ARTHUR ROSE, JERRY A. ACCIARRI, R. CURTIS JOHNSON’, and W. W. SANDERS2
The Pennsylvania State University, University Park, Pa.
Automatic Computation of Antoine Equation Constants Caproic and Caprylic Acids and Methyl Esters An IBM-card-programmed calculator can automatically compute Antoine equation constants for any substance when vapor pressure-tern pera tu re data are available
AN
AUTOhiATIC calculation procedure for obtaining Antoine equation constants uses a n IBM card-programmed calculator (CPC). By this means the method devised by Taylor (7, 9) can be carried out with great facility for any substance for which vapor pressuretemperature data are available. Results for four compounds are reported in this article. About 15 minutes of computer time is required to obtain the constants giving the best possible fit to a group of 20 experimental data points. Card punching for this requires about 10 minutes. Testing a n d checking before the first computation can be done in less than a n hour, Standard wiring is used, so there is no time required for wiring. Taylor’s method requires an initial rough estimate for three constantsa,, bo, and t o . From these better estimates (a, b, and c) of these same constants are obtained. T h e latter may then be used to obtain still better estimates, which are used to calculate the conventional Antoine constants of Equation 1.
loglo P = A - B / ( t
+ C)
(1)
Essentially the same procedure is used herein, with modifications made possible by machine computation. T h e procedure of this paper requires a choice of three loglo P, t data points: one each a t a low, intermediate, and high pressure. These are substituted in the following modified form of the Antoine equation. at f 6 f c loglo P
- t loglo P
= 0
(2)
1 Present address, Department of Chemical Engineering, Washington University, St. Louis 5, Mo. Present address, Standard Oil Co. of Indiana, Whiting Research Laboratory, Whiting, Ind.
1 04
where a = A;
6 = (AC
- B ) ; and c
= -C ( 3 )
This gives three simultaneous equations in a, 6, and c. T h e computer then calculates approximate values of the constants a, b: and c. These are designated a,, bo, and c,. T h e logl,#, t data are then fed to the calculator on standard IBM cards to carry out the calculations indicated in Tables I, 11, and 111. I n principle, these repeated trials are not necessary, as the Taylor equations can be solved explicitly for the desired constants, if a large enough number of significant figures are retained in all the steps of the calculations. I n practice it is more efficient to use fewer significant figures and repeated trials. T h e weighting factors of Taylor’s method have not been included in the present application, in the sense that a weight of unity is given to each d a t a point. I n general, the effect of these weighting factors on the results is small, a n d in some cases questions can be raised as to the propriety of the assigned factors. T h e calculation, as actually performed in the work described, is divided into a number of sections, each of which requires from one to a dozen or more cards. Table I gives the over-all sequence and interrelation of the sections, including the repetition pattern. Table I1 indicates the various operations in some more detail, while Table I11 is a condensed CPC planning chart. This chart indicates the various arithmetic and other operations, and the punching on the cards to achieve the numerical results. Table IV summarizes the meaning of the numbers in columns 1 through 12 of the planning chart (Table 111). These numbers signify the punchings tQ be used in the successive cards, to supply instructions a n d numerical values to the
INDUSTRIAL AND ENGINEERING CHEMISTRY
computer. The calculator was wired for eight-position arithmetic floating-decimal operation ( 4 ) .
Discussion When the root-mean-square temperature difference converges to a constant on successive trips around the sequence of Table I , the Antoine constants A , B , and C that are obtained give the best possible fit to the experimental data. When the final value of the root-meansquare temperature difference is not within the precision of the temperature measurements, the quantities fo% (step D, card D07) are examined. Irregularities in the data are indicated by large values of fez. and the corresponding experimental point or points should be discarded and the entire computation procedure repeated without the use of these data points. Table V gives the results of the detailed calculations for methyl caproate a n d Table V I gives the vapor pressuretemperature data for integral values of vapor pressure for methyl caproate as calculated by the computer. Table VI1 gives the values of Antoine constants A , B, and C for caproic and caprylic acids and their methyl esters along with the convergent values of the root-mean-square temperature difference.
Source of Experimental Data A modified self-lagging still ( 6 ) was used to obtain the vapor pressure-temperature data (7, 8) for the four compounds (Tables I X and X). Almost all the values agreed within I-mm. mercury pressure with the corresponding values given by Pool and Ralston ( 5 ) and by Bonhorst, Althouse, and Triebold (2). T h e acids and esters were especially purified by precision distillation, and
START1 LA
-
Clear
I
I
I
RI 3 B l Jvalues of.log 4 ,ti
- - -
182
N and 1.0
Calcd. le4
Calcd.
Products 109 9,fi
00,
bo* co
Clear Storog e s
21 -28
1R I
Clear stor. 21 to 28 in prep. for next trial
temp. from
RI
7
Eq. 13
I
Calcd. 1st sums of coefficients in Eq. 7
t R I last integral value of log P v
L Calcd. equil.
Complete automatic printing of P vs. t
temp. from Eq. 13
of coefficients
as in Table VI A
IM'
R I 1st integral value of log P
4 Calcd. A t RMS RMS calcn. is stopped here when this converges to a constant
E
Calcd. final sum
-I
.ISTOP
E
o f diff. between ti
and ti'
Calcd.
t/;
Table I.
IJN
See Table II
for details and equations
from Eq. 13 A
I
m
R I last values of log pNfN
4
..+
Calcd. sum of square of difference between ti and f,!
E
RI first
Calcd. f,f r o m A,B,C
*
and Eq. 13
exptl. values log
P/Jt/
i
VOL. 49, NO. 1
JANUARY 1957
105
Table II.
Explanations of Calculation o f Sequence
Explanations
Section
Section
A
All storage registers are cleared of numbers from previous calculations.
I1
B1
Values of log P - t at a low, intermediate, and high pressure are read in.
J1
B2
Value of N (the number of experimental observations) and the constant 1.0 are read in.
B3
Values of the products t log P are calculated.
B4
These cards operate the computer to calculate numerical values of a,, bo, and co of Equation 2. Through the use of reduced matrix transformations ( 3 ) ,Equation 2 may be solved to obtain m11w
co =
m11m32
m11m33 ml1m32
-
m3m4
- m12m31
- m1m4 m l m 3
- mlzmzl - m1m1
m11mz2
-
m1lm22
- m 1 3 m ~- m12m31
(4)
m12mz1
The various m's and n's are constants of Equation 3 as indicated in the table of nomenclature.
B5
Storage registers 21 to 28 are cleared in preparation for subsequent calculations.
C1
Values of loglo P and t are read in
B1
These cards operate the computer to calculate and sum the coefficients for the equations
+ ( 2 : ti)@ + ( 2 : ti log = - 2 + ( N ) P + ( 2 : log = - 2 : f o ; log Pc)a + (z: log P i ) @ + [ Z (log
( Z ti%
Pi)Y
( Z tOa
( 2 : ti
The difference between the calculated and corresponding temperature is calculated. Card KO3 provides for calculating the sum of the square of the difference.
I2
Steps I, J, and K are repeated for each of the other data points by introducing cards T02, P02, a duplicate set of JO1 to 504, a duplicate set of KO1 to K03, etc.
fa
J2
53 K3
L
The root-mean-square temperature difference is calculated from the equation
M1
The first integral value of vapor pressure is made ready for the subsequent calculations. The equilibrium temperature is calculated from Equation 13 for the first value of integral vapor pressure. Steps M and J are repeated for each of the integral log vapor pressure points by introducing card M02, a duplicate deck of JO1 to 504, card M03, another duplicate deck of JO1 to J04, etc. Storage registers 21 to 28 are cleared in preparation for repetition of the calculations. Steps C to R are repeated until the root-mean-square temperature difference converges to a constant value.
J1 M2 J2 M3 J3
R
(7)
Pi)*]Y =
log Pi
Table 111. Steps in Calculations, CPC Program Details, and Instructions to Machine for Planning Chart
where foi
= a,ti
+ 6, + c,,
log Pi
a n d a = a - a,; B = b - 6,;
E
" I
- [c + log Pi - A
K1
tifd
f'i)~
- 2: foi
C2 D2 c3 D3
ti' =
K2 I3
mzln14
Explanations These are reintroduced at this point to make available the experimental data needed in steps J and K. The numerical value o f t ' is calculated from the Antoine equation constants, the equation being written as
- t i log Pi y =
(8)
c - c,
(9) Steps C and D are repeated for each of the other data points by introducing cards P02, T02, a duplicate set of DO1 to D18; P03, T03, another duplicate deck of DO1 to D18, etc.
Card Column
Sec. Operation A 72-77 Clear B1
Y =
-
ai1634
a11a33
- a12a31 - a13a31
ana32
-
a1m2
a12a81
-
a21614
-
-
al1a23 alia22
-
6 7 00
00
21
1 X
R I log P I
BO2
00
00
22
1 X
R I log P H
BO3
00
00
23
1 X
RI
alibi4
- aizaz1 - (113a21
a11a22
A0 1 A02 BO 1
X
These cards operate the computer to calculate numerical values of a, P , and y of Equation 7. Through the use of reduced matrix transformations ( 3 ) ,Equation 7 may be solved to obtain a31614
11-28 Clear RIlog PI,
Normal Field Entrv 45 78 910 12 14-23 24-33 A B C OP A R
123 Card No.
tL
BO4
00
00
24
1 X
RI tr
(10)
BO5
00
00
25
1 X
RI
a12a21
f H
BO6
00
00
26
1 X
P =
RI N
(11)
R I 1.0
BO7
00
00
75
1 X 1
BO8
00
00
77
BO9
24
21
27
3
and this is the computation done by the computer at this stage. The various a's and b's are functions of the constants and sums in Equation 7 as indicated in the table of nomenclature.
B10
25
22
28
3
B11
26
23
76
3
The numerical values of a,P , and y are used to calculate corresponding values of a , b , and c according to Equation 9. The values of a, 6 , and c are also placed in the storage counters that until now were holding a,, bo, and c,. Thus a, b, and c are available for the repetition of steps C to F to obtain better estimates of a, b , and c.
so1 so2 SO3 SO4 so5 SO6 SO7
24
77
77
26
11 12 13 14 15 16 17
3 3 3 3 2 2 3 3 3 3 2
B2
X
F
H
106
Numerical values of Antoine constants B and C are obtained by the use of Equation 3. Constant A = a, and is therefore already available. Card H05 clears register 11 in preparation for subsequent steps.
INDUSTRIAL AND ENGINEERING CHEMISTRY
X X
SO8 SO9 s10 Sll
24
77
77
25 12 14 76 27 28 27 18
11 13 24 26 24 25 17
18
11 12 00
b
Steps in Calculations, CPC Normal Field Card Column 123 45 78 910 Sec. Operation Card No. A B C B4 (contd.) 15 13 Sll/S05 512 11 12 00 so9-s10 S13 514 16 18 Sl3/S06 13 14 S12-Sl4 S15 24 23 11 S16 miimaa 21 26 12 517 mlamsi 24 22 13 Sl8 m11mB 21 25 17 s19 mm21 11 12 00 s20 S16-Sl7 15 11 s 21 S20/ SO5 13 17 00 518-519 s22 16 17 523 S22/S06 11 00 S21-S23 524 Table 111.
co
si5
Co(S23)
S26
bo
527 528 S29 S30 S31 S32 s33
m12/m11
rndmli n14/mll
bo(S28) Co(S29) S30-S32 a,, = 533
+ S31
14
74 17 00
18
77 21 27 72 74 15
24 24 24 13 14
72 13 14 15 16 00 00
16 73
s34
B5
Clear 28 Clear 27 Clear 26 Clear 25 Clear 24 Clear 23 Clear 22 Clear 21
R01 R02 R03 R04 R05 R06 PO7 RO8
28 27 26 25 24 23 22
28 27 26 25 24 23 22 21 21
28 27 26 25 24 23 22 21
c1
RI log P ,
P O1
00 00
17
Program Details, and Instructions to Machine for Planning Chart (Continued) Normal Field Entry Entrv 12 14-23 24-33 Card Column 123 45 78 910 12 14-23 24-33 A B OP A B Seo. Operation Card No A B C O P E (contd.) E10 26 22 12 3 4 a'Llbl4 18 17 00 2 Ell 2 E08-EO7 E12 15 13 4 4 Ell/E05 I 2 I1 00 2 E13 2 ElO-EO9 E14 3 16 18 4 E13/E06 E15 13 14 2 3 E12-El4 24 23 11 3 E16 3 Uiiaa8 28 28 12 3 3 E17 a1aas1 2 24 25 13 3 E18 a11aPs E19 4 28 26 17 3 a1aa21 E20 2 11 12 00 2 E16-El7 E2 1 4 15 11 4 E20/E05 E22 13 17 00 2 2 E 18-El 9 X E23 E22/E06 16 17 4 4 11 E21-E23 E24 00 2 3 X 14 11 4 X E25 2 E26 17 00 3 X 4 18 12 2 4 E27 26 24 13 4 4 E28 3 28 24 14 4 E29 22 24 15 4 3 E30 13 12 16 3 2 E3 1 14 11 00 3 X E32 2 E33 15 00 1 E34 16 00 1 2 X c 2 00 13 3 E35 a 2 2 X 2 a = A FO 1 F 83 73 1 2 X 2 FO2 12 82 72 1 b 2 X X 84 11 74 1 F03 c 1
H
X
TO1
RI ti D1
00 00
16
aoti c o log Pi DO2 bo ti log Pi
DO1 DO2 DO3 DO4
73 74 17 72 16 17
z t i log Pa DO3 DO1
DO5 DO6
28 28 12 11 00
+
1
11
3
00
3 1
12 13
D
73 00 72 00
H03
B
00
21
DO7
13
14
1 c
3
H04 H05
74 00 22
3
Clear 11
11 11 11
2
RI ti
TO 1
00
C
X
I1
00
16
X f o b
3
X
3 1 1
H01 HOZ
+b
X
X
+
ac ac
1 X
2
RI log Pi
PO1
log Pi-A
JO 1
17
1
73 00
2 4 1
00 00
X
Zf o i
DO8
27 27
Z ti
DO9
16 26 26
z log Pi
D10 Dll
17 25 25 16 16 00
D12 D 13
24 24 17 17 00
1 X
t
it
z t: (log Pi)'
z (log P i ) * td0i
D14 D15
23 23 16 14 00 22 22 14 17 00
2: f o i log Pi
D18
21 2 1
1
1 3 1
3
D3 E
1
PO2 DO1 PO3 DO1
c3
011(1a2 ailaai a11a22 allazl
E01-EO2 EO3-E04 aiibac aaibit aiiba
K1
3
X
c2 D2
00 00
t!
104
00
00
3
- ti (ti' - t i ) * 2: ( I / - t / ) 2
KO1 KO2 KO3
16 00
2
00 11 11
3 1
EO1 E02 E03 E04 E05 E06 E07 E08 EO9
+ J02
3
X
D16 D17
tzfoi
21 22
C
1 X
log Pi
Z
JO2 J03
WJO1
X
X foi
J1
1
ti'
E
X
TO2
I2 J2 K2 I3
JO1
KO1 TO3 JOl
J3
K3
L
X
KO 1
z (ti' N
ti)*
LO1
75 00
4
M2
00
5
LO3
00 00
00
X
24 26 24 26 11 13 24
28 24
25 28 75 26 12 14 21 22 27
11
12 13 14 15 16 17 18 11
3 3
X
3 3
M1
2 2
J1
log Pi
M01
JOl
J02 JO3
3
3 3
1
00
21 22
00
1
73 00
2 4
00
00 00
a
1 X
1:
~ 0 4
00
VOL. 49, NO. 1
00
3
c
JANUARY 1957
107
,special precautions (dry nitrogen atmosphere) were used to prevent absorption of moisture from the air into the purified materials. T h e raw materials used for these distillation operations were about 90% pure, the impurities consisting mostly of water a n d the next highest and lowest naturally occurring homologs. T h e samples actually used for the vapor pressure measurements were taken from heart cuts of the distillations a t high reflux ratio. T h e refractive index variation of these heart cuts was less than 0.00005 unit. Table VI11 gives the re-
fractive indices of the materials used and also acid numbers, saponification numbers, a n d iodine values for these compounds.
U. S. Department of Agriculture, for their aid and interest in connection with the purification and the determination of data for the fatty acids and esters.
Acknowledgment
Nomenclature
Acknowledgment is made to Webb T. Comfort a n d William S. Dye 111, University Tabulating Department, for assistance a n d encouragement with the computer programming, a n d to H. E. Knight, Daniel Swern, and W. C. Auk, Eastern Regional Research Service,
A B
Table 111. Steps in Calculations, CPS Program Details, and Instructions to Machine for Planning Chart (Confinued) Normal Field Card Column Sec. Operation
45
78 910
12 14-23
CardNo.
A
B
OP
C
A
a
b c
a, bo
V.
CPC Calculations for Methyl Caproate Vapor Pressure-Temperature Curve Fitting (8)
Entry
123
First Trial
24-33
A B
B
M2 J2 M3 53
MO2 JO 1 M03 JO 1 R01 28 28 28 2 N Clear 28 R02 27 27 27 2 Clear 27 R03 26 26 26 2 Clear 26 R04 25 25 25 2 Clear 25 Clear 24 R05 24 24 24 2 R06 23 23 23 2 Clear 23 R07 22 22 22 2 Clear 22 R08 21 21 21 2 Clear 21 a This card is punched in columns 14 to 23, to indicate the numerical value being read into the computer through channel A. Numerical value of + l . O entering computer through channel A or B. C Numerical value of - 1.0 entering computer through channel A or B.
*
Significance of Notations on CPC Planning Chart
Table IV.
Table
C
= constant in Antoine equation = constant in Antoine equation = constant in Antoine equation = defined by Equation 3 = defined by Equation 3 = defined by Equation 3 = first approximation of a = first approximation of b
Card Column
P 15.5 25.5 30.0 36.0 44.0 54.5 62.5 67.5 84.5 88.0 117.0 153.5 180.5 211.5 234.0 275.5 316.0 395.0 472.0 730.0
&lrs
c
= 7.28597 = 1596.083 = 211.920
Second Trial
Third Trial
-4 = 7.28598 B = 1596.082 C = 211.920
A = 7.28598 toalod.
B = 1596.083 C = 211.920
texptl.
toalcd.
tcalcd.
49.9 60.0
49.92 59.55 62.85 66.64 70.95 75.68 78.80 80.58 85.91 86.89 93.97 101.05 105.42 109.83 112.70 117.45 121.55 128.44 134.15 148.97
49.92 59.55 62.85 66.64 70.95 75.68 78.80 80.58 85.91 86.89 93.97 101.05 105.42 109.83 112.70 117.45 121.55 128.44 134.15 148.97
49.92 59.55 62.85 66.64 70.95 75.68 78.80 80.58 85.91 86.89 93.97 101.05 105.42 109.83 112.70 117.45 121.55 128.44 134.15 148.97
0.2179
0.2179
9.2179
62.7
66.6 70.6 75.5 78.8 80.4 86.1 87.0 94.1 101.0 105.4 109.9 112.7 117.5 121.5 128.6 134.4 148.6
Card identification by a letter and number Source of number entering channel A of computer. Double zero indicates number read from columns 14 to 23 of the card. Source of number entering channel B of computer
Table VI. CPC Calculations of Equilibrium Temperatures for Methyl Caproate from Antoine Equation at Integral Values of Vapor Pressure lOgloP = 7.28598
-
1.596.082
t
Indication of storage unit to which result of computation is to be sent. Double zero indicates answer remains in counter. Indicates kind of calculation computer is to perform on entering numbers.
Punch
,
2 3 4
5 6 7 8 9 0 11 (or x)
12 (or Y)
1\08
Other Operation Addition A + B = C Subtraction A - B = C Multiplication A X B = C Division A / B = C Square root 42 = c Clear storage 72-77 Clear storage 11-28
1
i
Summary punch Print answer Eject page
INDUSTRIAL AND ENGINEERING CHEMISTRY
P 1.0
5.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0
200.0 300.0 400.0 500.0 600.0 700.0 760.0
+ 211.920 tdcd 7.14 30.39 41.99 54.76 62.8 68.89 73.76 77.86 81.43 84.59 87 44 90.03 108.26 119.98 128.84 136.04 142.15 147.49 150.40 e
Table VII.
Antoine Equation Constants for Caproic and Caprylic Acids and Their Methyl Esters
Compound Caproic acid Caprylic acid Methyl caproate Methyl caprylate
Table Vlll.
1511.752 1530.726 1596.083 1266.983
7.22192 7.12793 7.28598 6.50998
C
Caproic acid Caprylic acid Methyl caproate Methyl caprylate
Experimental Vapor Pressure-Temperature Data for Caproic and Caprylic Acids ( 7 )
P,mm. Hg
AtRM8
143.805 123.056 211.920 152.770
9.40 17.5 26.6 36.3 41.4 49.4 67.3 82.1 94.3 99.8 141.8 180.3 202.6 222.4 255.9 298.2 348.8
0.21 0.26 0.22 0.18
Properties of Caproic Acid, Caprylic Acid, and Their Methyl Esters
20.00°
IX.
Caproic Acid
Antoine Constants B
A
Table
Refractive Indices 25.00°
* O.05OC.
C.
Caprylic Acid t,
P,mm. Hg 12.0 13.1 20.9 24.2 29.8 33.5 40.6 51.2 55.6 77.9 78.0 103.4 103.5 153.0 198.4 248.7 305.5
98.1 108.8 117.0 123.3 126.2 129.8 136.6 141.1 144.0 145.5 154.0 160.8 163.5 166.7 170.3 174.7 179.1
Methyl Caproate
P , mm. Hg Theor.
Exptl. 344 279
15.5 25.5 30.0 36.0 44.0 54.5 62.5 84.5 88.0 117.0 153.5 180.5 211.5 234.0 275.5 316.0 395.0 472.0 730.0
344 278
Saponification Numbers Methyl caproate Methyl caprylate
430 350
430 354
Iodine Values (Wijs) Caproic acid Caprylic acid Methyl caproate Methyl caprylate
0.37 0.01 0.00 0.00
0.00 0.00 0.00 0.00
= first approximation of c
N
=
P
=
zt Z zti z ti log10 Pi
= = zt,
= N = = =
z log10 Pi z ti log10 Pt z log10 Pf
= L: (log10 Pill - 2 tifa
= = = = =
- 2 f o i log10 Pi defined by Equation 5 logarithm to base 10
=
tL
-2fot
= 1.0 = log10 P L
= tl = 1.0 = log10 PI =
tH
= 1.0 = log lop^ = tL log10 P& = tt log10 PI tH
log10 P H
RI t
number of observations = vapor pressure, mm. mercury = read i n = experimental t e m p e r a t u r e , p-
O
t’
c.
= calculated temperature, O C. cy = correction factor, defined by Equation 9 /3 = correction factor, defined by Equation 9 Y = correction factor, defined by Equation 9 2 = summation over all points AtRM8 = defined by Equation 14
Subscripts i = index
L
= = H = 0 = RMS =
r
lowpressure intermediate pressure highpressure first approximation root-mean-square difference
Superscript = calculated quantity
’
C.
Table X. Experimental Vapor Pressure-Temperature Data for Methyl Caproate and Methyl Caprylate (8)
Acid Numbers Caproic acid Caprylic acid
O
130.2 131.9 140.1 142.8 147.8 150.2 154.0 159.6 161.5 169.3 169.4 176.5 176.6 186.8 193.8 200.3 206.3
* 0.05OC.
1.41459 1.42597 1.40314 1.41490
1.41656 1.42793 1,40528 1.41708
t,
t,
Methyl Caprylate C.
P , mm. Hg 8.00 10.5 14.0 17.0 21.0 24.0 28.5 33.5 40.0 46.5 49.0 54.0
49.9 60.0 62.7 66.6 70.6 75.5 78.8 86.1 87.0 94.1 101.0 105.4 109.9 112.7 117.5 121.5 128.6 134.4 148.6
t,
O
C.
73.2 78.1 83.3 86.9 91.5 94.6 97.9 101.6 105.5 108.8 110.0 112.2
literature Cited (1) Acciarri, J. A., M.S. thesis, Pennsylvania State University, June 1955. (2) Bonhorst, C. W., Althouse, P. M., Triebold, H. O., IND.ENG. CHEM. 40, 2379 (1948). (3) Hankam, E. V., I.B.M. Technical Newsletter No. 3, p. 26, December 1951. (4) MacMillian, D.B., Stark, R. H., Zbzd., No. 2, Februaryl951. (5) Pool, W. O., Ralston, A. W., IND. ENG.CHEM.34. 1104 (1942). Rose, Arthur, Williams,‘ E. T., Zbid., 47,1528 (1955). Rossini, F. D., Mair, B. J., Streiff, A. J., “Hydrocarbons from Petroleum,” ACS Monograph Series, Reinhold, New York, 1953. Sanders, W. W.. Ph.D. thesis, Pennsylvania State University, .August 1954. Willingham, C. B., Taylor, W. J., Pignocco, J. M., Rossini, F. D., J. Research Natl. Bur. Standards 35, 219 (1945).
RECEIVED for review March 21, 1956 ACCEPTED June 20, 1956 VOL. 49, NO. 1
JANUARY 1957
109
