INDUSTRIAL AND ENGINEERING CHEMISTRY
October 1955
that the rate equation does not adequately describe the exchange of sodium in solution with potassium in the resin. CONCLUSIONS
The mathematically derived equation, based on the premise that the rate of ion exchange is limited by mass transfer, can represent the observed break-through curves for the exchange of potassium for sodium in Dowex 50 if the proper value of the rate coefficient is selected. The experimental rate coefficients for the exchange of potassium for sodium in Dowex 50 behave in a manner predictable from the theoretical derivation. The rate equation does not describe the exchange of sodium for potassium in Dowex 50.
2199
pi
= equilibrium resin-phase concentration of exchanging
po
= total exchange capacity of the resin, meq. per ml.
Re
= Reynolds number, D,vp/p, dimensionless
t
= time, measured from start of process a t inlet of column,
u
= flow modulus, dimensionless, = k
X
=
c
= = = = = = =
ion a t liquid-solid interface, meq. per ml.
seconds
V v‘ y ~ U Z
p
p
coy/V distance from input end of column expressed as volume of exchanger contained therein, ml. bed modulus, dimensionless, = kq,z/KV volume rate of flow through the column, ml. per second superficial liquid velocity, em. per second volume of solution that has passed through bed, ml. value of y a t c/co = ml. liquid density, grams per ml. liquid viscosity, poise
ACKNOWLEDGMENT
The authors wish to thank E. I. du Pont de Nemours & Co. for the postgraduate fellowship under which this work was accomplished. Thanks are also given t o the Dow Chemical Co. for the Dowex 50 ion exchange resin it supplied. NOMENCLATURE
A, C
= outside area of resin sphere, sq. cm. = bulk liquid-phase concentration of the exchanging ion,
c;
= equilibrium liquid-phase concentration
meq. per ml.
of exchanging
ion at liquid-solid interface, meq. per ml. = total normality of feed, meq. per ml. = fractional nonremoval of exchanging ion = Darticle diameter. cm.
cg
c/co
D,
&1
H ( z ) = error function = -
e-n’dp
H ’ ( x ) = derivative of error function
=
2 7; e-%’
Io(z) = imaginary Bessel funct,ion of first kind K = equilibrium constant for ion exchange reaction, dimensionless k = over-all rate coefficient as defined by Equation 3, (ml.1 (see.) (meq.) = liquid transfer coefficient, sq. em. per second k; k~ = liauid transfer coefficient defined as k:/A,. l/sec. k,’ = resin transfer coefficient, sq. cm. per sicorid ‘ = resin transfer coefficient defined as k,’/A,, l/sec. k, = bulk resin-phase concentration of the exchanging ion, q meq. per ml.
LITERATURE CITED (1) Baddour, R. F., Ph.D. thesis in chemical engineering, Massachusetts Iristitute of Technology, 1951. (2) Baurnan, W. C . , from Sachod, F. C., “Ion Exchange,” Academic Press, New York, 1949. (3) Bauman, W. C., and Eichhorn, J., J . Am. Chem. Soc., 69, 2830 (1947). (4) Beaton, R. H., and Furnas, C. C., IND.ENG.CHEM.,33, 1501 (1941). (5) Bonner, 0. D., Davidson, A. W., and Argersinger, W. J., J . Am. Chem. Soc., 74, 1044, 1047 (1952). (6) Boyd, G. E., Adamson, A. W., and Myers, L. S., I b i d . , 69,2836, 2849 (1947). (7) Brown, G. G., “Unit Operations,” Wiley, New York, 1950. (8) duDomaine, J., Swain, R. L., and Hougen, 0. H., IND.ENG. CHEX.,35,546 (1943). (9) Gilliland, E. R., and Baddour, R. F.,Ibid., 45,330 (1953). (10) Hiester, N. K., and Vermeulen, T., Chem. Eng. Progr., 48, 505 (1 952). (11) Kunin, R., and Myers, R. J., “Ion Exchange Re ins,” Wiley, New York, 1950. (12) Selke, W. A , , and Bliss, €I.,Chem. Eng. Progr., 46,509 (1950). (13) Sujata, A. D., Ph.D. thesis in ckenical engineering, University of Michigan, 1952. (14) Thomas, H. C., from Nachod, F. C., “Ion Exchange,” Academic Press, Xew York, 1949. (15) Thomas, H. C., J . Am. Chern. Soc., 66,1664 (1944). (16) Walter, J. E., J . Chem. Phys., 13, 229 (1945). (17) Whitcornbe, J. A . , Banchero, J. T., and White, R. R., Chern. Eng. Progr. Symposium Ser., No. 14, 50, 73 (1954). RECEIVED for review January 7, 1954.
ACCEPTED May 11, 1955
An Accurate Simple Formula for the Activity Coefficient of Vapors H. G. ELROD, JR. Case Institute of Technology, Cleveland, Ohio
T
HE activity coefficient (fugacity-pressure ratio) is a widely
used analytical tool for calculations of chemical equilibrium involving gases and vapors. The following formula relates this coefficient t o the compressibility factor ( p V / R T )for all pressures less than critical pressure.
The accuracy of this formula may be partially judged by inspection of Table I, where values from Equation 1 are compared
with those obtained by Thomson (4)in a precise analytical manner from a generalized p-V-V-T correlation of Cope, Lewis, and Weber ( 1 ) . For T I T , < 1, p / p o < 1, Thomson showed that their correlation leads t o a unique relation between the activity coefficient and the compressibility factor, the wet region excluded. Discrepancies of 1% or more occur in Table I only when p V / R T < 0.5. Such low values of pV/RT occur only in the immediate vicinity of the critical point. In this region the variation of p,V,/RT, from substance t o substance invalidates a generalized correlation of compressibility factor, and all of the discrepancy should not be attributed t o error in Equation 1.
INDUSTRIAL AND ENGINEERING CHEMISTRY
2200
Vol. 47, No. 10
An easily remembered approximation for relating the activity coefficient and compressibility factor of a vapor at pressures less than critical
A well known approximation due t o Lewis and Randall states that Y
=
(i)
=
(
(7) p
