carbide. The small values of the copperconstantan difference thermocouple readings for the manometer cells and for the main mercury column were converted to temperature differences by use of a "standard value" of the thermoelectric power, 40.5 /LV/K [1 5], and the reference temperature was measured by a calibrated standard platinum resistance thermometer.
In order to calculate the pressure differences from the gage readings of the diaphragm, its sensitivity was determined from the pressure changes calculated to have resulted from imposed changes of gas thermometer temperature.

The Thermal Expansion of the Bulb
The term in the gas thermometer equation second in importance to the pressure ratio is the term for the thermal expansion of the bulb. The thermal expansion coefficients of both 80 percent Pt-20 percent Rh and 88 percent Pt-12 percent Rh have been carefully measured from -25 to 550 °e. The constants entered into SUMMA are those for the thermal expansion of 88 percent Pt-12 percent Rh where the length ratio is expressed as a polynomial, The estimate of the standard deviation of a predicted point varies from 0.08 ppm in mid range to 0.14 ppm at either extreme. The estimate of the residual standard deviation of 0.14 ppm is consistent with our expectation of the average imprecision of a single measuremen t.

The Oeadspace
The values used in the deadspace calculations were obtained from various sources. The volume of the bulb was calculated from its dimensions except for the last gas load. In the bakeout preceding the final set of measurements, the bulb was deformed by an external overpressure. When the bulb was removed from the apparatus, its changed volume was determined from the contained weight of water to be 4.2103 X 10 5 mm 3 at 0 °e, and this value was used in the calcubtions for the final set.
To determine the volume of the tube, its diameter is required as a function of position. This was deter-mined as described in [7] with an estimated total uncertainty that is insignifi can t in th e fin al results.
Temperatures were derived from thermocouple readings for the calculation of 'IT and Vr . It m ay be useful to remark in advance th at con sid erable enOl' in the thermocouple values will not significantly affect the accuracy of the fin al r esul ts. The final arrangement illustrated in fi gure 3 was appli cabl e to the last 30 states, where thermocouples were added about the h eader , H . Th e couples 0 6, 0 7, and 011 comprise copper-con stan tan legs (T ype '1'), and 09 has Pt-lO percent Rh /Pt legs (Type S) . The couple 06, with its reference junction a t 0 DC, was used to measure the temper ature of the side of the header, and inferentiall y of the refer ence j unctions of Al ... A12, and Bl, becau se it is expected that the temperatUTe of th e reference rin g D is nearly the same as the side of the h eader. The differ ence couples, 0 7 and 0 11 were used to determine the increment of temperatUTe between the top and side of the head er and between the end of the tube at th e constant volume valve and the sid e of the he ader , respectively. 0 9 was a difference coupl e runnin g between th e b ath and th e sid e of th e header. It was not needed for the Jast 30 states, but was a necessary link with earlier meflsurements to evaluate th e earlier data. All copper-constantan r eadin gs were converted to temperatures by use of a standard tabl e [15] . Th e temperatures determined by all Pt-lO percen t Rh /Pt thermo coupl es wer e evalu ated from calibrations of similar couples which could be r epresen ted by a quartic, (9) with a standard deviation of the values at each in terval of 25 DC up to 500 D C of 0.47}.LV wh en B 1 = 5.45846 X 10-3 }.LVr C, B2 = 1.l3497 X lO-5 }.LVWC)2, B3= -1.52447 X lO-8 }.LVWC) 3 and B , = 9.0603:3X lO-12 }.LVW C) 4. When the emf of the Pt-10 per cent Rh/Pt thermo couple B1 was meas ured a t 400 DC (under special conditions where the depth of immersion wa,; thought to be adequ a te), however , i t was found to differ from th e equ ation by -28 }.LV ; the cause of the difference appeared to be work-h ardening of the platinum leg during in stall ation that was not subsequently relieved by ann ealin g. This difference was assumed to be typical of all the deadspace couples and was taken into account in th e gas thermometer calculations by multiplyin g all valu es of Al ... A12, and B1 by 1.00895.
In an earlier arrangment of th e thermocouples applicable to the preceding 42 states, the 0 9 junction was located on the top of the header , and there was no 06, 07, nor C11 couple. Furthermore, 0 9 was calibrated at 400 ° C and found to differ from the value calculated from eq (9) by -51 }.LV. Consequently, the measured emf 's for 0 9 were multiplied by a factor , 1.01644, to account for the differen ce.
To carry out th e calcul ati on described in the preceding paragraphs it was n ecessary to predi ct values of A1 (5), C7 (5), and 0 11 (5), for t he ea rlier measurements. This could be don e from the information available from later measurem en ts. In on e switch position, the thermocouple emf's could be read with respect to B1 as the reference junction (position 4) and were designated as Al (4), etc. In anoth er swi tch position, the thermocouples were referenced to their own j unctions with copper (position 5), and measurements with this reference were design a ted as A l (5), etc. B ecause the temperature of the referen ce rin g is uniform, A 1(5) -A1(4) = Bl (5) . Position 5 of th e switch was lI sed to read the thermo coupl es for the l as t 30 states. As shown in figure 4, the valu es of A 1 (5) and 0 7 (5) are wellbeh aved fun ction s of the bath temperat ures as expressed by 0 9 (5) . As for 0 11 (5) , its values depend not onl y upon the temperature of the bath , but also upon th e temperature of the room and of the plate direc tly b elow it which is cooled by tap water. For const an t room and cooling water tempera tures , C1l deer'eases linearly as th e b ath in creases in temperature, over a r an ge of about 40 MY. Th e cooling water temperature vari es seasonally from less than 8 D C in th e winter to over 20°C in th e summer; this expl ains th e large differ en ces in 0 6(5). The 40 }.LV r ange of 0 11 (5) tends to b e more negative as the cooling water temperature increases. Th e v alu es of 0 11 (5) used in tabl e 2 were derived on the basis of a linear variation from 40 or 41 }.LV at 0 DC to 0 MV at 457°C. A uniform ch ange of 20 }.LV in th e values of 0 11 (5) causes an error in the calculated gas thermometer temper ature varying from 0.2 ppm at 140 °c , up to 0.5 ppm at 457°C. Therefore high accuracy in predicting these 0 11 (5) v alues is not a requirement. To summarize: For all the earlier data for which only values of 09 (5) referenced to the top o~ t~e header arc available, we find the temperature dlstnbution along the tube by the following steps: (1) R ead 07 (5) and Al (5) from figure 4. Calculate 011 (5) by E-40 (457-tbath)/457 p.V.
(2) Find the emf of a standard couple referred to ice for the top of the header in p.V for Pt-l0 percent Rh/Pt [rom V~ (ttath) -E¢ X 1.01644, where Vq, is calculated from the quartic equation and E¢ is 09(5).
(3) Deduct 0 7(5) X O.125 /lV to account for the difference in emf corresponding to the difference in temperature between the top of the header and the aluminum reference ring. The equivalent emf for 06 (5) can be found if desired .
(4) Add Al (5) X 1.00895 to give a synthesized value of emf referenced to ice.
(5) Find tAl from the derived emf. Similarly, the temperatures can be found for all other positions including Bl.

Determination of Thermomolecular Pressures
Using the apparatus developed for this purpose, and described in section 2, we measured some thermomolecular pressure differences for helium at temperatures of 140, 260, 374, and 457 DC at pressures varying from 2.5 X 10 3 to > 1 X 10 5 Pa. The results were correlated by an equation based on Weber's treatment [16), where the product of the average pressure p = (PI + pz) /2 and the pressure difference flp is expressed in relation to T z , the upper temperature; TI , the lower temperature (296 K); and To, the reference temperature (273.15 Ie). The reference temperature is involved only in that the exponent n is defined by the assumption that the temperature dependence of the viscosity of helium can be expressed by (11) For the temperature range 0 °C <t < 460 DC, n

Tabulation of Results
The measurements which will be presented in subsequent tables are subdivided according to experimental groupings labeled Set A to Set 1. More information by group is given in table 1. The values of the calculated quantities for the states of the gas th ermometer are tabulated in tables 2 and 3. The results have been divided into those in table 2, in which temperatures of the header were derived by calculation, and those in table 3 for which the thermocouple emf' s were experimentally observed. The deviations of international temperatures from gas thermometer temperatures were calculated from the equation (12) where T j is the absolu te tempertLture of the chosen reference sttLte with the correspondiJig value of Z =Zi' The difference T' GT /K' GT-T68/K68 was further modified for the effect of the lower level of the gas thermometer bulb in the high temperature bath; the smaller pressure head requires a decrease in all the gas thermometer temperatures of 0.5 ppm except                  those measured in the low temperature bath. These changes are smaller than the thermomolecular pressure effects that were calculated by eq (10), and that were included in the results in column 15, head ed T CT / K CT -T68/ K 68.

Review of Previously Published Results
The results r eported in our last paper [10] were calculated by programs similar to those used for the present data. Th e P YE and DEDSP S programs derived temperatures in the manner described as our "original method," alld the SUMMA program differed in the form of the equation and the constants used for the thermal expansion of the bulb, with a trivial effect on the results. Both of these differences can be expected to produce a differen ce less t han 0.5 mK from t he present approach. The original data and some of the quantities derived from intermediate calculations are given in table 4, columns (1) to (10), and the values reported previously are given in column (11). As stated in section 2, the actual volume of the bulb was found to be different from the value used in the original SUMMA program. The change necessitated by correcting the volume of the bulb is given in column (12). A further adjustm ent for the effe cts of th ermomol ec ular pressur e, as determined by eq (10), is given in column (13).

The Effect of Gas Imperfection
When the numerical differences, T CT/K cT-T68/K 68, for the same temperature obtained at nearly cons tan t pressure ratio but over differ ent ranges of pressure are extrapolated vs the upp er pressure to zero pressure, the intercept is T /K -T68/K 68' Alternatively, if the equa tion of state for 1 mol of gas is consid ered in the form pV= RT+Bp the thermodynamic temperature T is related to' a gas thermometer temperature T CT by (13) where j's r efer to the measuring state, and i 's to the reference state, an d R to the molar gas constan t. e~msid e~ed in terms of this eq uation, the accuracy Wlth whlch the thermodynamic temperature can be found by extrapolation will be optimized when the range of p /s is large, so long as the lowest Pi is still large enough to be realized with the same r elative accuracy as large valu es. However, at low pressures, thern;lOmolecular pressure effects become important. For mstance, two pressure ranges were used at 100°C. For p j= I X 10 5 Pa , the combined th ermo-mo~ecular pr essure effects for p j and Pi are 0. 34 ppm, whlle for p j= 18374 Pa, the combined thermomolecular pressure effects are 10.6 ppm. For all values in Sets A, B, e, G, H, and I all p/s were abo ve 5 X 10 4 P a, so th at the thermo~olecular pressure affects the gas thermometer temperature by at most 2mK.

T/K = TcT /K oT-{[B J -B i(pdpi) (T j/Tt)]p j R }/K
Unfortunately, the number of measurements at vary:ing pressure .ranges was insufficient to permit preClse extrapolatlOn to zero pressure. An alternative ~o extrap.olating i~ to calculate the effects of gas ImperfectlOn from mdependently determined second virial coefficients of helium. Our original decision to r ely upon extrapolation involved our appraisal that the use of virials from the literature would not give as accurate thermodynamic tem peratures. This remains our conclusion, provided there are sufficient gas thermometer measurements; 6 in the interim we are compelled to use literature values of virials to derive thermodynamic temperatures, but with more uncertainty than we would expect when our own measurements are sufficiently complete.
Second virial coefficients calculated from compressibility measurem ents are probably more accurate than from ('ther sources; the Burnett method [17] is probably the most accurate of the experimental compressibility techniques. We have chosen a set of virial coefficients based upon the values obtained at the NBS by the Burnett method for temperatures up to 150°C [18] . The values at higher temperatures are those of Y ntema and Schneider [19], who also used the Burnett method. The interpolation and smoothing were mad e consistent with acoustic r esults of Gammon and Douslin [20]. The results from these three sources differ from each other substantially more than their respective estimated uncertainties. The values used and the equation derived by a least squares fit to the data are given in The effects of gas imperfection increase with increasing temperature and, of co urse, with increasing pressure. It amounts to 0.0227 K for T2=730 K and p = 10 5 Pa, and only 0.0002 K for T z= 29 3.81 K and p = 14500 Pa. Th e multiplying factor on Bl , j = Pt/PrTj/ Ti, is always greater than unity above o °e, therefore the sign of the difference of th e , However, another approach using low pressures corrected by carefull y determ ined t hermomolecular pressures is probably better. It is discussed ill the fllla1 section of t his paper. virials B2 -fBI, is always negative. Consequently, the thermodynamic temperature calculated from eq (13) is always greater than the gas thermometer temperature.
The results corrected for gas imperfection are given in the final column of tables 2, 3, and 4, and represent the difference b etween international and thermodynamic temperatures. An equation giving the deviation as (15) was fitted to all the values in tables 2, 3, and 4 by least squares. The constants found are given in table 6.  The estimate of the standard deviation of the fit, (16) where N is the number of points and M is the number of constants, is 1.52 mIL The estimate of the standard deviation of a predicted point varies from 0.75 mK near the triple point, to 0.34 mK in the middle of the range, to 0.46 mK near the upper end. The values and the calculated curve are shown graphically in figure 5. The least squares computer program is given in appendix 6 and the computer printout of the least squares solution is given in appendix 7. values versus t68 was derived from columns 2, 3, 4, and 5 of tables 2, 3, and 4 and is shown in figure 6. Subsequent to all the reported gas thermometer measurements, we carried out special measurements at 408 and 414°C to study this apparent contradiction. It has already been noted that the triple point resistances of the thermometers usually increased with time of immersion in the stirred liquid thermostat of the gas thermometer. But we have also observed that the triple point resistance, if already in substantial excess of its calibration (annealed) value, decreases initially when placed in the thermostat at 457°C. We now interpret this fact as resulting from a balance between the rate of increase of the resistance from work hardening of the platinum by the mechanical acceleration from the stirring and the rate of decrease of the resistance by annealing of the platinum at 457°C. At somewhat lower temperatures (350 to 420 °C) the rate of increase from work hardening remains high, bu t the annealing rate is substantially smaller, so that after the thermometers had been in the thermostat, a rise in the triple point resistance was observed. We found the triple point resistances to increase progressively wi th succeeding periods in the thermostat at 408°C, and, although the average temperature remained ne arly constant-both in fact and as caleul ated from the average of the values determined from the thermometers--the average deviation from the mean increased successively from 1.63 mK to 1.89 mK to 2.1 3 mIL When annealed before each measurement, the average deviation from the mean became 0.65 mK and remained essentially constant with further cycles. The precision of the values of t68 was the same after 7~ h anneals as after 2X h anneals. Although we also used different annealing temperatures, the best choice, on the basis of Berry's isochronal step depression results [13], is probably 450°C. Some conclusions one may draw:

Study of IPTS Realization Above 300 DC
A. Platinum resistance thermometers become very sensitive to mechanical shock at temperatures in excess of 350°C; and up to 457 °C, the calculated temperature determined by them tends to drift in value with progressive work hardening. For relatively minimal lengths of exposure these increases were limited to steps of resistance corresponding to less than 0.30 mK.
B . The direction of change of the resistance of a PRT at the zinc point from its annealed value because of work hardening is unpredictable on an a priori basis. For the thermometers we happened to use, the average of the values of t6 8 calculated from the measurements were a good approximation (within 1 mK) of the true value of t68 , whether determined from the annealed thermometers, or from thermometers of the same group which exhibited the effects of work hardening. C. The values of t68 calculated for a given temperature as measured with a group of thermometers kept as closely as possible in an annealed condition, are in good agreement with one another. The best choice of annealing temperature is probably 450°C, and at least for thermometers only slightly work hardened, a half-hour of annealing is sufficient. We believe under our conditions of measurement, the thermometers should be annealed each time b efore measurement is made, and, as has become our practice, the time during which the thermometers are subjected to mechanical acceleration should be kept as short as possible.
Other causes of measured temperature difference, investigated and found insignificant, were possible temperature gradients in the gas thermometer bulb case and possible imprecision of the thermostat temperature.

Estimation of Total Uncertainty
The stated total uncertainty consists of the limits, at 99 percent confidence level, for the random errors; and the systematic errors, estimated conservatively enough to warrant about the same confidence level. We discriminate between errors which fall into three categories: A. Sources of random error which contribute to the imprecision observed in the results. B. Sources of error which produce a constant bias in the results, but which are estimable in terms of random errors determined from other experiments. C. Sources of "systematic" error which prod~ce a constant bias in the results, but for whIch there is inadequate experiment and the~ry: to permit their evaluation by accepted statIstlCal techniques. The estimates of the values of the random errors (A and B) are expressed initially as one standard deviation.
The error of the reported quantity, the deviatio.n of international temperatures from thermodynamIc temperatures, T/K-T68/K68' can be esti~a~ed by combining in quadrature the errors of realIzmg t~1e international temperatures with the errors of realIzing the thermodynamic temperatures. The errors for each depend upon the range of temperature, and for the thermodynamic temperatures, also upon the range of pressures. There are t\\"O values of temperature for which a larger number of values was determined-373 and 730 K. We propose to assess the errors at these two temperatures, being values at or near the extremes of the r ange, as well as to consider the statistical evaluation of the imprecision derived from fitting the curve in term s of the estimate of the standard deviation of a predicted point.
The errors in the realization of the international tem peratures are those from : (1) The calibrations. The systems of measuring instrum ents, fixed points and thermometers is sufficiently exact that for t he foul' standard PRT's used in the measurements, the constants A and B in the equation (17) determined in four successive calibrations over a period of two years defined the value of t he temperu,ture at the zinc point with an estimated stu,ndard deviu,tion of the mean of the thermom eters of 0.20 mK. "These values arc not to be added with A and B on the basis that their combination \\'ith random err or~ is not ph il oso phically acceptable. They are ~tated at 1/3 the \'alue we estimate, for easy compariso n with t he error~ of Type A and B.
bThese arc valu es, sugge~te d by experim ent, for t he ae effects of shuntin g by de leads, i.e., a lossy capacitor.
The approximate gas thermometer temperature was calculated from eqs (2) through (5a) and the thermodynamic temperature was then found by accounting for the effects of thermomolecul ar pressure and gas imperfection. The approximate gas thermometer temperature can be expressed in terms of the ratio in a way useful for error analysis as The value of the second term is small relative to Tc T , so that the relative errors of the products in t he first term are directly refl ected as relative errors in the gas thermometer temperature. In the order of the quantities in eq (18), the elTors are (1) The relative uncertainty of the press ure ratio, o (~~)/(~:} The error h u,s Lwo main components, an impreclslOn depending 1I pon the regulation of the press ure and the variu,bility of the diaphragm, u,ncl a fixed uncertainty that can be evalll u,Led from the imprecisions in p, g and h. The pressure difference between Lhe gas thermometer u,ncl the manometer was m eu,s urecl by the diaphragm tru,llsc!lICer before and u,fter the bath tem perature readings . The estimates of the standard deviation of Lhe mean \\"ere calculu,ted from the deviations for the experimental pressures at 100 and 457°C and also at the fiducial point. For 457 °C, the estimate of the standard deviation of the gas thermometer tem pera ture from this source, for a pressure of 10 5 Pa is ± 0.23 mKCT' To this should be added a possibl e random f1u ctuu,tion of the regulated press ure, estimated in terms of a variance from the manometer se tting. This is estimated to amount to ± 0.033 Pa aL 1 u,tm (101,325 Pa) and 0.02 Pa at 3.5 X lO" Pu,. These effects combined in quadrature with the variu,biliLy of the diaphragm are, for one standard deviation, 0.~33 mKCT at 10 5 Pa and 0.52 mKCT at 3.5 X 104 Pu,.
For 100°C, the estimate of the standard deviation of the mean of the diaphragm readings affects the gas thermometer temperature by only 0.34 mKCT for 10 5 Pa, and 0.17 mKCT at 1.8X 10 4 P a. An estimate of random flu ctuation in gas t hermometer temperature, from the manometer setting at 10 5 P a of 0.033 Pa is equivalent to 0.12 mKCT and at 1.8X10 4 Pa of 0.02 Pa is equivalent to 0.41 mKCT' These errors combined in quadrature are 0.12 mKCT for 10 5 P a and 0.44 mKCT for 1.8 X 10 4 Pa.
The total uncer tainty of the pressure ratio at the manometer is discLissed in [5]. It was stated to be ± 1.5 ppm at the 99 percent confidence level, or about 0.5 ppm for 1 standard deviation of the mean. The error can be evaluated from measured quantities, so is of a "B" type.
(2) The contribution of the pressure head to the measured pressure in the present temperature range varies between 4.5 ppm and 11.5 ppm and is unlikely to be in error by more than 2 or 3 percent. Therefore, the error of the pressure head correction is relatively insignificant. (3) The errors in the thermal expansion measurements u sed to calculate the thermal expansion of the bulb. It is believed that the systematic effects are relatively small. Therefore, the error can be evaluated from the imprecision of the measurements, and is of type "B". The volume expansion is calcu- (20) lo is the length at 0 DC, and t68 is the temperature at which l was determined. As stated in section 6, a fifth order power series was fitted by least squares to the data. The estimate of the standard deviation of a predicted point varied from a minimum of 0.8 part in 10 7 from 0 to 54 DC, 1.9 part in 10 7 at 457 DC, and was 1 part in 10 7 at 100 DC. Therefore, the error contributed to the thermodynamic temperature, expressed as the estimate of a standard deviation is 0.38 ppm at 100 DC and 0.62 ppm at 457 DC.
(4) The uncertainty in the dead space calculation is primarily due to the uncertainty in the determination of temperatures along the connecting tube. These were measured by thermocouples, the measurement problems of which have been discussed earlier in this paper. The effect of the temperature uncertainty can be evaluated from an approximation for the value of V T, which is (21) where T =(t+23) j2+273.15 is the absolute temperature averaged between the temperatures of the room and the thermostat. We have evaluated the error in the gas thermometer temperatures resulting from a total uncertainty in T of ot= ± 0.4 % X (t-23) DC. For instance at 457 DC, the average temperature, T, is assumed uncertain by 1.74 DC for the measuremenots (corr~sponding to an uncertainty of about 2.5 C at t Itself). A summa.ry of the error calculations, which become insignificant below 141 DC is given in table 8.
' The value of one standard deviation of the mean S m, is about % of the total uncertainty given in th~ final column.
The remaining sources of error come from uncertainties in the evaluation of thermomolecular pressure and gas imperfection effects. The thermomolecular pressure correction was largest for states 10.01 and 10.02 at 457 DC, where it amounted to 16.4 mKGT. The only significant error in the correction derives from the imprecision of the measurements for determining thermomolecular pressure. The estimate of the standard deviation of the mean for the results at each temperature measurement was 1.1 percent at 140.5 DC, 0.68 percent at 260 DC, 1.1 percent at 374 DC and 0.92 percent at 457 DC. The error is of type B, and amounts to about 1 percent of the calculated correction, which in general is not significant.
The effects of gas imperfection were calculated from the virial coefficients for helium, selected as we described earlier. The largest correction for gas imperfection was made for states 9.04 and 9.05 at 400 DC, but it was almost as large for the numerous measurements at 457 DC with p=10 5 Pa. We postulate that the instrument error in the measurements to evaluate Z =pVjRT=l+BpjRT produces a total uncertainty in Z of at least ± 5 ppm, and is that small only for exceptional work. On the basis of a constant error in the virial coefficient of ±0.1l cm 3 jmol, this in turn makes an error for the combination of B2 -B1 of ± 0.16 cm 3 jmol, which is perhaps that small only in the range 0-150 DC. At least twice that total uncertainty should be assumed for the measurements at 400 DC (states 9.04 and 9.05), and for the measurements at 457 DC. These uncertainties amount to ±4.4 mK for states 9.04 and 9.05 and ±3.7 mK for the higher pressure measurements at 457 DC.
We can summarize the errors in the thermodynamic temperature as being The total can be found by addition in quadrature of the values given in table 9. • The errors of Classes A a nd B are given as one st andard devi a ti on. The estim<lted v<l lues of CI<lss C arc di vided b~' 3 in ord er to give a " coJnlTIon bas is" of comparison with A <l nd B.
Thus th e es timate of th e standard deviation of th e mean of T/ K -T68/K68 is th e combination of A, B, and G from t ables 7 and 9, as follows : The observed imprecision should be consistent with the sum of "A" of hardening of the PRT's) . These latter are, in quadrature, the following: 5.2 X 10-4 at 10 5 Pa and 457 °e 6.6 X lO-4 at 3.5X10 4 Pa and 457 °e 1.9 X 10-4 at 10 5 Pa and 100 °e 4.8 X 10-4 at 1.8 X 10' Pa and 100 °e.
The corresponding "observed imprecision", the standard deviation of the mean of the experimental values of T/K-T68/K68' at each temperature is 6.6 X 10-4 for 10 5 Pa and 457 °e 1.02XlO-3 for 10' P a and 100 °e. The large experimental value at 100 °e may indicate sources of random error that were not adequately appraised (as discussed in [8]) . The observed imprecisions can also be evaluated more broadly in terms of the deviations of the 72 values of T/K -T68/K68 from the least-squares fit. The standard deviation of a predicted point for T/K -T68/Ke8 is 4.6 X 1)-4 at 457 °e and 3.5 X 10-4 at 100 °e.
The estimates of the total uncertainty can be derived by adding the estimates of type A and B in quadrature, and multiplying by 3 as the approximate factor from Student's  2.7 X 10-3 random error 4.6 X 10-3 system atic error 2.8 X 10-3 random error 3.1 X 10-3 systematic crror 1.2 X 10-3 r andom error 2.0 X 10-3 systematic error 1.8 X 10-3 random error .54 X 10-3 systematic error

Discussion
All data, including those published previously, have been interpreted in the light of fur ther information and improved understanding of the apparatu s. The reevaluation of the volume of the bulb makes an appreciable difference in the dead space calculation for the data published previously (amounting to as much as 2.8 mK at 414 .75 K). I n contrast, the difference in composition of the top of the bulb from the remainder causes no net change in bulb volume with temperature (compared with an all 88 percent Pt-12 percent Rh composition) in excess of 1 ppm, ?ecause the shape of the distortion is compensating In the volume to the first order. The problem is minimized because the total thermal expansion of the two alloys becomes equal for t"-'360 °e. Inasmuch as we had used the values for 88 percent Pt-12 percent Rh in the previous calculations there is no error exceeding 1 ppm from this cause in any of the results, old or new.
All the evidence of system cleanliness-R GA scans, diaphragm, and gas thermometer stability, reproducibility before and after gas thermometer bakeouts-supports the proposition that the results are free of significant effects of sorption (except se t G which was not included in the final results).
Above 0.01 °e, gas thermometer temperatures are increased by the effects of sorption . It is not generally recognized (although it was mentioned in [9]) , that our earliest measurements yielded "thermodynamic temperatures" in excess of 373.15 K at the normal boiling point of water; these values declined as the gas thermometer became cleaner. We believe we could adjust the purity of the gas thermometer to produce a selected value of the steam point over a range of about 0.1 K. In the less clean condition, we expect that there would be less precision and stability, but the gas thermometer would probably be deceptively stable even with enough sorption effect to give "Tth"=373.15 K.
Operating in the higher temperature range emphasized certain problems, viz., the measurement and treatment of dead space temperatures fl,nd of thermomolecular pressures, the evaluation of the effects of gas imperfection, and the realization of international temperatures, all of which can be handled better in the future with higher resulting accuracy. We have installed a new gas thermometer that h as a much improved thermocouple installation for measuring the dead space temperatures. Special annealing and testing equipment was built to assure that these couples were uniform and had emf's in close agreement with accepted values. The effects of thermomolecular pressure and gas imperfection are both encountered in any of our gas thermometer measurements, but they can be evaluated with little loss of accuracy by a modification of the customary measuring techniques: (1) Independent measurement of thermomolecular pressures. The results of some measurements have been reported. Further work is planned where conditions are more exactly defined and more accurate values are obtained at lower average pressures.
(2) Use of relatively small fidu cial pressures. Our ability to determine the virial coefficients is limited by our ability to determine thermomolecular pressures. Therefore , the highest accuracy results when only one factor is involved. The m easurement of thermomolecular pressures and the theoretical justific ation for the transfer of results to the gas thermometer will be the subj ect of a separate paper.
It has recently become possible to operate the gas thermometer with no loss of accuracy in a pressure r ange mu ch lower than was the limit ten years ago. Of special benefit in this respect is the ten-fold improvement in the calibration of gage blocks. No new manometer errors become significant in a low range of pressure based on a fiducial value of 2.54 cm of Hg. The great advantage of this lower pressure range is that the correction for gas imperfection is sufficiently small that the uncertainty is less than 1 ppm at 457 °0 for a pressure of 6.78 cm Hg, and the thermomolecular pressure correction, though large, is measurable with such accuracy that its uncertainty is about the same.

Summary and Conclusions
The deviation of international temperatures from thermodynamic temperatures has been evaluated by a constant volume helium gas thermometer b etween 0 and 457 °0. Previously reported values were modified for a correction in the dead space effect, and al so in light of a reinterpretation of the effects of gas imperfection and thermomolecular pressure. The results are consistent with independent measurements of each of these effects. As examples, the total uncertainty, con sisting of the estimate of the random errors at the 99 percent confidence limits and the systematic errors, intended to b e conservatively enough s tated to warrant about the same confidence, was calculated for each of two pressures at 100 and 457 °0. The estimate of the systematic error in each case was appreciably larger for the higher pressure r ange, but the resul ts agree with those mea ured in a lower pressure range. Hence the uncertainties for the lower pressures are regard ed as the proper ones, and are given following the value of the deviation :