A Simple Method for Determining the Volume of Closed Containers C. A. Seitz and David E. Emerson Helium Research Center, Bureau of Mines, U . S . Department of the Interior, Amarillo, Texas. 79106 THISREPORT describes a simple method for accurately determining the volume of closed containers and a procedure for routine volume determinations of sample containers that has been used to determine trace impurities in helium ( I ) since August 1960. A method for determining trace impurities in Grade A helium by Kirkland, Brandt, and Deaton does not contain a reference to procedures for determining the volume of the metal sample containers. The volume of each metal sample container used in the trace impurity determination is approximately 10 cc; a n apparatus was designed and built to determine the volume within 2 parts in 1000. The procedure for volume calculation is given.
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L,
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I I
I I
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T--’- - I
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I
SPECIFIC CONSIDERATIONS
I
Derivation of equations used to determine the volume of a sample container under isothermal conditions is as follows. In Figure 1, the volume between valves 1, 2, and 3 and the height of mercury is termed “working volume” (VZ). This volume is evacuated to the extent that the pressure (P) can be considered to be zero (less than 0.1 mm of Hg). The contents of the known volume ( Vl) are then expanded into the Vz. The volume of Vl is based on the total volume Vl weight of mercury it will contain. When starting at a pressure of Pi in VI, the final pressure in the total volume ( Vl VZ) is designated as PlZ. Using Boyle’s law and accounting for the mass of gas before and after expansion, Equation 1 can be written, where Vrr, represents the first time the working volume is used. Thus,
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”2
I
I
II
-I
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-1-
+
+
P12Vw1 = P1Vl
Figure 1. Diagram for volume calibration apparatus
- P12Vl
where the temperature ( T ) is the same for each container and remains constant, and the compressibility factors ( Z ) are equal to unity. Solving for VW,gives
If a pressure measuring device were used which did not change volume as a function of its measurement, then VW, would equal VW,. However, the mercury manometer changes volume with pressure and a relationship is established between VW,and Vw,. Thus, VZ = mh
Next, the container whose volume is to be determined is opened to the atmosphere and V z is re-evacuated. The contents of V3 are expanded into the total volume VZ V3. When starting at a pressure of Pain V3,the final pressure in the total volume ( Vz V 3 )is designated as P23. Using Boyle’s law, Equation 2 can be written, where VW, represents the second time that the working volume is used. Thus,
+
+
p23vW2
=
p3v3
-
p23v3
at the same conditions as above for T and Z . Solving for Vw, gives
where m is a factor associated with a particular manometer, in units of cubic centimeters per millimeter, h is the manometer reading in millimeters, and VT, is the working volume a t zero mercury height. The factor m can be calculated from the manometer tube diameter or determined experimentally by a series of expansions into the working volume from Vl. The expansions are performed by starting with various pressures in VI. The value of m is determined from the slope of the curve obtained by plotting the volume V2 as a function of the height of mercury reading h. Thus, the relationship for VW,and Vw, is shown as VW, = mh,
(1) C. G. Kirkland, L. W. Brandt, and W. M. Deaton, BuMines R e p . Inrest. 5644, 1960.
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ANALYTICAL CHEMISTRY
+ VW,
+ VW,
(3)
where ha and hb are the values read from the manometer and used t o determine P ~and z P 2 3 , respectively.
VW,is determined from Equation 1. The volume of the container V 3 can be determined after Vw, is known. The difference between Vw,and V,, is obtained t o eliminate the undetermined quantity Vrv, and to establish a relationship between the two working volumes:
-
V,,
v,,., = m(hb - I?,)
(5)
The simultaneous solution of Equations 1, 2, and 5 yields the relation of Va and known quantities as shown in Equation 6.
The values of pressure in the equation are determined by the height of mercury above zero corrected for scale irregularities, temperature, and gravity. Scale corrections, if significant, must be made prior to applying correction factors for temperature and gravity. The scale corrections were assumed t o be insignificant for this study. Pressure can be calculated from P
=
(hR
- h0)K
where P = pressure in mm of mercury hR = height of mercury reading ho = zero mercury height K = KgKI K , = gravity correction K I = temperature correction The corrections due to temperature and gravity are normally given as a subtraction factor but are best suited as a multiplication factor because the temperature and gravity corrections are proportional to the height of mercury (2). For clarity, alphabetic subscripts are used with the manometer readings, / I , because each pressure is determined by using the difference between two readings. Each manometer reading is defined as follows: hc = the manometer reading with the working volume (V,) evacuated prior to making the expansion from VI into V2. ho = the manometer reading with the working volume at atmospheric pressure. h, = the manometer reading after expansion from V1 into V2. hT = the manometer reading with the working volume ( V2) evacuated prior t o making the expansion from V 3into V2. hD = the manometer reading after expansion from V 3 into V2.
- ho)K (hc - h,)K
Notice that K cancels when temperature is constant and is not needed for volume determinations.
Equation 8 is used t o calculate the container volume ( V 3 ) . Neglecting the compressibility factor results in a n error of less than 4 parts in 10,000. The volume change in the movement of the stem of valve 3 is included in the volume of the container determined. It has been found to be less than 0.01 cc when opened one-fourth turn, which is sufficient to allow pressure equilibrium in less than 15 seconds using the described procedure. GENERAL CONSIDERATIONS
A general approach to the determination of the volume of an unknown container is given considering temperature, pressure, and compressibility factors. The three containers are designated as 1, 2, and 3 as shown in Figure 1. In each case, we are assuming that the starting pressures are zero or above. I n the first case, allow the pressure in Y l and V2to equalize, starting with VI at P I , T I , and Z1,and V2,at P z , T2, and Z2, resulting in a pressure of P l z . Two compressibility factors, Z A and Z B ,are introduced such that Za is the compressibility factor used for the pressure P12a t T l , and Z B is the compressibility factor used at P 1 2and T2. The relationship between two volumes, Vl and V2, in terms of the temperature and compressibility factors is shown in Equation 9 in accounting for the mass of gas before and after expansion.
(9)
Next, allow the pressure in V2 and V 3 t o equalize, giving a final pressure of P23when starting with the conditions V2 a t P ~ AZ2a, , and T2,and V3at Pa,T 3 ,and Z 3 . When accounting for the total mass before and after expansion, Equation 10 is written where Zc is the compressibility factor for the pressure P23 a t temperature Tz, and Zu is the compressibility factor for the pressure P23 at temperature T 3 .
Therefore, P 1 = (hc Pi2 P3
= =
(hT
P23 = ( h r
- ho)K -
Equation 1 1 results when eliminating V2between Equations 9 and 10.
h0)K.
Substitution of the above into Equation 6 gives Equation 7 and when reduced, gives Equation 8.
If Tl = TZ= T 3and the compressibility factors are taken as unity and P I A = P 2 = 0, which were the assumptions made in deriving Equation 6 for the volume of traps, Equation 12 results, (2) W. G. Brombacher, D. P. Johnson, and J. L. Cross, Nurl. Bur. Std. , Monograph No. 8 (1960). VOL 40, NO. 1, JANUARY 1968
261
which is equivalent to Equation 6 if rn were to equal zero. The value of in equals zero when the volume of the pressure measuring device remains essentially constant. CONCLUSIONS
The method described here has been used successfully for the past 6 years. If greater accuracy is required, the following improvements could be made by using: z more accurate pressure measuring device ; a constant temperature bath for
all three containers; a dry pure gas whose compressibility is accurately known; and the temperature of each container, if different, and compressibility factor at each condition as shown in Equation 11. This method can be used to determine the volume of any gastight container. Best results can be obtained when the volumes, VI, V2,and V 3 ,are approximately equal. RECEIVED for review September 8,1961. Accepted November 6, 1967.
A Microcalorimeter Especially Suited for the Study of Small Quantities of Materials William J . Evans, Emile J. McCourtney, and William B. Carney Seed Protein Pioneering Research Laboratory,’ New Orleans, La.
FORSOME TIME we have been using a modified version of the Tian-Calvet heat conduction microcalorimeter. The original version of this calorimeter consists essentially of a thermostated metal block containing twin microcalorimetric elements (1, 2). Each of these elements is covered by an array of closely fitting thermocouples which are connected in opposition. One of the elements serves as a tare with the other containing the process under investigation. The modified version incorporated the main features of the original calorimeter with one important exception-namely, the fabrication of the thermocouple sensing elements. These elements were made by a process of electroplating. In this manner the chief constructional difficulty of the original version of a calorimeter of this type was greatly simplified. Although the modified calorimeter which we have previously described (3), has proved useful (3-6), it has the disadvantage, among other things, of requiring a relatively long time (approximately four hours) for thermal equilibration. In view of the vast potential of calorimetry as an analytical tool in all phases of chemistry, we describe here a calorimeter which possesses features not incorporated in our earlier version. Chief among these features are : relatively short equilibration time (about one hour), Peltier compensation, reduction in the overall physical size of the calorimeter and in the amounts of materials required for its operation, ability to mix equal volumes of reagents, and automatic integration of the EMF-time curves. EXPERIMENTAL
Apparatus and Procedure. The two cell holders were machined from a copper rod with an 0.d. of approximately 0.7 inch and an i.d. of about 0.65 inch. The overall length of the cell holders was 21/4 inches, Two flanges, l/8-inch thick with an 0.d. of 0.95 inch, were machined a t the extreme ends of each holder. Ten l / d n c h holes, 36” apart, (1) E. Calvet, Compt. Rend. Acad. Sci., 226, 1702 (1948). (2) E. Calvet and H. Prat, “Microcalorimetrie.” Masson, Paris, 1956. (3) W. J. Evans and W. B. Carney, Anal. Biochem., 11,440 (1965). (4) H. D. Brown, W. J. Evans, and A. M. Altschul, Life Sciences, 3, 1487 (1964). ( 5 ) H . D. Brown, N. J. Neucere, A. M. Altschul, and W. 3. Evans, Ibid.,p. 1439.
(6) H. D. Brown, W. J. Evans, and A. M. Altschul, Biochim. Biophys. Acta, 94, 302 (1965). 262
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ANALYTICAL CHEMISTRY
were drilled in each of these flanges in such a fashion so as to have a spacing of 0.04 inch between the center of the holes and the outer wall of the cell holder (see Figure 1 .). The thermocouple elements were made in the following manner: enameled constantan wire of 0.005-inch diameter was wound on plastic tubing (acetate composition) with a n 0.d. of ‘/l inch and wall thickness of 0.01 inch. The tubing, approximately 2 inches in length, was held on a mandrel in the lathe and exactly 400 turns of the constantan wire were wound on it. While the tube with the wire in place was held on this same mandrel in the lathe, the enamel was removed from the outer periphery of the wire by means of a finegrained emery cloth. Great care is necessary in this operation to avoid abrading or cutting of the wire. Once the enamel had been removed from the wire, the tubing was clamped in a V-bottom jig (brass construction) so as to make good electrical contact with the wire. With the tubing held in this manner, a copper coating, from acid cupric sulfate solution, of about 0.001-inch thickness was formed on one half of the circumference of the wire coils on the tubing. Each line of demarcation between the copper plate and the constantan represents a thermo junction. Thus, each tube with the plated wire constituted a battery of 400 thermocouples. Twenty such batteries of thermocouples were constructed in this manner, 10 for each calorimetric unit. The tubes containing the thermocouples were arranged around the cell holders, taking care that the junctions were in proper line, with one row of junction being in contact with the cell holders. Prior to this operation the tubes as well as the cell holders were coated with an epoxy varnish. AS each tube was placed on the cell holder, as described above, two small pins, approximately inch long, were pushed through the holes in the flanges of the cell holders. As mentioned previously, 10 such thermocouple assemblies were fastened to each cell holder, resulting in a thermopile of 4000 copper-constantan thermocouples for each calorimetric unit. The individual thermocouple assemblies were wired in series through a double-pole, 6-position, 6-deck switch with gold contacts. The two complete thermopiles were also wired in series, but opposed, through this same switch, thereby effecting the differential arrangement, Further, by means of this switch, 4 of the individual thermocouple assemblies, in series (1600 thermocouples), could be isolated from the detecting circuit of each calorimetric unit for Peltier com1 One of the laboratories of the Southern Utilization Research and Development Division, Agricultural Research Service, U. S. Department of Agriculture.
