IDEAL SOLUTIONS, FUGACITY, ACTIVITY, AND STANDARD STATES

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TOPIC 3

• IDEAL SOLUTIONS, FUGACITY, ACTIVITY, AND STANDARD
  STATES

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I. PARTIAL AND APPARENT MOLAR PROPERTIES
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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.

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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!

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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)!

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• 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

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Schematic plot of the molar volume of aqueous
NaCl solutions as a function of mole fraction of
NaCl.
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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.
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PARTIAL MOLAR FREE ENERGY -THE CHEMICAL POTENTIAL
Chemical potential

• The previous relationships also apply

It can also be shown that
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Schematic plot of chemical potential vs. mole
fraction for a binary system
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COMPOSITIONAL CHANGES

• The Master equations that we developed previously
  for one-component systems can now be written as

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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.

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• 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

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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

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GIBBS-DUHEM EQUATION - CONTINUED

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

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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

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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.

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Total volume of a solution as a function of
solute concentration. Illustrates the difference
between partial and apparent molar volume.
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• 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.

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Comparison of apparent molar and partial molar
volumes
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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

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II. IDEAL SOLUTIONS
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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

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• 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.

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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!
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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
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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.

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MEASUREMENT OF FUGACITY
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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.

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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

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

A function of pressure and temperature!
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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
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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.

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• 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.

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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

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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

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• 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.

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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.

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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.

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• 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.

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• 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.

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Partial pressure in the mixture
acetone-chloroform at 35.2C. This mixture
exhibits negative deviations from Raoults Law
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Partial pressure in the mixture carbon
disulfide-acetone at 35.2C. This mixture
exhibits positive deviations from Raoults Law
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THE GIBBS-DUHEM EQUATION REVISITED

• Previously we derived the Gibbs-Duhem equation
  for a binary solution
• This can be rearranged to give

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• 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.

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Chemical potentials in solutions of carbon
disulfide and acetone.
Gibbs-Duhem Equation
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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

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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
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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.

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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.

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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.
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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.

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Positive deviation from Raoults Law
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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

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• 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

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• 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!
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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

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• 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.

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IV. STANDARD STATES
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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.

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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.

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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.

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• 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

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• ?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.

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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.

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• 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.

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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)

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• 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)