CHAPTER 3: PHASE EQUILIBRIA

CHAPTER 3: PHASE EQUILIBRIA 3.1 Introduction

Multiphase and solution thermodynamics deal with the composition of two or more phases in equilibrium. Thus, the maximum concentration of a species in an aqueous stream in contact with an organic stream can be estimated by these calculations. This can establish the contaminant levels obtained in various wastewater streams. A second major application is in partitioning of a pollutant into various phases in the environment. These multiphase thermodynamic calculations are important in design of heterogeneous reactors. In this section, we provide some basic definitions and illustrate the applications of thermodynamic models to waste minimization.

First, we provide various definitions for thermodynamic equilibrium and then illustrate them with applications.

3.2 Vapor-Liquid Equilibrium The ratio of the composition measure such as (mole fraction) in the vapor phase to that in the

liquid phase at equilibrium is referred to as the K-value. Note that K y is dimensionless.

K yi

=

yi xi

eq

(1)

where yi is the mole fraction of species i in the vapor phase and xi is the liquid.

For ideal solutions, the Raoults law applies. This can be stated as follows. At equilibrium the

partial pressure of a species in the gas phase, Pi , is equal to the mole fraction of the species in the liquid

phase,

xi , multiplied by its vapor pressure,

P vap i

,

at

the

given

temperature.

It is also equal to the product

of the mole fraction in the gas phase, yi , and total pressure, P.

Pi = xi Pivap = yi P

(2)

Hence, Raoults law can also be stated as:

yi

=

xi

P vap i P

(3)

Therefore, the K-factor for ideal mixtures is:

K yi

=

P vap i P

(4)

For non-ideal solutions, the K-factors can be calculated using the activity coefficients. However,

for many environmental applications, one can use experimentally reported K-factors. These calculations

are, for example, useful to find the composition of the vapor phase in contact with the liquid in the

reactor. If there is a fugitive emission, then we can estimate the amount of the toxic chemicals that has

inadvertently escaped to the atmosphere. Vapor-liquid equilibrium is also useful in estimating the

maximum concentration of VOC in a mixture.

For dilute solutions, or for gaseous species, Henry's law, given by Eq. (5) is more convenient.

xi = H i yi P

(5)

where Hi is the Henry's law constant, (atm-1) and P (atm) is the total pressure. With this definition, we find that Ky-factor is

1

K yi

=

1 HiP

(6)

Vapor pressure data is often needed to estimate the levels of VOC emissions at various

temperatures. The data for pure liquids are well represented by the Antoine equation.

log 10

P*

=

A-

T

B +C

(7a)

or by an empirical extended Antoine is equation:

ln P vap

=

k1

+

k2 k3 + T

+

k4T

+ k5

ln T

+

k6T k7

(7b)

3.3 Gas-Liquid Systems

Phase equilibrium between a dissolving gas, A, and its dissolved concentration,

[ A H2O] or [ A] , in water is often expressed (for dilute systems) through the equilibrium constant KA

( ) [ ] K A =

A H2O PA

mol L-1 atm -1

(8)

which is a function of temperature

d lnK A = H Aab

dT

RT 2

(9)

where

H = H% Aab

Asolution

- H% Agas

is the heat of absorption.

Typically

H Aab < 0

so that as temperature increases the equilibrium constant decreases. Actually, this equilibrium constant

is called Henry's constant in environmental chemistry and is independent of composition for dilute

solutions.

[ A H2O] = H APA

H A M atm-1

(10)

Note that the units of H A are now (mole/L atm) and M means mole/liter.

Large HA implies a very soluble gas. Low HA is a slightly soluble gas. (Attention: some books define the reciprocal of HA as Henry's constant). The solubility of various gases is indicated in Table 3.1.

A dimensionless Henry's constant, H^ A , is obtained if gas concentration is used in its definition

instead of partial pressure, i.e [A]g = P A /R T

H^ A

=

HART

=

[ A H2O] [ A]

(11)

g

Note that depending on the composition measures used to define equilibrium, Henry's constants appear

( ) ( ) with different units e.g. H A atm-1 , H A M atm-1 , H^ (-). It is unfortunate that the chemical

engineering and some environmental engineering literature use the reciprocal of the above defined Henry's constant under the same name! We have adopted here the definitions prevalent in the

2

environmental chemistry literature although the chemical engineering approach is the older and the better established one. A website posted as part of the course gives the various definitions and conversion factors ().

The following relationship holds between the above constants

HA

=

KA C L TOT

=

HA C L TOT

= H^ A CL TOT RT

(12)

Where CL TOT is the total liquid molar concentration. TABLE 3.1: Henry's Law coefficients of atmospheric gases dissolving in liquid watera

3.3.1 Water Ionization

The ionization or recombination reaction for water is for all practical purposes infinitely fast

H 2O H + + OH -

(13)

so that equilibrium is always established

K

' w

=

H + OH

- / [H2O]

= 1.82

x 10-16

M

at

298 K

(14)

Since molar concentration of water is constant, we get Kw = H + OH - = 1.0 x 10-14 M 2 at 298 K (15)

3

For pure water [H + ] = [O H - ] = 1.0 -7 M. Recall that

p H = - log [H + ]

(16)

so that p H = 7 for pure water at 298K.

3.3.2 Carbon Dioxide/Water Equilibria

Carbon dioxide is prevalent in the atmosphere and undergoes the following equilibrations in

water (with equilibrium constant for each step indicated in parentheses)

CO2 (g) + H 2O CO2 H 2O

(KCh)

(17a)

CO2 H 2O H + + HCO3-

(KC1)

(17b)

HCO3 H + + CO32- (KC2)

(KC2)

(17c)

We can express now the concentration of dissolved carbon containing species (ions) in terms of these constants

[ ] CO2 H2O = Kch pCO2 = H p CO2 CO2

[ ] HCO

- 3

=

KC1

CO2 H2O H +

= H K p CO2 C1 CO2 H +

CO

2 3

-

=

KC2 HCO H +

- 3

=

H CO2

KC1KC 2 pCO2 H + 2

(18a) (18b) (18c)

We also must always satisfy the electroneutrality relation

H

+

=

HCO

- 3

+

2

CO

2- 3

+

OH

-

(19)

Upon substitution of the above expressions into the electroneutrality equation we get:

H + =

H K p CO2 C1 CO2 H +

+ 2 HCO2 KC1KC 2 pCO2 H + 2

+

Kw H +

(19a)

Upon rearrangement a cubic equation for [H + ] results

H + 3 - Kw + HCO2 KC1 pCO2 H + - 2 HCO2 KC1KC2 pCO2 = 0

(20)

Given the temperature, T, and partial pressure of CO2, pCO2 , the hydrogen ion concentration [H+ ] can be calculated from the above equation and pH obtained. (For temperature dependence see Table 3.2).

At pCO2 = 330 ppm (= 3.3 x 10-4 atm) at 283oK the solution pH = 5.6 ([H +] = 2.51 x 10-6 (mol/L)). This is often called the pH of 'pure" rain water.

4

TABLE 3.2: Thermodynamic data for calculating temperature dependence of aqueous equilibrium constants

3.3.3

Sulfur Dioxide/Water Equilibrium

The scenario is similar as in the case of CO2 absorption

SO2

(

g

)

+

H 2O

SO2

H2O

H SO2

SO2

H

2O

H

+

+

HSO

- 3

KS1

HSO

- 3

H+

+

SO

2- 3

KS2

(21a) (21b) (21c)

The concentrations of dissolved species are

[ ] SO2 H2O = HSO2 pSO2

HSO

- 3

=

H K p SO2 s1 SO2 H +

SO

2 3

-

=

H SO2

KS1KS 2 H + 2

pSO2

(22a) (22b) (22c)

Satisfying the electroneutrality relation requires

( ) H + 3 - Kw + H SO2 KS1 pSO2 H + - 2 H SO2 KS1KS 2 pSO2 = 0

(23)

The total dissolved sulfur in oxidation state + 4 is S(IV) and its concentration is given by

( ) S

IV

= H SO2 pSO2

1 +

KS1 H +

+

KS1KS 2 H + 2

=

H

* S ( IV )

pSO2

(24)

where

H

* S ( IV )

is the modified Henry's constant which we can express as

H

* S ( IV )

=

H SO2

1 + 10 pH

KS1

+ 10 2 pH KS1KS 2

(25)

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download