DG 5 2 278 J Would this be considered a spontaneous process? Because the pressure is not kept
5.4 Solutions and Condensed Phases
Up to this point the equilibrium constants have been expressed in terms of partial pressures. However, for real gases the fugacities of the species should be used. If the pressures are low enough, the pressures themselves can be used, because at low pres- sures the pressure is approximately equal to the fugacity. But many chemical reactions involve phases other than the gas phase. Solids, liquids, and dissolved solutes also par- ticipate in chemical reactions. How are they represented in equilibrium constants?
We answer this by defining activity ai of a material in terms of its standard chemi- cal potential mi° and its chemical potential mi under nonstandard pressures:
mi5 m°i1RT ln ai (5.11)
Comparison of this equation with equation 4.59 shows that for a real gas, activity is defined in terms of the fugacity as
agas5 fgas
p° (5.12)
Reaction quotients (and equilibrium constants) are more formally written in terms of activities, rather than pressures:
Q5
i productsq ai0ni0
j reactantsq aj0nj0 (5.13)
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5.4 | Solutions and Condensed Phases 143
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This expression applies no matter what the state of the individual reactant or product.
For condensed phases (that is, solids and liquids) and dissolved solutes, there are different expressions for activity, although the definition from equation 5.11 is the same for all materials. For condensed phases, the chemical potential of a particular phase at a specified temperature and standard pressure is represented by mi°. In the last chapter, we found that
a'mi 'pb
T
5Vi
where Vi is the molar volume of the ith material. We rearrange this into dmi5Vi dp
The differential of equation 5.11 at constant temperature is dmi5RT (d ln ai) Combining the last two equations and solving for d ln ai:
d ln ai5Vi dp RT
Integrating both sides from the standard state of ai5l and p5l atm (or bar):
3
a
l
d ln ai5 3
p
l
Vi dp RT
ln ai5 l RT3
p
l
Vi dp
If the molar volume Vi is constant over the pressure interval (and it usually is to a good approximation unless the pressure changes are severe), this integrates to
ln ai5 Vi
RT(p2l) (5.14)
ExamplE 5.7
Determine the activity of liquid water at 25.0°C and 100 bar pressure. The molar volume of H2O at this temperature is 18.07 cm3.
Solution
Using equation 5.14, we set up the following:
ln aH2O5
a18.07 cm3
molb a 1 L 1000 cm3b a0.08314 L#bar
mol#Kb(298 K)
(100 bar 2 1 bar)
Solving:
ln aH2O5 0.0722 aH2O 5 1.07
100 bar is about 98.7 atm.
Note a conversion factor between L and cm3, as well as the appropriate value and units on R.
Notice that the activity of the liquid is close to 1, even at a pressure that is 100 times that of standard pressure. This is generally true for condensed phases at pressures that are typically found in chemical environments. Therefore, in
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144 Chapter 5 | Introduction to Chemical Equilibrium
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most cases, the activities of condensed phases can be approximated as 1 and they make no numerical contribution to the value of the reaction quotient or equilib- rium constant. Note that this is not the case in conditions of extreme pressures or temperatures.
For chemical species that are dissolved in solution (usually water), activities are defined in terms of the mole fraction:
ai5 gixi (5.15)
where gi is the activity coefficient. For solutes, the activity coefficient approaches 1 as the mole fraction approaches zero:
limxiS0gi5l limx
iS0(ai)5xi
Mole fractions can be related to other defined concentration units. The strictest mathematical relationship is between mole fraction and molality, mi, and is
mi5 1000xi (l 2xi)#Msolv
where Msolv is the molar mass of the solvent in grams per mole, and the 1000 factor in the numerator is for a conversion between grams and kilograms. For dilute solu- tions, the mole fraction of the solute is small compared to 1, so the xi in the denomi- nator can be neglected. Solving for xi, we get
xi5mi# Msolv 1000
Thus, the activity for solutes in dilute solution can be written as ai5 gi#mi#Msolv
1000 Using equation 5.11, we substitute for the activity to get
mi5 m°i 1RT ln agi#mi#Msolv 1000b
Because Msolv and 1000 are constants, the logarithm term can be separated into two terms, one incorporating these constants and the other incorporating the activity coefficient and the molality:
mi5 m°i 1RT ln aMsolv
1000b 1RT ln (gi#mi)
The first two terms on the right side of the equation can be combined to make a
“new” standard chemical potential, which we will designate mi*. The above equation becomes
mi5 mi*1RT ln (gi#mi)
Comparing this to equation 5.11 gives us a useful redefinition of the activity of dis- solved solutes:
ai5 gi#mi (5.16)
Equation 5.16 implies that concentrations can be used to express the effect of dis- solved solutes in reaction quotient and equilibrium constant expressions. In order that ai be unitless, we divide the expression by the standard molal concentration of 1 mol/kg, symbolized by m°:
ai5 gi#mi
m° (5.17)
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