Algorithms for the Vapor Pressure of Water
Over Aqueous Solutions of Salt and Caustic Soda
R. B. M a c M u l l i n
R. B. MacMullin Associates, Niagara Falls, N e w Y o r k
ABSTRACT
Incidental to development of a digital computer program for the performance of electrolytic chlor-alkali cells is the necessity of general formulas
for calculating the vapor pressure of the aqueous solutions used for the electrolytes. This paper offers algorithms for vapor pressure of pure water, pure
NaCl solutions, pure NaOH solutions, and aqueous solutions containing both
NaCI and NaOH, within the range of conditions usually encountered in commercial cells of both the mercury and the diaphragm type.
Vapor Pressure Lowering in Pure Salt Solutions
Incidental to d e v e l o p m e n t of a digital c o m p u t e r prog r a m for t h e p e r f o r m a n c e of electrolytic c h l o r - a l k a l i
cells is t h e necessity of a g e n e r a l f o r m u l a for calculating t h e v a p o r pressure of the aqueous solutions used
for the electrolytes. T h e electrolytes usually en co u n t e r ed are:
Diaphragm cell
Anolyt.e NaCI,>20%, 90~176
Catholyte NaOH.10-14%, 93~176
NaCl, 14-20%, 93~176
Source of data: I n t e r n a t i o n a l Critical Tables (3),
Vol. III, p. 370.
The fractional l o w e r i n g of v a p o r pressure due to
the solute (in this case NaC1) is expressed
Mercury cell
R .
NaCI >20%, 50~
NaOH,30-60%, 70~176
(Po - - P)
~P
PoM
PoM
.
.
.
[4]
Since the concentration of NaC1 in t h e table is e x pressed as w / o (weight p er cent), the Molality is
The algorithmsI which follow cover a range of conditions which include the above ranges of interest.
I. Vapor pressure of pure water over the range i0 ~
150~
II. Vapor pressure lowering due to th e dissolved
NaC1 o v e r the r a n g e 3M to saturation, and 30~176
III. Vapor pressure l o w e r i n g due to the dissolved
NaOH: (a) o v e r th e r a n g e 0-12.5M, 20~176
(b)
o v e r the r a n g e 12.5-25M, 20~176
IV. Combinations of II and I I I ( a ) .
In this paper, t e m p e r a t u r e is expressed ~ concentration, as Molality (g.f.w./kg H 2 0 ) ; M for NaCI, N
for NaOH; v ap o r pressure, as Torr ( m m Hg).
w / o NaC1
M:
+ 0.0584428
w / o H20
Values of i00 R w e r e calculated for e a c h of the data
points given in t h e ICT table quoted above. In Fig. 1,
100 R vs. t is plotted for the various concentrations,
and R appears to be a linear function of t. In Fig. 2,
100 R vs. M. is plotted for the various temperatures,
and above 3 M, R appears to be a linear function of M.
The lines appear to intersect at the point 100 R ~ 3.5
at M ~ 3.0. That is, above 3 M, R m a y be expressed as
follows:
Vapor Pressure of Pure Water
R---- ( M - - 3 ) (0.0019772--0.00001193 t) -~ 0.035 [5]
The steam tables of K e e n a n and K e y e s (1) are based
on two formulas t a k e n f r o m the paper by Smith,
Keyes, and G e r r y (2). The second f o r m u l a covering
t h e r a n g e 10~176
is the simpler of the two:
4O
3,e
log~0 pc-
x [ a + bx ~-cx3 ]
p =}-
w , "/. NaCl
o
[l]
3a
l+dx
p
---- v a p o r pressure in int arm
Pc = critical pressure, 218.167 int a t m
T ---- degrees Kelvin, t~ ~- 273.16
T~ ---- critical t e m p e r a t u r e , 647.27~
x
=
a
b
c
d
= 3.2437814
---- 5.86826 X 10-3
= 1.1702379 X i0 -s
= 2.1878462 X 10-3
3~
35
Tc--T
3,4
30
40
50
60
,,'C.
70
80
eO
100
Fig. I. Vapor pressure lowering, pure aqueous NaCI. Range 15%
NaCI to saturation; 40 ~176
Expressing p as Torr and t as ~ the above formula
reduces to the following, which are programmed in
sequence:
374.11- t = x
[2]
W f . ~ NQCI
3.9
38
antiloglo (5,219603--
I
37
t -I- 273.16
1 -b d x
----
Po
~6
[3]
35
This f o r m u l a reproduces published tables for t h e
va p o r pressure of w a t e r to w i t h i n ~- 0.02 Torr.
. . . . . . . . .
Fig. 2. Vapor pressure lowering, pure aqueous NaCI. Range, 3M
z I-Iere used as the art of reducing data involving three or more
variables to a single algebraic equation for computer use.
to saturation; 40%110~
416
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V o l . 116, N o . 3
Sincep
VAPOR
~ po-
PRESSURE
OF
(1--RM)p~
RMpo ~
WATER
417
C a l l i n g t h e s l o p e of t h e s e l i n e s - - d R / d r
p+ ~ [1--((M+--3)(1.9772
• 10-3-1.193 • 1 0 - s t +) + 0.035) M +] po +
~ m,
R = 0.0317 - - m ( t - - 174)
[6]
or
R = 0.0317 -}- m ( 1 7 4 - - t)
where the superscripts indicate conditions applicable
to t h e a n o l y t e .
T h e a b o v e f o r m u l a r e p r o d u c e s m o s t of t h e d a t a in
t h e I C T t a b l e s , at c o n c e n t r a t i o n s a b o v e 15 w / o NaC1
a n d t e m p e r a t u r e s f r o m 40 ~ to l l 0 ~
t o a n a c c u r a c y of
a b o u t ~ 0.5 T o r r . S e v e r a l r a n d o m i n c o n s i s t e n c i e s i n
the ICT data are apparent.
B e l o w 15 w / o NaC1, it will g e n e r a l l y b e s u f f i c i e n t l y
a c c u r a t e f o r m o s t p u r p o s e s , t o t a k e R = 0.035 a t a n y
temperature. For the anolyte problem, the concentrat i o n w i l l n e a r l y a l w a y s b e a b o v e 15% NaC1, w h e r e t h e
sophisticated formula above applies.
V a p o r P r e s s u r e L o w e r i n g in P u r e Caustic
Soda Solutions
S o u r c e of d a t a : I n t e r n a t i o n a l C r i t i c a l T a b l e s ( 3 ) ,
Vol. III, p. 370.
T h e f r a c t i o n a l l o w e r i n g of v a p o r p r e s s u r e d u e to t h e
s o l u t e ( i n t h i s c a s e N a O H ) is e x p r e s s e d
T h e s l o p e of e a c h l i n e w a s t h e n m e a s u r e d , as p e r
the table which follows:
M o l a l i t y , N"
S l o p e 100 m
--h
100 R / A t
0
2.5
5
7.5
10
12.5
0
0.0021~,
0.00636
0.01000
0.01227
0.01299
I n Fig. 5, t h e n e g a t i v e of t h e slope, 100 m is p l o t t e d
a g a i n s t M o l a l i t y . T h e c u r v e a p p e a r s to b e c l o s e to a
q u a d r a t i c w i t h a m a x i m u m a t N = 12.5. H o w e v e r , a t
l o w e r c o n c e n t r a t i o n t h e c u r v e r e v e r s e s , as s h o w n . A
s u i t a b l e c u r v e fit is o b t a i n e d b y a s s u m i n g a p o l y n o m i a l
of t h e f o r m
m=
a + b N + cN~ + d / N
[8]
Ap
Solving this polynomial for the data given, the following constants are formed
R=
poN
S i n c e c o n c e n t r a t i o n of N a O H i n t h e I C T t a b l e is e x p r e s s e d as C ~ g N a O H / 1 0 0 g H 2 0
a :
--8.6715 • 10 -'~
b=
C
- -~ g . f . w . / k g H 2 0
3.99971
N
100 R h a s b e e n c a l c u l a t e d f o r a l l d a t a p o i n t s g i v e n
in t h e I C T t a b l e u p t o C ~ 120 ( 5 5 % N a O H ) , a t 20 ~,
40 ~, 60% 80 ~, a n d 100~
I n Fig. 3, 100 R vs. N is p l o t t e d f o r v a r i o u s v a l u e s of c o n s t a n t t, a n d o n e n o t e s t h a t
100 R goes t h r o u g h a m a x i m u m a t a b o u t N : 12.5
Molal.
Case A . - - C o n c e n t r a t i o n s < 12.5 M o ] a l (33 w / o of
N a O H ) . I n Fig. 4, 100 R vs. t is p l o t t e d a t v a r i o u s v a l u e s of N. A t e a c h v a l u e of N, R a p p e a r s t o b e a l i n e a r
f u n c t i o n of t. T h e l i n e s a p p e a r to i n t e r s e c t a t a c o m m o n p o i n t , 100 R = 3.17, t ~ 174~
S5
I
i
I
i
I
I
TEUp'C.
I
I
I
I
r
I
~
i
so
3.368
•
-~
c =--1.354
• 10 - "
d =
• 10 -,~
7.88
I-[ence,
R :
0.03170 + ( 1 7 4 - - t )
(a + b N + c N 2 + d / N )
[9]
174--t)
(a + b N + cN 2 + d / N ) ] N po
[10]
and
Ap~
[(0.03170 +
and
p--
(1 - - R N ) Po = (1 + [ ( t - -
174)
(a + b N § cN 2 -}- d / N ) - - 0.03170]N}po [11]
T h e c u r v e s s h o w n i n Fig. 3, u p t o N ~ 12.5, a r e
a c t u a l l y a p l o t of Eq. [11]. A l l t h e d a t a p o i n t s f a l l
o n t h e s e c u r v e s , e x c e p t two, a n d t h e s e a r e b e l i e v e d
to b e e r r o r s o r m i s p r i n t s i n t h e d a t a t a b u l a t e d i n ICT.
Case B . - - N > 1 2 . 5 (33% N a O H ) to N : 30 (55%
N a O H ) . A p p l i c a b l e to c a u s t i c m a d e i n d e c o m p o s e r of
a m e r c u r y cell.
4.S
40
I n Fig. 6, 100 R vs. t is p l o t t e d a t v a r i o u s v a l u e s of
N. A t e a c h v a l u e of N, w i t h i n t h e t e m p e r a t u r e r a n g e
of 60~176
R a p p e a r s to b e a l i n e a r f u n c t i o n of t.
T h e l i n e s a p p e a r t o i n t e r s e c t a t a c o m m o n p o i n t , 100
R ~ 2.83, t = 198~
C a l l i n g t h e s l o p e of t h e s e l i n e s - - d R / d t = m ,
S~
N, MOLALITy
OF NOOH
R = 0.0283 + m ( 1 9 8 - - t )
Fig. 3. Vapor pressure lowering, pure aqueous NaOH. Range,
0-30 Molal; 20~176
[12]
.014
o,~
5.:
/
f
.012
.010
+
joo8
~po7
-g,ol
~
004
~03
pOZ
o
~~
,'o
~'o
~01
o'~
,;o
,~o
, 9174 ~
Fig. 4. Vapor pressure lowering, pure aqueous NaOH. Range,
0-12.5 Malal NaOH; 40~176
N MOLALITY OF N O H
Fig. 5. Temperature coefficient of 100 R for pure NaOH. Range,
0-12.5 Molal NaOH; 40Ll10~
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418
J. E l e c t r o c h e m . Sac.:
i
i
i
ELECTROCHEMICAL T E C H N O L O G Y
March 1969
Fo r the usual cell liquor composition range, M - will
be > 3 Molal and N - <12.5 Molal. The assumption is
now made that the APM- and APN- are additive, as
indicated from Eq. [17].
F r o m II
i
5o
-%PM- ---- [ ( M - -
3) (0.0019772--0.00001193 t - )
/ - 0.035]M- p~
[18]
From IIIA
30
~ p ~ - = [0.03170 W (174-- t - )
3.o
610
40
810
t, -c.
,~
'
120
140
~f~TERSECTION
IOOR 9 2.B3
, . 198"C
(a 4- b N - 4- c ( N - ) 2 4- d / N - ) ] N -
Fig. 6. Vapor pressure lowering, pure aqueous NaOH. Range,
12.5-30 Molal NaOH; 40~
Molality,
4- 0.035]M- -- [ ( 1 7 4 - - t - )
Slope 100m
-- & 100R/At
N
0.01319
0.01268
N-
0.01160
0.00990
M-
[20]
D
-- - -
[21]
1--D
0.00804
25
30
0.00616
0.00254
In Fig. 7 the n e g a t i v e of the slope 100 m is plotted
against Molality. The c u r v e shows a m a x i m u m at
N ~ 12.5. As before, a suitable c u r v e fit is obtained
by assuming a p o l y n o m i a l of the f o r m
m = a' 4- b ' N 4- c'N 2 4- d ' / N
Solving this p o l y n o m i a l for the
following constants are found
data
[13]
given, the
6.2066 X 10 -4
b' ~-- --2.3155 X 10 -~
2.1860 X 10 -7
d' ~- --2.920
X 10 -'~
Hence,
R ~- 0.0283 -5 (198 - - t) (a' -5 b ' N -5 c'N 2 -5 d ' / N )
[14]
~p ~ [0.0283 4- ( 1 9 8 - t ) ( a ' -5 b ' N -5 c'N 2 -5 d ' / N ) ] N Po
[15]
p :
4- 0.0317]N-}Po-
Defining the fractional decomposition of salt in one
pass t h r o u g h the d i a p h r a g m cell as D, then
17.5
20
22.5
c' =
( a 4 - b N - 4- c ( N - ) 2
4- d / N - )
12.5
15
a'=
[19]
p - = {1 - - [ ( M - - - 3) (0.0019772 - - 0.00001193 t - )
10~
The slope of each line was t h e n measured, as p er
the table wh i ch follows
Po
( 1 - - R N ) p o : {1 -5 [ ( t - - 198)
(a' 4- b ' N -5 c'N ~ 4- d ' / N ) --0.0283]N}po
[16]
The curves shown in Fig. 3, f r o m N = 12.5 to
N --~ 30, are actually a plot of Eq. [16]. These curves
fit the data from the ICT table v e r y well.
V a p o r P r e s s u r e L o w e r i n g in D i a p h r a g m Cell L i q u o r
Let -%PM be the v a p o r pressure l o w e r i n g due to the
NaC1
~PN be t h e v a p o r pressure l o w e r i n g due to t h e
NaOH
so that, if D is known, t h er e is only one concentration
variable, either N - or M - .
It is also w o r t h noting that, in a d i a p h r a g m cell,
the catholyte t e m p e r a t u r e is always g r e a t e r than t he
anolyte t e m p e r a t u r e . Fo r Hooker cells at rated load,
tt + = approx 3~
-
-
Example
F i g u r e 8 shows the calculated v a p o r pressure of
d i a p h r a g m cell liquor at various concentrations of
N a O H and salt decomposition w i t h i n t h e r a n g e of i nterest, for a typical t e m p e r a t u r e of 100~
Since the
solubility of NaC1 in N a O H solutions is limited, the
saturation line is shown. O n l y that portion of the
plot lying above t h e saturation line is valid.
Comparison of calculated v ap o r pressure along the
saturation line w i t h e x p e r i m e n t a l data indicates an
accuracy within 3%. This is sufficiently good for most
en g i n eer i n g purposes. G r e a t e r accuracy would r e q u i r e
taking into account t h e interaction b e t w e e n NaC1 and
NaOH; that is, the slight d e p a r t u r e from the assumption that the v ap o r pressure lowerings of the t w o
solutes are additive.
Programming
For input to a digital computer, the equations can
be simplified. The f o r m a t w i l l depend on t h e c o m p u t e r
used and its p r o g r a m logic. F o r example, for t he
M a t h a t r o n 8-48 (Mathatronics Division of B a r r y
W r i g h t Corporation, Waltham, Massachusetts),
Eq. [3]
c.x.x+b)
x+a)
P - ----PoApM-- ApN[17]
w h e r e the m i n u s superscripts r e f e r to the catholyte at
t-at.
-
-
-
x--
( t + 2 7 3 . 1 6 ) -- ( l + d . x )
- - 5.219603)
(--1) = log Po, antilog = Po
-
ioo~
o,~
~o
.014
.oq~ i
.o~
.OlC
+
5ci
~
//
//
//
- /j
/
/
~oo;
~,oo~
~+
_~
Joo~
.oo,
//
~o
N, MOLAL~TY OF N=OH
Fig. 7. Temperature coefficient of 100 R for pure NoaH. Range,
12.5-30 Molal NoaH; 40~ ]0~
Fig. 8. Vapor pressure of catholyte from a diaphragm cell vs.
salt decomposition. Ordinate, Torr. Abscissa, fractional decomposition D. Heavy dashed line shows saturation with NaCI. Valid values
of composition and vapor pressure lie above this llne.
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Vol. 116, No. 3
VAPOR
PRESSURE
Eq. [6]
1.9772. 10 -3 - - 1.193. 1 0 - 6 . t +) ( 3 - - M +)
-- 0.035)
M + -f- 1 ) P o - ~ p *
Eq. [11]
cN-bb) N§
(t--174)
- - 0.0317) N ~- 1) Po = P
Manuscript submitted Oct. 22, 1968; revised m a n u script received Dec. 6, 1968. This paper was presented
at the Tripartite meeting of the A.I.Ch.E., Montreal,
Sept. 25, 1968.
A n y discussion of this paper will appear in a Discussion Section to be published in the December 1969
JOURNAL.
Eq. [16]
c'N % b') N ~- a' nu d' -- N) ( t - - 198)
--0.0283) N W 1) Po = P
Eq. [20]
c.N-+b)
N-+ad-d--N-)
( t - - - 174)
- - 0.0317)N- ~ ((3 - - M - ) (1.9772 9 I0 - J
- - 1.193 9 I0 -~ 9 t - ) - - 0.035) M - + l) Po = p -
419
OF WATER
REFERENCES
1. K e e n a n a n d Keyes, " T h e r m o d y n a m i c Properties of
Steam," J o h n Wiley & Sons, New York (1962).
2. L. B. Smith, F. G. Keyes, and H. T. Gerry, Proc. Am.
Acad. Arts and Sci, 69, 137 (1934).
3. I n t e r n a t i o n a l Critical Tables, M c G r a w Hill Book Co.,
V o l . III (1926).
Erratum
There was an error in the paper "Accommodation of
Lattice Mismatch at Heterojunctions" by R. S. Mroczkowski, A. F. Witt, and H. C. Gatos, 115, 750 (1968)
which was published in the J u l y 1968 issue of the
J o u r n a l (115, 750-752).
In considering the theoretical dislocation density
required for the accommodation of a n a b r u p t lattice
mismatch in single crystal growth (~ = Sao2/5), i n advertently, the lattice t r a n s l a t i o n in the a(l~o~ direction in the z i n c - b l e n d e lattice was t a k e n as ar
1
= k/2a~ instead of a(11o) =
ao; the theoretical dis-
location density is smaller by a factor of 2 t h a n that
indicated in the article. Consequently, a comparison
with the e x p e r i m e n t a l l y observed dislocation density
suggests that part of the lattice mismatch is accommodated by strain. In the light of the p r e s e n t correction, the results seem to confirm the validity of the
Van der Merve model.
The authors are grateful to Dr. G. O. Krause, of
Texas I n s t r u m e n t s , Inc., for b r i n g i n g this error to
their attention.
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