Made i n United States of America Reprinted from THEJOURNAL OF PHYSICAL CHEMISTRY Vol. XXXVII, No. 8, November, 1933
SIMPLIFIED FORMULAS FOR T H E CALCULATION O F EXPANSION COEFFICIENTS AND COMPRESSIBILITIES OF GASES AT LOW PRESSURES FROM T H E BEATTIE-BRIDGEMAN EQUATION OF STATE J. E. M. COPPOCK' Department of Inorganic Chemistry, The University of Leeds, Leeds, England
Received February 34, 1938
J. A. Beattie and 0. C. Bridgeman (1) have developed an equation of state that relates p , V , and T in a simple form involving only five constants, each set of constants being specific to a gas in addition to the gas constant R. The equation may be written in the virial form in terms of either p or V and is expressed: nRT
n2P
v
vz
p=-+-+-+-
nay V8
n46 v 4
where @ = RTBo y =
- Ao - Rc/TZ
- RTBob+ Aoa - RBoc/T2 6 = RBobc/TZ
(2)
(3) (4)
Ao, a, Bo, 6, and c are the characteristic constants for each gas. From equation 1 Beattie (2) derives the value of V in terms of p and obtains: no ny . p + n6 v = nRT - + -+ . p2
P
RT
R2T2
R3Ta
(5)
I t will be observed from an examination of the equations as written in the virial forms that the values of p or V a t 0°C. and 100OC. may be calculated by insertion of the correct values of p, y , 6, the values of these coefficients a t the two temperatures stgted being calculated from equations 2, 3, and 4. Values of the five constants for thirteen gases have been conveniently tabulated (3) and from these values p1 and y1 a t 100°C. have been calculated, the values of the virial coefficients a t 0°C. having been obtained from computations made by Beattie and Bridgeman. 'Present address, Robert Gordon's Colleges, Aberdeen, Scotland. 995
996
J. B. M. COPPOCK
From a knowledge of these values it is possible to calculate the coefficients of expansion a t constant pressure ( a v )and constant volume (ap), for p = conat.
where vo, pa represent' the volume or pressure of the gas at 0°C. and dv dp
- - the rate of increase of volume or pressure with temperature. dt' dt The values of av,a#,here derived are for the temperature interval 0°C. to 100°C. and at a constant pressure of 1 atmosphere in the first case and an initial pressure of 1 atmosphere in the second case. If we consider 1 gram-molecule of gas (that is, n = 1) then we have from equation 1 PO
=
YO 7+ Pov*+ -
RTo
a t 0°C.
v3
and from equation 5 when the pressure is 1 atmosphere we have: Vo = RTo
++ x2a t 0°C. R To
Vi
a t 1°C. + RTI -+ RZTi2
= RT1
YO
PO
Yl
P1
~
Taking as our units those cited by Beattie and Bridgeman, Le., moles, liters, atmospheres and " Kelvin, we may calculate values of av a t a constant pressure of 1 atmosphere and ap a t an initial pressure of 1 atmosphere over the range 0°C. to 100°C. from the following equations:
CY"
where p
=
=
1 atmosphere,
where V = const.
=
To= 273.i3"C., TI= 373.13"C., and
22.4131 liters a t N.T.P.
997
BEATTIE-BRIDGEMAN EQUATION O F STATE
Substituting values of p which contribute the major portion to the divergence from the perfect gas law, and values of y which contribute about 1 in 15,000 as a maximum to the value of the coefficient, we arrive at the values given in table 1, which are compared with those calculated by Leduc (4)and those determined experimentally. The experimental figures have been taken from the International Critical Tables, and where necessary the values a t 1 atmosphere have been interpolated from the data given. Values not obtained from the International Critical Tables have a definite reference given. It may be claimed that the performance of the equation is extremely good: it shows clearly the phenomenon that a P for hydrogen, nitrogen, TABLE 1 COEFFICIENTS' AT CONSTANT PRESSURE
-
GAS aD
(calculated)
He ........................... 3659.1 Ne ........... .,., 3660.6 A ............................ 3672.4 3660.3 3670.9 3674.6 ........................... Air. . . . . . . . . . . . . . . . . . . . . . . . . . . 3671.1 coz .............. 3721.7 CHn. . . . . . . . . . . . . . . . . . . . . . . . . . 3689.4 CZH4 . . . . . . . . . . . . . . . . . . 3724,2 NHs . . . . . . . . . . . . . . . . . . . . . . . . . . 3790.0 co. . . . . . . . . . . . . . . . . . . . . . . . . . . 3670.9 3721,7 NzO. ..........
Leduc (calculated)
3662 3671 3673
-
3723 3681 3735 380 3672 3732
Experi. mental
3659 3660t 3673 3660 3670. 3674 3671 3723 3683
-
38476 3669' 3720'
COEFFICIENTS AT CONSTANT
VOLUME
I -~a,,
(calmlated)
3661.3 3662.8 3671.7 3662.7 3671.8 3673.5 3671.6 3710.0 3688.1 3710.5 3767.8 3671.8 3710.0
Leduc (oalculated)
-
3664 3672 3672
-
3712 3678 3722 377 3673 3719
Experi-
mental
3661 3661j 3672 3662 3671.5 3674 3672 3712 3680
-
376g6 3668'
-
* All these coefficients have been multiplied by lo8 except Leduc's figures for ammonia, which are multiplied by 105. and helium is greater than a v , The values for nitrogen, oxygen, and air are certainly reliable and that for carbon dioxide agrees well with the predictions of Leduc and the experimental figure. The values for methane arc? a little unsatisfactory and those for ethylene are disappointingly low compared with the calculations of Leduc; investigation into this point seems desirable. For ammonia the predictions are satisfactory in the case of the pressure coefficient, the value calculated agreeing almost exactly with that experimentally observed, but, it seems curious that the value of aVshows such a discrepancy. It would appear that values of the volume coefficient are more susceptible to external factors, such as the increase in volume caused by the evolution of adsorbed layers from the surface of the
998
J. B. M, COPPOCK
containing vessels with rise in temperature and thereby vitiating the true values of the increase of volume with temperature. The values for carbon monoxide and nitrous oxide are assumed by Beattie and Bridgeman (8) to be the same as those for nitrogen and carbon dioxide. This assumption seems fully justified in the case of carbon monoxide, but for nitrous oxide this is probably not quite true, as shown by the divergence between the predicted values of Leduc and those calculated from the Beattie constants for carbon dioxide. The compressibility of a gas is given by the equation 1 + x = povo PlVl
at constant temperature, where povo is the product at zero pressure and plvl the product a t 1 atmosphere. From the Beatt,ie-Bridgeman equation of state we have at any temperature, T,,
for
where VI is the corresponding volume of 1 gram-molecule a t T10Absolute and p1 is of the appropriate value. The omission of the further terms in the Beattie equation is justifiable at low pressures in this case, as values
of
2L only contribute about 1part in 100,000 to the final result. V2 As VI = RTl
81 + R(approximately) TI
then (1
+x ) n = 1 -
81
[
RT1 RTi
81
+-;TI]
+
R*Ti2 81 =
l-
This simple form of equation allows the compressibilities of any gas to be calculated extremely rapidly over a wide temperature range, and in the case of the more compressible gases, such as carbon dioxide, the error in-
999
BEATTIE-BRIDGEMAN EQUATION O F STATE
volved is not greater than 1 part in 10,000. The agreement between the theoretical derivation of 1 X for temperatures between -100°C. and +lOO"C. and the experimental figures of Holburn and Otto (9) for nitrogen is satisfactory; and the assumption of Beattie and Bridgeman previously mentioned as regards the similarity of carbon monoxide and nitrogen seems fully borne out in the comparison of the values obtained from the data of Bartlett and his coworkers (10) on carbon monoxide and those calculated for nitrogen. In table 2 the values (1 A), are those calculated from the Beattie constants for nitrogen and those from Smith and Taylor's data (1) are given in the column designated (1 X)B. (1 A), values are the experimental determinations of Holborn and Otto for nitrogen, and the values A), are the figures obtained by Bartlett and coworkers for given for (1 carbon monoxide.
+
+
+
+
+
1°C.
- 100 -50 -25 0 +25 +SO 100
+
(1
1.00369 1 .CUI146 1.00090 1.00051 1 ,00025 1.00006 0.99979
+ X)C
+ VB
(1
-
1.00367 1,00144
1.00082 1.00047 1.00024 1.00006
-
-
1 ,00046 1.00020 1.00000 0.99980
(1
+
1.00135 1 ,00077 1.00040 1.00018 1 ,00008 0.99978
The author wishes to express his thanks to Dr. W. Wild and Professor R. Whytlaw Gray for helpful criticism and advice, and also to Professor J. A. Beattie for communications received. REFERENCES (1) BEATTIEAND BRIDGEMAN: Proc. Am. Acad. Arts. Sci. 63, 229 (1928); J. Am. Chem. SOC.49, 1665 (1927); 60,3133 (1928). (2) BEATTIE:Proc. Nat. Acad. Sei. 16, 14 (1930). (3) BEATTIE AND BRIDGEMAN: Z. Physik 62,95 (1930). (4) LEDUC:Ann. phys. 6, 180 (1916). (5) HEUSE:Ann. Physik [5] 4,778 (1930). (6) PERMAN AND DAVIES:Proc. Roy. SOC.London A78,28 (1906). (7) REQNAULT: Ann. chim. phys. 4, 5 (1842); 6 , 5 2 (1842). (8) BEATTIE AND BRIDGEMAN: J. Am. Chem. SOC.60,3151 (1928). AND OTTO:Z. Physik. 33, 1 (1925). (9) HOLBORN (10) BARTLETT, HETHERINQTON, KVALNES, AND TREMEARNE: J. Am. Chem. SOC.62, 1374 (1930).
