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TOPIC 3

- IDEAL SOLUTIONS, FUGACITY, ACTIVITY, AND STANDARD

STATES

I. PARTIAL AND APPARENT MOLAR PROPERTIES

MOLAR VS. PARTIAL MOLAR QUANTITIES

- Molar values of state functions are defined as

follows - etc. These are useful only in the case of

single-component systems and dependent only on

pressure and temperature, not composition. - Partial molar quantities are defined according

to - These are dependent on T, P, and composition.

PHYSICAL INTERPRETATION OF PARTIAL MOLAR VOLUMES

- The partial molar volume of component i in a

system is equal to the infinitesimal increase or

decrease in the volume, divided by the

infinitesimal number of moles of the substance

which is added, while maintaining T, P and

quantities of all other components constant. - Another way to visualize this is to think of the

change in volume on adding a small amount of

component i to an infinite total volume of the

system. - Note partial molar quantities can be positive or

negative!

SUMMING PARTIAL MOLAR QUANTITIES

- The total value for a state function of a system

is obtained by summing the partial molar volumes

of its components according to - We can manipulate partial molar quantities in a

manner identical to the way we manipulate total

quantities. - As with total state functions, we cannot know

absolute values, only differences (except for V

and S)!

- We can also express the summations in terms of

molar state functions and mole fractions - In the case of the volume of a two-component

system, e.g., NaCl-H2O, we can write

Schematic plot of the molar volume of aqueous

NaCl solutions as a function of mole fraction of

NaCl.

HOW TO DETERMINE PARTIAL MOLAR VOLUME

- Refer to the previous diagram. Triangles A and B

are similar, so it is true that

but

so

comparison with

shows that

So the partial molar volumes can be determined

from the intercepts of a line tangent to the plot

of volume vs. mole fraction.

PARTIAL MOLAR FREE ENERGY -THE CHEMICAL POTENTIAL

Chemical potential

- The previous relationships also apply

It can also be shown that

Schematic plot of chemical potential vs. mole

fraction for a binary system

COMPOSITIONAL CHANGES

- The Master equations that we developed previously

for one-component systems can now be written as

AN ADDITIONAL REQUIREMENT FOR EQUILIBRIUM

- Consider a system with components i, j, k, l,

distributed among phases ?, ?, ?, ?, - At equilibrium it must be true that
- ?i? ?i? ?i? ?i?
- ?j? ?j? ?j? ?j? ...
- ?k? ?k? ?k? ?k? ...
- ?l? ?l? ?l? ?l? ...
- etc.

- Chemical potentials represent the slope of the

Gibbs free energy surface in compositional space.

Thus, a component will move from a phase in which

it has a high chemical potential, to one in which

it has a low chemical potential, until its

chemical potential in all phases is the same. - Specific example Consider a silicate melt in

equilibrium with forsterite (Mg2SiO4), and

enstatite (Mg2Si2O6). At equilibrium the

following must be true - ? Mgmelt ?MgFo ?MgEn
- ?Simelt ?SiFo ?SiEn
- ?Omelt ?OFo ?OEn

GIBBS-DUHEM EQUATION

- For a homogenous phase of two components, A and

B, the Master Equation becomes - If we now specify equilibrium at constant T and P
- Now, we have shown above that
- Differentiating this we obtain

GIBBS-DUHEM EQUATION - CONTINUED

- At equilibrium
- Substituting the previous expression
- we obtain
- In the general case we get the Gibbs-Duhem

equation

GIBBS-DUHEM EQUATION - CONTINUED

- Starting with the expression
- If we divide through both sides by dXA we get
- And now dividing by nA nB we get

APPARENT MOLAR QUANTITIES

- Although in principle, partial molar quantities

can be measured from intercepts of lines tangent

to a plot of state functions vs. mole fraction as

outlined previously, they are not determined this

way in practice. - In practice, apparent molar quantities are

determined. For a state function like volume, the

apparent molar volume, ?V, is given by - ?V (V - n1V1)/n2
- where n1, and n2 are the number of moles of

solvent and solute, respectively, and V1 is the

molar volume of pure solvent.

Total volume of a solution as a function of

solute concentration. Illustrates the difference

between partial and apparent molar volume.

- The apparent molar volume is the volume that

would be attributed to one mole of solute in

solution if it is assumed that the solvent

contributes the same volume it has in the pure

state. - Starting with the definition of apparent molar

volume - ?V (V - n1V1)/n2
- we can rearrange to get
- V n1V1 n2 ?V
- and dividing by (n1 n2),
- V X1V1 X2 ?V
- Thus, the volume of solution can be calculated

knowing ?V instead of the partial molar volume.

Comparison of apparent molar and partial molar

volumes

PARTIAL MOLAR VOLUMES FROM APPARENT MOLAR VOLUMES

or

- If ?V measurements are fit by an equation of the

type ?V a bm cm2 - then we have V2 m(b 2cm) a bm cm2 or
- V2 a 2bm 3cm2

II. IDEAL SOLUTIONS

THERMODYNAMICS OF IDEAL SOLUTIONS

- An ideal solution is one that satisfies the

following equation ?i - ?i RT ln Xi - where ?i is the chemical potential of some

component i in a solution and ?i is the chemical

potential of that component in the pure form. - Recall that
- substituting we get

- These equations tell us that the free energy of

an ideal solution is the sum of two terms the

free energy of a mechanical mixture, and a free

energy of ideal mixing.

ENTHALPY AND VOLUME OF AN IDEAL SOLUTION

If i is a pure substance.

- There is no volume or enthalpy change upon ideal

mixing. In other words

However, there is a change in entropy upon ideal

mixing. Because the solution is more disordered,

entropy increases!

ENTROPY OF MIXING

- but we have ?Gideal mix ?Hideal mix -T?Sideal

mix - and ?Hideal mix 0
- so

Thus, the only contribution to ?Gideal mix is an

entropy contribution!

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III. FUGACITY AND ACTIVITY

FUGACITY

- Starting with dG VdP - SdT
- at constant T this becomes dG VdP
- For an ideal gas dG (RT/P)dP RT dln P
- This is true for ideal gases only, but it would

be nice to have a similar form for real fluids. - dG RT dln ? where ? is the fugacity
- ? ?/P ? 1 as P ? 0
- ? is the fugacity coefficient
- ? ?P
- Fugacity may be thought of as a thermodynamic

pressure it has units of pressure.

MEASUREMENT OF FUGACITY

Alternatively, we can begin again with

But we now define the compressibility factor Z

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- The above equation is the basis of the

experimental determination of fugacities from

P-V-T data. - We can substitute into the integral (Z-1)/P

calculated from any equation of state, or we can

integrate graphically.

CALCULATION OF EQUILIBRIUM BOUNDARIES INVOLVING

FLUIDS

- A number of mineral reactions involve only solid

minerals and either H2O or CO2, e.g. - KAl3Si3O10(OH)2 ? KAlSi3O8 Al2O3 H2O
- or
- CaCO3 SiO2 ? CaSiO3 CO2
- or in general
- A(s) ? B(s) C(fluid)
- ?rV VB VC - VA (VB - VA) VC
- ?rV ?sV Vfluid

- The pressure integral for the solids is then

evaluated using the constant ?sV approximation

and that for the fluid is evaluated using

fugacities.

For the muscovite breakdown reaction above, we

can start with the equation

- ?rV VKspar Vcor VH2O - Vmusc
- ?rV (VKspar Vcor - Vmusc) VH2O
- ?rV ?sV VH2O

A function of pressure and temperature!

FUGACITIES IN GASEOUS SOLUTIONS

- Starting with the following, in terms of partial

molar volumes

We obtain the expression for the fugacity

coefficient of a component in a solution

ALL CONSTITUENTS HAVE A FUGACITY

- The expression dG RT dln ?
- may be integrated between two states 1 and 2 to

give - G2 - G1 RT ln (?2/?1)
- This equation applies to a pure one-component

system. For a solution we must use chemical

potentials and we write - ?i - ?i RT ln (?i/?i)
- This equation makes no stipulation as to the

state of component i, and can therefore refer to

solid, liquid or gas.

- Solids and liquids therefore are also associated

with a fugacity. In some cases, this fugacity can

be thought of as a vapor pressure. Fugacity can

also be thought of as an escaping tendency. - However, in some cases, a vapor phase may not

exist, but a fugacity always exists. One must

realize that the fugacity is a thermodynamic

model parameter, not always an approximation to a

real pressure. - Fugacities of solid phases or individual

components of solid solutions are not generally

known. - Fugacities are absolute physical properties.

ACTIVITIES

- The absolute values of the fugacities of solids

and liquids cannot always be determined, but

their ratios can be. - Consider ?i - ?i RT ln (?i/?i)
- If we let one of these states be a reference

state, this can be rewritten - ?i - ?i RT ln (?i/?i)
- We now define the activity of constituent i to be
- ai ?i/?i
- Thus
- ?i - ?i RT ln ai

DALTONS LAW

- Dalton (1811) discovered that, at low total

pressures, a mixture of gases exerts a pressure

equal to the sum of the pressures that each

constituent gas would exert if each alone

occupied the same volume. - Strictly true only for ideal gases, but is

approximately true at low total pressure where

real gases approach ideality. For each gas we

have - P1V n1RT
- P2V n2RT
- etc.
- For the gas mixture we have

- If we divide the expression for each constituent

by the expression for the mixture we obtain - etc.
- or
- P1 X1Ptotal
- P2 X2Ptotal
- etc.
- Ptotal P1 P2 P3
- P1, P2, etc. are called the partial pressures.

HENRYS LAW

- Henry (1803) was studying the solubility of gases

in liquids. He found that the amount of gas

dissolved in a liquid in contact with it was

directly proportional to the pressure on the gas,

i.e., - Pi Kh,iXi
- Kh,i is a constant called the Henrys Law

constant. - In practice, this law holds only at relatively

low values of Pi.

RAOULTS LAW

- Raoult (1887) studied vapor-liquid systems in

which two or more liquid components were mixed in

known proportions and the liquid was equilibrated

with its own vapor. The composition of the vapor

was then determined. The total vapor pressure of

the system was low, so the vapor behaved ideally

and conformed to Daltons law. In such systems,

the partial pressures of the gaseous components

were found to be a linear function of the their

mole fraction in the liquid.

- Thus, for a binary system A-B, he obtained
- PA XAPAº and PB XBPBº
- where PAº and PBº are the vapor pressures of pure

components A and B, respectively.

- The only way that such a simple relationship as

Raoults law can hold is if the intermolecular

forces between A-A, B-B, and A-B are identical.

Solutions in which this is the case are called

ideal solutions. - The most general way of expressing Raoults Law

is - Pi XiPiº
- Very few systems follow Raoults Law over the

entire range of composition from Xi 0 to Xi

1. However, Raoults Law often applies to the

solvent in dilute solutions, whereas the solute

in dilute solutions follows Henrys Law.

Partial pressure in the mixture

acetone-chloroform at 35.2C. This mixture

exhibits negative deviations from Raoults Law

Partial pressure in the mixture carbon

disulfide-acetone at 35.2C. This mixture

exhibits positive deviations from Raoults Law

THE GIBBS-DUHEM EQUATION REVISITED

- Previously we derived the Gibbs-Duhem equation

for a binary solution - This can be rearranged to give

- This equation shows that the slopes of tangents

to curves of chemical potential vs. mole fraction

for binary solutions are not independent of one

another. - For example, if XB 0, and

has a finite - value, then .
- If XA 0.5, then

etc.

Chemical potentials in solutions of carbon

disulfide and acetone.

Gibbs-Duhem Equation

THE DUHEM-MARGULES EQUATION

- Starting with the Gibbs-Duhem equation
- If we recall that d?i RT dln ?i we can make

the substitution and obtain

When the vapors are nearly perfect gases, we may

substitute partial pressures for fugacities to

obtain the approximate relation

Realizing that dXB -dXA, and that d ln P d

P/P we can rewrite this as

Application of the Duhem-Margules equation.

Partial pressure is plotted on the Y-axis.

Duhem-Margules Equation

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THE LEWIS FUGACITY RULE

- This is a variation on Raoults Law
- fimixture Xifipure
- This states that the fugacity of a constituent in

a mixture is equal to its mole fraction times

its fugacity in the pure state. - Many substances that do not obey Raoults Law do

in fact obey the Lewis Fugacity Rule.

IDEAL MIXING AND ACTIVITY

- If we compare the definition of the activity
- ai ?i/?i
- and a rearrangement of the Lewis Fugacity Rule
- Xi fimixture/fipure
- we see that for solutions that obey the Lewis

Fugacity Rule - ai Xi
- We can also now write
- ?i - ?i RT ln Xi
- which is considered another form of Raoults Law.

Activity relations for an ideal binary system.

It turns out that these relations hold not only

for liquid and gaseous solutions, but also for

solid solutions.

NON-IDEAL MIXING

- As already discussed, most real solutions do not

conform to Raoults Law over the entire

compositional range. - However, whether solid, liquid or gas, in many

solutions, the component in excess (solvent)

follows Raoults Law and the minor component

(solute) follows Henrys Law over a limited range

at low mole fractions.

Positive deviation from Raoults Law

ANOTHER NIFTY APPLICATION OF THE GIBBS-DUHEM

EQUATION

- Task Prove that, if the solute in a binary

solution obeys Henrys Law, then the solvent

obeys Raoults Law. - Starting with the Gibbs-Duhem equation for a

binary system - and dividing through both sides by nA nB we get
- At low pressures we have d?i RT dln Pi

- So we can now write
- Now if component A is the solute and obeys

Henrys Law we have PA Kh,AXA - Taking the natural logarithm of both sides we

have - ln PA ln Kh,A ln XA
- Now differentiating we get
- d ln PA d ln XA
- so now

- Now -dXB dXA so
- Now we integrate from XB 1 to XB XB
- where PB is the partial pressure of B when XB

1, i.e., the partial pressure of pure B.

Raoults Law!

ACTIVITY COEFFICIENTS

- The ideal solution is useful as a model with

which real solutions are compared. - This comparison is effected by taking the ratio

of the activity of the real solution relative to

that of the ideal solution. This ratio is called

the activity coefficient. - Deviations from Raoults Law are expressed by the

Raoultian activity coefficient ?R - ai ?R,iXi

- Deviations from Henrys Law are expressed by the

Henryian activity coefficient ?H - ai ?H,iXi
- Activity coefficients, because they are ratios of

activities, are unitless. - A major difference between the two types of

activity coefficients is that - ?R ? 1 as X ? 1, but
- ?H ? 1 as X ? 0
- Thus, ?H is usually more useful for solutes in

dilute solutions.

IV. STANDARD STATES

STANDARD STATES

- Because the activity is the ratio of two

fugacities, i.e., - ai ?i/?i
- the value of the activity depends on the

reference state chosen for ?i. This state we

usually refer to as the standard state. - The choice of the standard state is completely

arbitrary. - The standard state need not be a real state. It

is only necessary that we be able to calculate or

measure the ratio of the fugacity of the

constituent in the real state to that in the

standard state.

A STANDARD STATE HAS FOUR ATTRIBUTES

- Temperature
- Pressure
- Composition
- A particular, well-defined state (e.g., ideal

gas, ideal solution, solid, liquid, etc.) - If desirable, we can permit T or P to be

variable, i.e., on a sliding scale.

STANDARD STATES FOR GASES

- Single Ideal Gas
- Starting with the relationship ?2 - ?1 RT ln

(P2/P1) - we can assign our standard state to be the ideal

gas at 1 bar and any temperature. In this case we

can write ? - ? RT ln P - and ? - ? is the difference in chemical

potential between an ideal gas at T and P, and an

ideal gas at T and 1 bar.

- Ideal mixture of ideal gases
- For such a mixture we can write
- ?1 - ?1 RT ln (X1P)/(X1P)
- If we choose our standard state to be the pure

ideal gas 1 at any temperature and 1 bar, then

X1 1 and P 1, so ?1 - ?1 RT ln (X1P)

RT ln P1 - Non-ideal gases
- For non-ideal gases we would write
- ?1 - ?1 RT ln (f1/f1)
- but recall that lim (fi/Pi)Pi?0 1. So if we

chose our standard state to be the pure, ideal

gas at any temperature and P 1 bar, we get

- ?1 - ?1 RT ln f1
- This equation is frequently written, but rarely

understood. It only has meaning if the standard

state is specified to be the pure ideal gas at

any temperature and 1 bar. - This is the most commonly chosen standard state

for gases and supercritical fluids. However,

there is no reason why this particular standard

state has to be chosen. We could just as easily

choose 1) the pure ideal gas at any T and 10

bars 2) a pure real gas at 25C and 1 bar 3) a

specific mixture of gases at any T and ? bars or

4) any other well-defined standard state.

LIQUIDS AND SOLIDS

- The following equation applies to liquids and

solids as well - ?i - ?i RT ln (fi/fi)
- Fixed pressure standard state
- The standard state is chosen to be the pure phase

at the temperature of interest and 1 bar. Then

fi 1, so ai fi. In this case, it is

necessary to know fi at each and every set of P-T

conditions of interest. - Variable pressure standard state
- The standard state is the pure phase at the

pressure and temperature of interest.

- Under these conditions, fi fi, so ai 1. The

only way the activity of a solid deviates from

unity under this standard state is when the solid

is not pure, but is a solid solution. - It may seem that the second standard state is

easier to deal with in terms of pressure

corrections. However, with the first standard

state, the pressure correction is applied to fi,

whereas in the second standard state, the

correction is applied to ?i. In either case,

volume data for the constituent are required to

make the correction.

AQUEOUS SOLUTIONS

- The activities of solutes in dilute solutions are

more closely approximated with Henrys Law than

Raoults Law. Thus, a somewhat different standard

state is applied. We start with the equation

expressing the difference in chemical potentials

between two solutions with different molalities - ?i - ?i RT ln (?Hm/?Hm)
- If we let one solution be the standard state, we

can write - ?i - ?i RT ln (?Hm)/(?Hm)

- We then define the standard state to be the

hypothetically ideal one-molal solution at the

temperature and pressure of interest. Under these

conditions ?H 1 and m 1, so (?Hm) 1, and

we write - ?i - ?i RT ln ?Hm
- This somewhat strange standard state is

necessary, because if we let the standard state

be the infinitely dilute solution, we would have

?H 1 and m 0, so (?Hm) 0, which would

result in an undefined value of - ?i - ?i RT ln (?Hm)/ (?Hm)

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