Thermodynamic Behavior of Electrolytes in Mixed SolventsâII

4 Error Analysis of Isobaric Liquid-Vapor Equilibrium Data for Mixed Solvents

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Containing Salts at Saturation D E R E K JAQUES Department of Applied Chemistry, Royal Melbourne Institute of Technology, Melbourne, Victoria, 3000, Australia

In the calculation of total pressure and vapor

composition

from boiling point data using the indirect

method, the

greatest source of error lies in the liquid-phase

composition.

We have attempted to characterize the frequency

distribu

tion of the error in the calculated vapor composition by the standard statistical methods and this has given a satisfactory result for the methanol-water

system saturated with sodium

chloride when the following estimates of the standard devia tion were used: x, 0.003; y, 0.006; T , 0.1°C; and π, 2 mm Hg. This work indicates that in the design of future experiments more data points are required and, for each variable, a reliable estimate of the standard deviation is highly desirable.

* 0 e c e n t l y there has b e e n c o n s i d e r a b l e interest i n t h e c h e m i c a l l i t e r a t u r e o n t h e subject o f t h e r m o d y n a m i c consistency, e v a l u a t i o n o f d a t a , a n d i n error analysis o f salt-free data.

T h e s e authors (1,2,3,4),

f o r reasons

of s i m p l i c i t y , chose i s o t h e r m a l d a t a w h e r e t h e i s o t h e r m a l - i s o b a r i c f o r m of t h e G i b b s - D u h e m e q u a t i o n c a n b e u s e d w i t h o n l y a v e r y s m a l l error. W e are i n t e r e s t e d i n i s o b a r i c d a t a c o n t a i n i n g salts a t s a t u r a t i o n because m o s t salt d a t a are m e a s u r e d u n d e r these c o n d i t i o n s . A l s o t h e m o d e l w e w i s h to use is b a s e d u p o n B a r k e r s m e t h o d ( 5 ) w h i c h p r e d i c t s v a p o r composition from b o i l i n g point data.

T h i s a p p r o a c h has b e e n d i s c u s s e d

p r e v i o u s l y i n some d e t a i l b y t h e present a u t h o r (6) a n d f o r a l c o h o l / w a t e r systems i t w a s p r e f e r a b l e t o t h e c o r r e l a t i o n o f excess free e n e r g y w h i c h i n c o r p o r a t e s t h e r e d u n d a n t (/-values a n d t h e i r associated errors. F u r t h e r 0-8412-0428-4/79/33-177-039$05.00/l © 1979 American Chemical Society Furter; Thermodynamic Behavior of Electrolytes in Mixed Solvents—II Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

40

THERMODYNAMIC BEHAVIOR OF ELECTROLYTES

more, a suggestion was made (7) rium

II

t h a t as O t h m e r l i q u i d - v a p o r e q u i l i b

stills h a v e b e e n u s e d e x t e n s i v e l y t h e r e is u n c e r t a i n t y i n t h e d e t e r

m i n a t i o n of t e m p e r a t u r e . H e n c e w e ask w h a t is t h e r e l a t i v e i m p o r t a n c e of t h e errors i n e a c h v a r i a b l e . I f t h e i m p o r t a n c e of t h e Γ-error w a s m u c h greater t h a n t h e {/-error t h e B a r k e r m e t h o d m i g h t n o t b e t h e best approach. T h e r e are t h r e e sources of error i n t h e c a l c u l a t e d v a p o r c o m p o s i t i o n w h e n these are c a l c u l a t e d f r o m b o i l i n g p o i n t d a t a : r a n d o m error i n e a c h e x p e r i m e n t a l o b s e r v a t i o n ; s y s t e m a t i c e r r o r i n one o r m o r e of t h e o b s e r v a t i o n s ; a n d t h e m o d e l is i m p e r f e c t ( t h i s is p a r t i c u l a r l y t r u e f o r i s o b a r i c Downloaded by GEORGETOWN UNIV on October 26, 2017 | http://pubs.acs.org Publication Date: June 1, 1979 | doi: 10.1021/ba-1979-0177.ch004

d a t a because use is m a d e of the G i b b s - D u h e m e q u a t i o n w h i c h w a s d e r i v e d for constant t e m p e r a t u r e a n d p r e s s u r e ) . I n the present w o r k w e s h a l l assume t h a t t h e o n l y e r r o r i n t h e d a t a is c a u s e d b y r a n d o m n e s s . T h e p u r p o s e of this w o r k is t o a t t e m p t t o a n a l y z e t h e r a n d o m errors in

e a c h i n d e p e n d e n t v a r i a b l e a n d assess w h i c h one

contributes

the

greatest e r r o r i n t h e c a l c u l a t e d q u a n t i t i e s w h e n use is m a d e o f

the

indirect method. Correlation

Procedure

T h i s p r o c e d u r e has b e e n g i v e n i n d e t a i l e l s e w h e r e (6)

so i t w i l l o n l y

b e d e s c r i b e d h e r e b r i e f l y for t h e sake o f completeness.

T h e function

2(π —

7T ) C

is m i n i m i z e d w h e r e t h e t o t a l pressure is g i v e n b y t h e e q u a t i o n :

2

*c — z p i ' y i * i +

(1 -

(1)

x) W72*2

T h e v a p o r p r e s s u r e of e a c h solvent is r e p l a c e d b y t h e v a p o r pressure of t h e solvent s a t u r a t e d w i t h salt at t h e o b s e r v e d t e m p e r a t u r e . T h e W i l s o n E q u a t i o n ( δ ) is u s e d to r e l a t e t h e a c t i v i t y coefficient o n a salt-free b a s i s :

l n

y

i

-

-ln(l - A

2

1

( l

-x))

+

(1 — rc)Ai2

(1

Ax 12

(2)

xA i l - A ( l - z ) 2

2

1

- » - - h P - * * » - ' { * î ^ - . - £ 5 - . , ) } (3) where A i = 2

l - ^ - e x p ( - Z Vι

1

/ R T )

Furter; Thermodynamic Behavior of Electrolytes in Mixed Solvents—II Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

(4)

4.

JAQUES

Isobaric Liquid-Vapor

41

Equilibrium

and A

=

1 2

l - ^ - e x p ( - Z v

2

/ R T )

(5)

2

O n e of t w o p r o c e d u r e s c a n b e u s e d n o w .

The equilibrium vapor com

p o s i t i o n is e v a l u a t e d u s i n g t h e o b s e r v e d temperatures a n d E q u a t i o n 6: In y — Ι η ^ ρ / Φ ι Α τ ο ) — l n ( l — A i ( l — x))

+

2

Π — r) / — ^) i2 \ 1 - A z

Â i

A


1

j

1 2

2

1 -

A

(l -s)

2 1

\

(6)

/

or a l t e r n a t i v e l y the e q u i l i b r i u m t e m p e r a t u r e is c a l c u l a t e d a n d s u b s t i t u t e d i n E q u a t i o n 7: 1η2/ = 1η(αρι'Φι/π) J

/i U

X )

ln(l -

A i(l -*))

(1 — a;)Ai2 1 - A

χ

1

2

s

£A i _ A

+

2

2

1

2 1

|

(7)

( l - s ) J

I n t h e present w o r k the second p r o c e d u r e has b e e n u s e d . Error We

Analysis b e g i n b y d e f i n i n g the e r r o r i n t o t a l pressure ( Ε ) π

for

each

m e a s u r e m e n t as: # π = π — 7T = τι- — χ γ ι ρ / Φ ι — (1 — ζ)γ2Ρ2 ϊ>2 /(

C

(8)

T h e i n d e p e n d e n t v a r i a b l e s are χ, T, a n d π. I n E q u a t i o n 8 t h e a c t i v i t y coefficients are f u n c t i o n s o f χ a n d T , t h e v a p o r pressures a r e f u n c t i o n s o f T , a n d t h e f u g a c i t y coefficients a n d m o l a r v o l u m e s are a s s u m e d free of r a n d o m error. H e n c e for the v a r i a n c e of the Ε

π

error w e h a v e :

T h r e e d i f f e r e n t i a l terms are r e a d i l y c a l c u l a t e d at e a c h d a t u m b u t t h e c o r r e s p o n d i n g set of χ, Γ, a n d π s t a n d a r d deviations are strictly u n k n o w n . H o w e v e r w e c a n m a k e a reasonable estimate of these values a n d also assume t h a t e a c h is i n d e p e n d e n t o f c o m p o s i t i o n . T h e v a p o r c o m p o s i t i o n s are c a l c u l a t e d f r o m the e q u a t i o n :

y

=


(10)

42


u s i n g t h e c a l c u l a t e d temperatures.

H e n c e w e h a v e the f o l l o w i n g v a r i

a b l e s : χ, Τ, π, Z i , a n d Z , b u t t h e y are n o t a l l i n d e p e n d e n t . 2

χ, π, Z i , a n d Z

2

II

So w e t a k e

as i n d e p e n d e n t v a r i a b l e s . T h e p r o b l e m s associated w i t h

t r y i n g to assess the error i n Z

x

and Z

2

w i l l b e d i s c u s s e d f u l l y later. H e r e

i t is sufficient to note t h a t the a c t u a l values of Z

x

and Z

2

d e p e n d u p o n the

r a n d o m errors associated w i t h χ, T, a n d π a n d h e n c e t h e p r o b l e m is c o m p l e x a n d not a m e n a b l e to a f u l l statistical t r e a t m e n t . T h e v a r i a n c e


of the j/-error is g i v e n b y :

(ID

T o c h e c k that t h e m e t h o d c a n b e u s e d for i s o b a r i c d a t a a set of p e r f e c t d a t a are g e n e r a t e d a n d r a n d o m errors a d d e d to x, y, Γ, a n d π i n t u r n a n d a l l together to see w h a t effect t h e y h a v e o n o u r s t a n d a r d p r o cedure.

F o r l a r g e samples w e expect 6 8 %

of the s a m p l e v a l u e s to l i e

w i t h i n one s t a n d a r d d e v i a t i o n of t h e p e r f e c t v a l u e of t h e selected v a r i a b l e . I n the case of s m a l l samples, e.g., t w e l v e d a t a , error b o u n d s are c a l c u l a t e d u s i n g b i n o m i a l p r o b a b i l i t i e s f o r e a c h of the a b o v e v a r i a b l e s so t h a t , w i t h p r o b a b i l i t y of 0.95, w e expect 4 1 - 9 5 % of t h e s a m p l e observations to l i e w i t h i n one s t a n d a r d d e v i a t i o n of the p e r f e c t v a l u e of t h e selected v a r i a b l e ( t h e n o r m a l d i s t r i b u t i o n is a s s u m e d ) .

T w e l v e is a c o m m o n n u m b e r of

d a t a p o i n t s w i t h s a l t - s a t u r a t e d solutions a n d this shows t h e d e s i r a b i l i t y of t a k i n g m o r e e x p e r i m e n t a l observations. Application

of Error

Analysis

I n a p r e v i o u s e v a l u a t i o n of salt-saturated d a t a , i t w a s f o u n d ( 7 ) t h a t t h e m e t h a n o l - w a t e r system s a t u r a t e d w i t h s o d i u m c h l o r i d e s h o w e d l i t t l e o r n o average bias for the c a l c u l a t e d v a p o r c o m p o s i t i o n f o r b o t h t h e Τ — χ fit a n d t h e G / R T — χ fit, i t p a s s e d t h e a r e a test q u i t e e a s i l y a n d E

s h o w e d satisfactory values of a l l s a m p l e d e r i v a t i o n s . H e n c e t h i s system w a s chosen for error analysis. Stage 1. T h e M e O H / H 0 / N a C l d a t a are s u b j e c t e d to t h e c o r r e l a 2

t i o n p r o c e d u r e d e s c r i b e d p r e v i o u s l y w h i c h gives v a l u e s of t h e W i l s o n e n e r g y constants ( Z

t

and Z ) 2

a n d a n e w set o f d a t a f o r t e m p e r a t u r e a n d

v a p o r c o m p o s i t i o n t h a t are i n t e r n a l l y consistent (see

Table I ) . T h e small

v a l u e s of the s t a n d a r d d e v i a t i o n a n d t h e b i a s i n d i c a t e g o o d q u a l i t y d a t a i n t h e salt effect

field.

F o r t h e analysis of s e r i a l c o r r e l a t i o n a m o n g t h e

r e s i d u a l s w e use the D u r b i n - W a t s o n test ( 9 ) .

A r u n of positive or

n e g a t i v e signs i n t h e series of r e s i d u a l s is some i n d i c a t i o n t h a t t h e m o d e l


4.

JAQUES


43

Equilibrium

Table I . Experimental and Calculated D a t a for the M e O H / H 0 / N a C l System at π = 762 m m H g Pressure


2

Xobs'

Yobs

Ycalc

T &

0.029 0.050 0.074 0.110 0.174 0.250 0.348 0.448 0.557 0.653 0.768 0.878

0.259 0.418 0.515 0.590 0.661 0.721 0.766 0.804 0.841 0.875 0.913 0.953

0.301 0.415 0.499 0.578 0.658 0.714 0.763 0.803 0.842 0.875 0.915 0.954

99.6 95.0 90.5 86.5 82.3 79.0 76.2 74.0 72.2 70.2 68.0 66.1

^--values y-values T-values

0

RMS Deviation

Bias

10.4 0.014 0.40

-3.0 0.0 0.11

s

rp a J- calc 99.05 94.55 90.84 86.95 82.58 79.29 76.30 73.88 71.61 69.77 67.70 65.82 DurbinWatson Test (D) 0.57 1.04 0.60

° New data set.

u s e d is i n a d e q u a t e .

I n the present case f o r i s o b a r i c d a t a w e w o u l d n o t

e x p e c t the m o d e l to b e perfect because of the use of t h e G i b b s - D u h e m e q u a t i o n w h i c h is n o t s t r i c t l y a p p l i c a b l e to i s o b a r i c d a t a a n d so a v a l u e close to t w o is n o t to be expected.

H o w e v e r , w e s h a l l use the test to g i v e

a r e l a t i v e measure of t h e a d e q u a c y of the m o d e l . Stage 2. W e p r o d u c e 99 e q u a l l y s p a c e d x-values e x c l u d i n g the t w o extreme values. found

i n S t a g e 1 a n d the e x p e r i m e n t a l t o t a l p r e s s u r e v a l u e f o r

MeOH/H 0/NaCl 2

and T.

between 0 and 1

B y u s i n g t h e v a l u e s of Z i a n d

Z

system w e c a l c u l a t e the c o r r e s p o n d i n g v a l u e s of

N e x t w e i n t r o d u c e n o r m a l l y d i s t r i b u t e d r a n d o m errors of

2

the y

zero

m e a n f o r e a c h v a r i a b l e b y s p e c i f y i n g t h e s t a n d a r d d e v i a t i o n of x, y, T , a n d π, r e s p e c t i v e l y , a n d a d d t h e m i n t u r n a n d t h e n a l l together to the generated data.

T h e f o l l o w i n g values o f t h e s t a n d a r d d e v i a t i o n

were

s e l e c t e d : x, 0.003; y, 0.006; Γ, 0 . 1 ° C ; a n d ττ, 2 m m H g . T h e l a t t e r w a s h i g h because J o h n s o n a n d F u r t e r ( 1 0 )

d i d not c o n n e c t a monostat to t h e i r

e q u i l i b r i u m s t i l l . T h e average v a r i a t i o n of a t m o s p h e r i c pressure q u o t e d f o r t h e i r d a t a set is ± 2 m m H g . T h e c h o i c e of t h e o t h e r v a l u e s w a s d e t e r m i n e d b y t h e r e q u i r e m e n t t h a t 6 8 % of the differences b e t w e e n t h e g e n e r a t e d d a t a p l u s r a n d o m error a n d t h e g e n e r a t e d d a t a m u s t l i e w i t h i n t h e specified confidence levels b a s e d o n E q u a t i o n 11. F i g u r e 1 shows S^



44


0

0.2

0.4

X0.6

0.8

II

1.0

Figure I . Effect of errors in χ, T , and π on the calculated standard devia tion of Ε in Equation 9: (1) 3 simultaneous errors; (2) x-error; (3) T-error; (4) π-error p l o t t e d against χ ( E q u a t i o n 9) w h e r e the effect of e r r o r i n e a c h v a r i a b l e separately a n d t h e n a l l together is s h o w n .

N o t e t h a t t h e effect of χ is

a l w a y s a p p r e c i a b l e a n d is t h e d o m i n a n t v a r i a b l e at l o w x-values. F i g u r e 2 shows S p l o t t e d against χ ( E q u a t i o n 11) f o r a l l f o u r v a r i y

ables (i.e., χ, π, Z i , a n d Z ) 2

separately a n d t h e n together.

A g a i n χ is t h e

d o m i n a n t v a r i a b l e at l o w χ a n d has a n a p p r e c i a b l e effect over t h e r e m a i n i n g concentration range. I t is p e r h a p s w o r t h m e n t i o n i n g t h a t i n E q u a t i o n 10 y is a f u n c t i o n of χ, π, Γ, Zi, a n d Z χ, 7Γ, Z i , a n d Z

2

2

b u t t h e y are n o t i n d e p e n d e n t v a r i a b l e s , b e c a u s e i f

a r e k n o w n Γ c a n b e c a l c u l a t e d . H e n c e t h e error i n t h e

m e a s u r e d Γ is i n c l u d e d i n t h e errors associated w i t h t h e e n e r g y p a r a m eters.

T h e f a c t t h a t t h e y also c o n t a i n χ a n d π errors c o m p l i c a t e s

the

statistical t r e a t m e n t . T h e l e v e l of e r r o r associated w i t h t h e W i l s o n e n e r g y p a r a m e t e r s is d i f f i c u l t to q u a n t i f y . T h e p r o b l e m arises b e c a u s e t h e v a l u e s of t h e p a r a m e t e r s a r e g o v e r n e d b y the errors i n χ, T, a n d π t h r o u g h t h e use of E q u a t i o n 8. W e e x a m i n e d t h e s u m of squares ( E q u a t i o n 8) f o r a r a n g e o f values of t h e t w o p a r a m e t e r s to see i f t h e y a r e r o b u s t , i.e., to see i f s l i g h t changes i n v a l u e c a u s e d l a r g e changes i n t h e s u m of squares a n d f o u n d this not to b e so. W e a s s u m e d a n e r r o r l e v e l of 2%

for each energy

p a r a m e t e r as b e i n g reasonable.


4.

JAQUES

Isobanc

Liquid-Vapor

45

Equilibrium

T a b l e I I gives the s t a n d a r d d e v i a t i o n s of pressure, v a p o r c o m p o s i t i o n a n d t e m p e r a t u r e , a n d the c o r r e s p o n d i n g bias a n d D - v a l u e as e a c h v a r i a b l e is c h a n g e d r a n d o m l y a n d t h e n as a l l f o u r are c h a n g e d s i m u l t a n e o u s l y . W e see t h a t t h e r a n d o m error of χ c o n t r i b u t e s c a . 75%

of the i n d u c e d

error i n t h e v a l u e o f t h e s t a n d a r d d e v i a t i o n o f b o t h t h e pressure a n d t e m p e r a t u r e w h i l e the r a n d o m error of Τ a n d π o n l y c o n t r i b u t e 12%

each.

about

O n the o t h e r h a n d t h e r a n d o m errors of χ a n d y c o n t r i b u t e

e q u a l l y to the i n d u c e d - v a p o r c o m p o s i t i o n s t a n d a r d d e v i a t i o n w i t h t h e pressure m a k i n g a n e g l i g i b l e c o n t r i b u t i o n . T h e bias v a l u e s are n e g l i g i b l y s m a l l except f o r t h e pressure s t a n d a r d d e v i a t i o n s w h e r e t h e y are s t i l l n o t Downloaded by GEORGETOWN UNIV on October 26, 2017 | http://pubs.acs.org Publication Date: June 1, 1979 | doi: 10.1021/ba-1979-0177.ch004

l a r g e . T h e final c o l u m n has D - v a l u e s at least e q u a l t o t w o a n d t h i s gives one c o n f i d e n c e i n the m o d e l a n d suggests i t is a d e q u a t e f o r g o o d q u a l i t y d a t a as i n this p a r t i c u l a r case t h e o n l y source of e r r o r is c a u s e d

by

random behavior.

3

0

2

^

Jt^-Â

0.2

^

r

~

•

_

0.4

DC

0.8

0.6

1.

Figure 2. Effect of errors in χ, T , Z and Z on the calculated standard devi ation of the y-error in Equation 11: (1) 4 simultaneous errors; (2) x-error; (3) -π-error; (4) Z error; (S) Z -error l9

t

r

t


46


Table II.

Effect of Random E r r o r on 99

Standard

7r-Value y-Value r-Value

X

y

5.7 0.006 0.23

0.006

—

H

Deviation

Τ

7Γ

AU 4

2.6

2.4 0 0.08

6.8 0.008 0.27

0.09

Stage 3. T h e t w e l v e c a l c u l a t e d d a t a f r o m T a b l e I are p r o c e s s e d b y Downloaded by GEORGETOWN UNIV on October 26, 2017 | http://pubs.acs.org Publication Date: June 1, 1979 | doi: 10.1021/ba-1979-0177.ch004

a d d i n g n o r m a l l y d i s t r i b u t e d r a n d o m errors of zero m e a n to e a c h v a r i a b l e i n t u r n a n d t h e n a l l together.

T h e results are s h o w n o n F i g u r e s 3 a n d 4

f o r Δπ a n d Δ , r e s p e c t i v e l y . T h e confidence regions also are s h o w n a n d ν

w e observe t h a t 9 2 % a n d 5 8 % of the c a l c u l a t e d differences, r e s p e c t i v e l y , f a l l w i t h i n these l i m i t s .

F o r 12 d a t a p o i n t s there is a 9 5 % p r o b a b i l i t y

that b e t w e e n 4 1 % a n d 9 5 % of the c a l c u l a t e d values s h o u l d l i e w i t h i n t h e confidence l i m i t s . T h i s w i d e r a n g e for a s m a l l n u m b e r of d a t a p o i n t s a g a i n

lô-

IOΔ1Γ 5Θ

ο

Θ

0-

ο

Θ

Θ

Ο

Θ

Θ

Θ

-5-

Ι

-10-

-15-

c) Figure

/ 3.

0.2 Pressure

0.4

χ

0.6

difference and the 68% confidence calculated data of Table I

0.8

1.

region for the 12


4.

JAQUES


47

Equilibrium

Simulated D a t a for M e O H / H 0 / N a C l 2

Durbin

Bias X

y

0.3 0 0.01

0

τ 0.16

—

—

-0.01

7Γ

0.06 0 -0.01

AU

4

0.11 0 -0.01

χ

y

1.4 1.4 1.5

1.7

—

Watson

Test (D)

Τ

π

1.6

2.1 1.9 2.3

—

1.8

h i g h l i g h t s t h e d e s i r a b i l i t y o f t a k i n g a l a r g e n u m b e r o f observations.

4

All 2.0 2.1 2.0

Table


I I I shows t h e s t a n d a r d d e v i a t i o n , b i a s , a n d D - v a l u e f o r t h e s i m u l t a n e o u s a d d i t i o n o f r a n d o m errors to a l l v a r i a b l e s . F i n a l l y t h e o r i g i n a l d a t a are s h o w n o n F i g u r e s 5 a n d 6 together w i t h t h e confidence regions. N o w w e see that 4 2 % o f t h e pressure differences l i e w i t h i n t h e confidence levels w h i l e 6 6 % o f t h e v a p o r

composition

differences a r e w i t h i n t h e levels. I n c l u d e d i n T a b l e I I I a r e t h e s t a n d a r d d e v i a t i o n s , b i a s , a n d results o f t h e D u r b i n - W a t s o n test. C o m p a r i s o n o f t h e t w o sets of results i n d i c a t e s a p p r e c i a b l y l a r g e r v a l u e s f o r s t a n d a r d d e v i a t i o n a n d bias f o r t h e e x p e r i m e n t a l results w h e r e a s f o r t h e D - t e s t t h e

0.04

-0.02

-0.04 1.0 Figure

4.

Vapor

^^^^^gy^^^^^^ggt^

c o n

fi^

e n c e

region for

Society Library 1155 16th St. N. W. Furter; Thermodynamic Behavior Washington, D.ofC.Electrolytes 20036in Mixed Solvents—II

Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

48

THERMODYNAMIC BEHAVIOR O F ELECTROLYTES

Π

\

15-

\°

10-

Δ1Γ

\

Θ

®

5-


Θ 0-

Θ -5-

Θ

G

-10-

-15ι

0

1

Figure 5.

Θ

1

0.2 m

•1

Θ • 1

0.4

1

0.6

1

0.8

1.0

Pressure difference and the 68% confidence region for the methanolr-water-sodium chloride system

reverse is t r u e . P a r t of the e x p l a n a t i o n f o r this lies i n t h e p a r t i c u l a r set of c a l c u l a t e d d a t a p l u s r a n d o m errors u s e d i n T a b l e I I I . O n e h u n d r e d sets of c a l c u l a t e d d a t a c o n t a i n i n g s i m u l a t e d r a n d o m errors w e r e p r o c e s s e d a n d the average v a l u e of the pressure s t a n d a r d d e v i a t i o n a n d its s t a n d a r d d e v i a t i o n c a l c u l a t e d . T h i s w a s 5.0 ± 3.4. H e n c e t h e p a r t i c u l a r set u s e d i n T a b l e I I I w a s o n t h e l o w side of the m e a n . T h e o t h e r p a r t of t h e ex p l a n a t i o n f o r t h e a p p a r e n t d i s c r e p a n c y is g i v e n i n t h e n e x t section.

Table ΠΙ. Comparison of 12 Experimental D a t a and Simulated D a t a Containing Random E r r o r Standard

π-Value y-Value r-Value

Deviation

Exptl.

Calc.

10.4 0.014 0.40

3.1 0.009 0.11

Bias Exptl. -3.0 0 0.11

12

D-value Calc. -0.8 0 0.3

Exptl.

Calc.

0.57 1.04 0.60

1.14 2.28 1.20



4.

JAQUES

Ο Figure 6.


0.2

49

Equilibrium

0.4

χ 0.6

0.8

1.0

Vapor composition difference and the 68% confidence region for the methanol-water-sodium chloride system

Conclusions I n the analysis o f t h e effect o n t h e c a l c u l a t e d q u a n t i t y o f r a n d o m errors i n m e a s u r e d q u a n t i t i e s i t is u n f o r t u n a t e t h a t t h e o n l y m o d e l s u s c e p t i b l e to a n exact statistical t r e a t m e n t is t h e l i n e a r one

(II).

Here we

have

a t t e m p t e d to c h a r a c t e r i z e the f r e q u e n c y d i s t r i b u t i o n of t h e error i n t h e c a l c u l a t e d v a p o r c o m p o s i t i o n b y the s t a n d a r d m e t h o d s

a n d have

i n c l u d e d a c o - v a r i a n c e t e r m for e a c h p a i r of d e p e n d e n t v a r i a b l e s

not (12).

O u r a p p r o a c h has g i v e n a satisfactory r e s u l t for t h e m e t h a n o l - w a t e r s o d i u m c h l o r i d e system b u t i t has not b e e n tested o n other systems a n d p e r h a p s of m o r e i m p o r t a n c e , i t has n o t b e e n p o s s i b l e , so f a r , to c o n f i r m t h e essential correctness of t h e m e t h o d b y a n i n d e p e n d e n t

procedure.

W o r k is c u r r e n t l y b e i n g u n d e r t a k e n o n t h i s project. Several conclusions c a n be

d r a w n f r o m this w o r k .

First, i n the

c a l c u l a t i o n o f t o t a l pressure a n d v a p o r c o m p o s i t i o n f r o m b o i l i n g p o i n t d a t a the greatest source of e r r o r lies i n the l i q u i d - p h a s e c o m p o s i t i o n , p a r t i c u l a r l y at l o w c o n c e n t r a t i o n . S e c o n d , t h e estimates of the s t a n d a r d d e v i a t i o n for v a p o r c o m p o s i t i o n a n d t e m p e r a t u r e of 0.006 a n d 0 . 1 ° C ,


50

T H E R M O D Y N A M I C BEHAVIOR OF E L E C T R O L Y T E S

II

r e s p e c t i v e l y , are q u i t e l o w a n d suggest t h a t t h e m a i n effort is r e q u i r e d i n r e d u c i n g the error i n the first c o n c l u s i o n . set of d a t a the m e a s u r e m e n t of Τ-π-χ

C e r t a i n l y w i t h t h e present

data only w o u l d have given very

satisfactory j/-values. T h i r d , unless a m o r e s o p h i s t i c a t e d a p p r o a c h is to b e u s e d f o r c o l l e c t i n g i s o b a r i c d a t a , d a t a d e t e r m i n e d at v e r y l o w x-values are g o i n g to b e subject to a v e r y l a r g e r a n d o m error a n d h e n c e i t w o u l d b e m o r e p r o f i t a b l e to o b t a i n e x t r a d a t a at h i g h e r x-values. F i n a l l y , i n t h e d e s i g n of f u t u r e e x p e r i m e n t s w e n e e d m o r e d a t a p o i n t s a n d , f o r variable,

a

reliable

estimate

of

the

standard

deviation

each

should

be


determined.

Nomenclature Subscripts: 1 =

alcohol

2 =

water

c =

calculated

A21, A12 = =

Ε

π

p/ = âr>

Sy, SiTy

s

=

T

constants i n W i l s o n e q u a t i o n pressure difference i n E q u a t i o n 8 v a p o r pressure of C o m p o n e n t i s a t u r a t e d w i t h salt estimate of s t a n d a r d d e v i a t i o n of the error i n the e x p e r i m e n t a l v a r i a b l e s x y, π, a n d T, r e s p e c t i v e l y y

Szi, Sz

2

=

estimate of the s t a n d a r d d e v i a t i o n of t h e error i n the c a l c u l a t e d energy p a r a m e t e r s

Sε =

c a l c u l a t e d s t a n d a r d d e v i a t i o n of Ε i n E q u a t i o n 9

S„ =

calculated

standard

deviation

of

the

error

in

y

in

E q u a t i o n 11 Τ = Vi =

temperature, ° C m o l a r v o l u m e of C o m p o n e n t i

χ =

m o l e f r a c t i o n of a l c o h o l i n l i q u i d p h a s e , c a l c u l a t e d o n a

y =

m o l e f r a c t i o n of a l c o h o l i n v a p o r p h a s e

salt-free basis Zi, Z

2

=

energy p a r a m e t e r s i n W i l s o n e q u a t i o n c o m m o n l y expressed as (λ,, -

yι = Δ7Γ

=

&y = IT = Φι =

λα)

a c t i v i t y coefficient of C o m p o n e n t i 7Γ —

7T

C

y — y t o t a l pressure c

c o r r e c t i o n t e r m f o r n o n i d e a l i t y of C o m p o n e n t i i n a n i n d e a l gaseous s o l u t i o n a n d is t h e r e c i p r o c a l of

the

coefficient.


fugacity

4.

JAQUES

Isobaric

Liquid-Vapor

Equilibrium

51

Acknowledgments T h e a u t h o r w i s h e s to t h a n k I . R . I . C o x o f t h e D e p a r t m e n t o f M a t h e m a t i c s at R . M . I . T . f o r m a n y h e l p f u l discussions a n d t h e C o m p u t e r C e n t r e at R . M . I . T . f o r t h e p r o v i s i o n o f facilities.


Literature Cited 1. Ulrichson, D. L., Stevenson, F. D., Ind. Eng. Chem. Fundam. (1972) 11, 287. 2. Van Ness, H. C., Byer, S. M., Gibbs, R. E., AIChE. J. (1973) 19, 238. 3. Fabries, J. F., Renon, H., AIChE J. (1975) 21, 735. 4. Peneloux, Α., Deyrieux, R., Canals, E., Neau, E., J. Chem. Phys. (1976) 73, 706. 5. Barker, J. Α., Aust. J. Chem. (1953) 6, 207. 6. Jaques, D., Ind. Eng. Chem. Process Des. Dev. (1976) 15, 236. 7. Jaques, D., Ind. Eng. Chem. Process Des. Dev. (1977) 16, 129. 8. Wilson, G. M., J. Am. Chem. Soc. (1964) 86, 127. 9. Kendall, M. G., "Time Series," p. 163, Griffin, London, 1973. 10. Johnson, A. I., Furter, W. F., Can. J. Chem. Eng. (1960) 38, 78. 11. Draper, N. R., Smith, H., "Applied Regression Analysis," Wiley, New York, 1968. 12. Kempthorne, O., Folks, L., Probability, Statistics, and Data Analysis," p. 129, Iowa State University, Ames, 1971. RECEIVED February 6, 1978.


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