Osmotic Pressure
Figure3.9shows a beaker containing the pure liquid A. Immersed in the liquid is a tube with its lower end closed to the liquid by a membrane. The membrane allows Ato permeate into the tube and vice versa. Inside the tube there is a binary mixture of two componentsAandB. The latter however is held back inside the tube by the membrane. What happens? As far as Ais concerned the two subsystems, the pure solvent outside the tube and the binary mixture inside the tube, do exchangeA-moles and therefore chemical equilibrium requires
μ∗A(T,P)=μA(T,P+,xA). (3.55) The left side is the chemical potential of pureAoutside the tube. The right side is the chemical potential ofAinside the tube. Because we do not consider the gas phase we may omit the index(l). The temperature is the same in both subsystems. The outside pressure isP, whereas the inside pressure is different i.e.P+. Why?
Initially the tube may contain B only. Chemical equilibrium therefore requires flow of A across the membrane into the tube. For simplicity we assume that the
Fig. 3.9 Sketch of a simple osmotic pressure experiment
h
μ μ
* A(P) membrane
A(P+ )
initial surface level inside and outside the tube is the same and the density ofAand Bis the same as well. The pressure difference across the membrane,, can then be determined by measuringh(at equilibrium) and computing the force of gravitation exerted by the mass of material above the surface level of the surroundingAsolvent.
The reason for the sustained pressure difference is the membrane, which does not allow the chemical equilibration ofBon both sides.
Making use of the Gibbs-Duhem equation (2.168) we have μ∗A(T,P+)≈μ∗A(T,P)+ 1
n∗AV∗(P)(P+−P). (3.56) Again it is assumed that incompressibility of the liquid is a good approximation, cf. (3.34). Combination of Eqs. (3.55) and (3.56) then yields
μ∗A(T,P+)− 1
n∗AV∗(P)≈μA(T,P+,xA), (3.57) and according to Eq. (3.45), ifxAxB, we obtain
μ∗A(T,P+)− 1
n∗AV∗(P)≈μ∗A(T,P+)+RTlnxA. (3.58) At first glance this may seem strange, because in Eq. (3.45) the pressure arguments of the chemical potentials are PA andPA∗, which are presumably different. In gen- eral the chemical potential of a particular species does depend on temperature, total pressure, and composition. In Eq. (3.45) the total pressure is the same on both sides of the equation and does not show up explicitly in the lists of arguments. The com- position dependence is expressed in terms of different partial pressures via (origi- nally)Dalton’s law. Here the total pressure is included in the argument of the chemical potential and the composition is expressed in mole fractions. Having explained this we can write down the final result forgiven by
≈ RT
VA,mol
xB, (3.59)
where we have used the molar volume of Ain the liquid state at pressure P and temperatureT, i.e.VA,mol =V∗(P)/n∗A, and lnxA =ln(1−xB)≈ −xB.
One final transformation of this equation is useful. Inside the tube we have xB= nB
nA+nB ≈nB
nA. (3.60)
In additionVsoluti on ≈VA,molnAand thus ≈ nBRT
Vsoluti on. (3.61)
This is the so called van’t Hoff equation (Jacobus van’t Hoff, first Nobel prize in chemistry for his work on chemical dynamics and osmotic pressure, 1901). Note that the osmotic pressure only depends on the molar concentration of component B and temperature (under the approximations we have made in the course of the derivation). Here osmotic pressure is another example for a colligative property.
Remark: Reverse Osmosis. According to our derivation leading to Eq. (3.61), it should be possible to apply extra pressure to the tube in Fig.3.9and by doing so reduce its solute content. A technical example is desalination of sea water, which is forced through a membrane using a pressure exceeding the osmotic pressure. This process is called reverse osmosis.
Example: Osmotic Pressure in Hemoglobin Solutions. As an application we consider the following problem. Use the osmotic pressure data (pobs.) from table X in Adair (1928) to estimate the molar mass of (sheep) hemoglobin.
Gilbert S. Adair was a pioneer of macromolecular biochemistry and succeeded in determining the correct molecular weight of hemoglobin from osmotic pres- sure measurements. He also supplied the horse hemoglobin crystals which al- lowed Max Perutz (Nobel prize in chemistry for his work on the structure of globular proteins, 1962) to obtain the first hemoglobin x-ray structures.
We may rewrite van’t Hoff’s equation as
c ≈ RT mH b.
Herec = (nH b/V)mH b andmH b is the molar mass of component B (Hb:
hemoglobin). Figure3.10shows the data from the above reference plotted in the original units. The solid line is a fit on the basis of a theory explained later in this book. Notice first that van’t Hoff’s equation describes the data only at
very low hemoglobin concentration. This is expected, because we have used the approximationxA xB. The deviation from van’t Hoff’s equation arise due to non-ideality, which basically means that there are complex solute-solute interactions—something we have no information about at this point. However, we can still determinemH b, i.e.
mH b ≈ RT /c,
in the limitc→0. From the figure we extract the value/c≈0.3 cmHg/(g/dl).
In additionT =0◦C=273.15 K. After converting the units, 1cmHg=1333.224Pa
1g/dl =10 kg/m3,
we obtainmH b≈57 kg/mol. This is roughly 10 % below the exact value—but not bad at all.
The example also shows that van’t Hoff’s equation is valid at small concentrations only. We continue our discussion of osmotic pressure on page 111 and a second time on p. 170 dealing with extensions of different origin. In particular we shall discuss the so called scaled particle theory behind the solid line through the data in Fig.3.10 beginning on p. 179. This theory allows to estimate the size of Hb (≈5.5 nm in diameter) based on its osmotic pressure data.
0 5 10 15 20 25 30 35
c [g/dl]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ccm Hggdl)/[(/-1 ]
Fig. 3.10 Concentration dependent osmotic pressure in hemoglobin solutions
Equilibrium Adsorption
Consider a gas in contact with a solid surface. Molecules from the gas may adsorb onto and subsequently desorb from the surface. Eventually an equilibrium develops characterized by a constant coverage depending on temperature and the pressure in the bulk gas. Coverage here refers to the net amount of gas adsorbed. We obtain the net amount adsorbed by counting the gas molecules in a column-shaped volume perpendicular to the surface. This column continues out into the bulk gas, where the surface is no longer felt by the gas molecules. Subsequently we subtract the (average) number of gas molecules present in an identical column when the surface is removed (The number is equal to the volume of the column multiplied by the bulk density of the gas). Just how long the column has to be, in order for it to extend into the bulk gas, depends on the interaction forces between the gas molecules and the surface as well as on thermodynamic conditions. In some cases the “interfacial thickness” to good approximation is just one molecular layer. One speaks of monolayer or even sub-monolayer coverage. In other cases the interface is “thicker” and more “diffuse”.
The examples in Fig.3.11show computer simulation generated gas density pro- files above an adsorbing surface atz=0.5The units used here are so-called Lennard- Jones units6—but this is of no particular interest to us at this point. What is shown is the gas number density,ρ(z), as function of distance,z, from the surface. In the top panel we recognize a first peak atz ≈1 (cut off at 1) and a second smaller one at z≈2. Beyond the second peak the density levels off (with fluctuations) indicating the bulk phase. The eventual drop atz=12 is merely due to the finite extend of the simulation box to which the gas is confined. The “gap” between the surface and the first peak is due to the finite extend of the atoms in the surface and the molecules in the gas—ρis a center of mass number density. This figure shows that at the given conditions there exists a dense layer of adsorbed molecules adjacent to the solid surface. A much less dense second layer is followed by a rather rapid transition to bulk behavior. Altered conditions do change the picture. In this case the temperature is reduced. In fact we approach the saturation line of methane at constant pressure (the transition temperature at this pressure in LJ-units is just about 1). We notice that the adsorbed layer thickness increases as more peaks emerge. However, at this point these graphs merely serve as illustration to bear in mind when we talk about adsorption on solid surfaces.7
An important quantity characterizing the interaction of the molecules with the surface is the isosteric heat of adsorption,qst, defined via
5The system is methane gas adsorbing on the graphite basal plane located atz=0. A computer program generating profiles like these is included in the appendix. The theoretical background needed to understand the program is discussed in Chap.6.
6In these units the gas pressure in Fig.3.11isP=0.04. The temperatures from top to bottom are T=2.0,1.2,1.05.
7We return to Fig.3.11in an example starting on p. 206.
0 2 4 6 8 10 12 z [LJ]
z [LJ]
z [LJ]
0.2 0.4 0.6 0.8 1.0
0 2 4 6 8 10 12
0.2 0.4 0.6 0.8 1.0
0 2 4 6 8 10 12
0.2 0.4 0.6 0.8 1.0
ρ[LJ]
ρ[LJ]
ρ[LJ]
Fig. 3.11 Computer simulation generated gas density profiles above an adsorbing surface at dif- ferent temperatures
qst =T∂μs
∂T
Vs,Ns −T∂μb
∂T
Pb. (3.62)
The indicessandbrefer to the surface and the bulk, respectively. Note that. . .|Vs,Ns
means “at constant coverage”, whereas. . .|Pb means “at constant (bulk) pressure”.
The temperature is the same in both cases.
Using the equilibrium conditionμs =μbwe may write dμs
Vs,Ns = ∂μs
∂T
Vs,Nsd T =∂μb
∂T
Pbd T +∂μb
∂Pb
Td Pb, (3.63) which yields
∂μs
∂T
Vs,Ns
= ∂μb
∂T
Pb
+ ∂μb
∂Pb
T Vb/Nb=ρ−b1
∂Pb
∂T
Vs,Ns
. (3.64)
Combination of this equation with Eq. (3.62) yields another, and perhaps the most common, expression forqst:
qst = T ρb
∂Pb
∂T
Vs,Ns = − Pb
ρbT
∂lnPb
∂(1/T)
Vs,Ns. (3.65) At very low gas pressure one may assume thatNsis proportional toPb, i.e.
Ns =kHPb+O(Pb2). (3.66)
The leading term is a “surface version” of Henry’s law Eq. (3.46). In this approxi- mation Eq. (3.65) becomes
qst(o)=R∂lnkH
∂(1/T)
Vs,Ns
. (3.67)
Hereqst(o)is the molar isosteric heat of adsorption in the limit of vanishing cover- age. Experimentally this quantity may be determined by measuring the amount of adsorbed gas (e.g., by weighing the sample) at a given (low) pressure. The general relation Ns(T,P) vs. P is called adsorption isotherm, the low pressure slope of which again iskH. On p. 206 we return to the isosteric heat of adsorption and discuss one explicit method how to calculate it theoretically.
Law of Mass Action
In the following we discuss an important application of Eq. (2.119), i.e.
d G
T,P≤0. (3.68)
At equilibrium we can use the equal sign and based on Eq. (2.107) (withμdnreplaced byK
i=1μidni) we have K i=1
μi(T,P)dni =0. (3.69)
This equation requires some thought. If K =1 then Eq. (3.69) impliesdn =0.
The inequality Eq. (3.68) applies to cases where, aside from keepingT andPat fixed values, we leave the system alone. In particular we do not change its mass content.8 IfK >1 there exists however the possibility of a suitable relation between thedni, developed by the system itself, allowing Eq. (3.69) to hold without requiringdni =0
∀i. For instance we may replacedniin Eq. (3.69) via
dni =νidξ, (3.70)
where of course not allνi do have the same sign. It turns out that chemical reaction equilibria may be described in this fashion. For a chemical reaction thedni obey according to experimental evidence
dni
dnj = νi
νj, (3.71)
whereνiandνjare integers.
We proceed writing the chemical potentials of the components as
μi(T,P)= ¯μi(T,P)+RTlnai. (3.72) This particular form is analogous to the special limiting forms Eqs. (3.27) and (3.45).
The quantityai, which is called activity of componenti, contains interaction and mixing contributions to the chemical potential of componenti, i.e. all effects due to the interactions of this component with all other components. Often the activity is expressed via
ai =γixi, (3.73)
8Potentially this may be disturbing. According to the steps leading from Eq. (2.164) to Eq. (2.165) one may be led to conclude thatG=0 all the time and everything falls apart. However this reasoning confuses two very different situations. Inequality Eq. (3.68) means that we prepare a system subject to certain thermodynamic conditionsT andPand leave this system alone until no further change is observed. This fixes the equilibrium value of the free enthalpy,G, for a particular pairT,P.
Repeating this procedure for manyT,P-pairs we map out the equilibrium values ofGabove the T-P-plane (cf. Fig.2.13). With this functionG =G(T,P)orG =G(T,P,n)we now can do calculations, differentiating or integrating, involvingT,P,nand possibly other variables. This is how we have obtained the Eqs. (2.164) and (2.165). Therefore there is no problem here!
whereγiis the activity coefficient. This is the usual terminology in condensed phases.
In the gas phase the fugacity,
fi =γiP, (3.74)
whereγiis the fugacity coefficient and P is the pressure, replacesai. We see that the special limiting forms Eqs. (3.27) and (3.45) correspond toγi =1. The reference chemical potential,μ¯i(T,P), may be identified if we letγi andxi approach unity.
Combining Eqs. (3.69), (3.70), and (3.72) we obtain K
i=1
(μ¯i(T,P)+RTlnai)νi =0 (3.75) or
K i=1
aiνi =K(T,P), (3.76)
where
K(T,P)=exp
− K
i=1νiμ¯i(T,P) RT
. (3.77)
Equation (3.76) is called law of mass action and K(T,P), not to be confused with the indexK, the number of components, is the equilibrium constant. The equilibrium constant is not really a constant. It depends onT andP. By conventionνi <0 for reactants andνi >0 for products.
The law of mass action in its present form provides little insight. Therefore we study the special case of a gas phase reaction assuming that the gas is ideal. Combining Eqs. (3.26) and (3.21) we obtain
μ(ig)(T,Pi)=μ(ig)(T,Pio)+RTln Pi
Pio. (3.78)
HerePi is the partial pressure of componenti, andPiois a standard pressure, which remains constant during the reaction. In addition we have
xi(g)= Pi
P(g), (3.79)
cf. (3.28), where P(g) is the total pressure. Combination of Eqs. (3.78) and (3.79) yields
μ(ig)(T,Pi)=μ(ig)(T,Pio)+RTln P(g)xi(g)
Pio . (3.80)
Inserting this into Eq. (3.69) we find the law of mass action K(T,Po)=Piνi
i
xiνi, (3.81)
where−RTlnPiois absorbed intoK(T,Po). Notice that we have omitted the index (g), and we assume that Po =Pio∀i. This equilibrium constant is independent of the gas pressure Pand the mole fractionsxi.
Example: A Chemical Reaction. In the following simple example of a chem- ical reaction,
2H2+O22H2O, (3.82)
we haveνH2 = −2,νO2 = −1, andνH2O =2. Thus Eq. (3.81) becomes K(T,Po)= 1
P x2H
2O
x2H2xO2
. (3.83)
Increasing the total pressure (at constant temperature) shifts the reaction equi- librium to the right. Analogously we can see what happens if the concentrations are changed.
Remark:What is different if in addition to H2,O2, and H2O another inert gas is present or there is an excess of one or more of the aforementioned components? The corresponding mole fractions do not appear explicitly on the right side of Eq. (3.83), but they do enter into the pressure,P.
At this point we may ask: What is a component? Thermodynamic knows nothing about atoms, molecules, and details of the interactions/reactions between them. But we know that there are even smaller building blocks than atoms—electrons, protons, and neutrons. And this is not the end. So what is a component? In principle we may apply thermodynamics on all levels. For the above it is important, however, that there exists a meaningful chemical potential for everything we want to call component. That is a component must exist long enough (on average) under well defined thermodynamic conditions like equilibriumT andP.
This requires us to rethink our derivation of the phase rule. Consider the following example for a chemical reaction:
3AA3. (3.84)
If we consider A and A3as components, then the phase rule Eq. (3.12) allows up to four coexisting phases. However, we have an additional equilibrium constraint imposed by Eq. (3.69), reducing the degrees of freedom by one and the maximum number of coexisting phases to three. The modified phase rule therefore is
Z =K −Q−+2 (≥0), (3.85)
where Qis the number of additional constraints imposed via Eq. (3.69). Notice that Qis not necessarily one all the time. There may be independent chemical reactions occurring simultaneously in which case the summation in Eq. (3.69) breaks up into independent parts, e.g.
3AA+A2A3.
Here we have a system containing three components according to our definition. But for each reaction we have to fulfill Eq. (3.69). ThereforeK =3 andQ=2.
Example: Critical Micelle Concentration. Figure3.12shows a sketch of a system containing typo-amphiphilic molecules. Typo-Amphiphilic molecules consist of two covalently bonded moieties—one, depicted as zigzag-line, does not like to be in contact with water, not shown explicitly, whereas the other, de- picted as solid circle, does like to be in contact with water. An example of such a molecule is Hexaethylene-glycol-dodecylether (C12H25(OC H2C H2)6O H).
In this case it is theC12H25-moiety that does not like to be in contact with water. The natural thing to happen therefore is a clustering of the zigzag “tails”
into droplets shielded on the outside by their water loving “head” groups. In a sense this is a phase separation, which we study in the next chapter, on a molecular scale. Because of this the molecular “shape” strongly couples to the “shape” of the drop or aggregate and in fact determines it (The aggregates we have in mind can be spherical, cylindrical or transform into layered struc- tures with complicated topology. It also is possible to extend this approach to vesicles. But this is not our topic here.). The type of droplet aggregate we just described is called a micelle. However, our current approach covers other types of aggregates as well.
As our starting point we choose the “chemical reaction equation”
s A1As. (3.86)
HereAsdenotes a s-aggregate containingsmolecules or monomersA1. We put
“chemical reaction” in quotes, because the bonding forces between monomers considered here are different from chemical bonds within molecules. In prin- ciplescan be any integer number and therefore Eq. (3.86) represents many
“reaction equations”. Expressing this in terms of the chemical potential yields
sμ1=μs. (3.87)
Assuming low monomer concentration we may use Eq. (3.48), i.e.
sμ¯1+s RTlnx1= ¯μs+RTln(xs/s). (3.88) Note that the quantityxs/sis the mole fraction s-aggregates. Thereforexs is the mole fraction of monomers in s-aggregates. We may solve forxs, i.e.
xs =s x1eαs
(3.89) with
α= 1 RT
¯ μ1−1
sμ¯s
. (3.90)
We assume thatμ¯s is an extensive quantity in terms ofsand that thereforeα is independent ofs(see also the next example). Equation (3.89) has an inter- esting consequence. To see this we note that the total monomer mole fraction is given by
x=x1+ ∞ s=m
xs =x1+ ∞ s=m
s x1eαs
. (3.91)
Herex1is the mole fraction of free monomers, whereas the sum is the mole fraction due to all other monomers bonded inside aggregates. We note thatm is a minimum aggregate size. In the case of spherical micelles for instance, it accounts for the fact that a certain number of head groups are required to form a closed surface avoiding contact of the tail groups with water. This number may be large—saym ≈ 50—depending of course on the type of monomer.
Butm=2 also is possible. This is the case of linear aggregates (chains of monomers—These monomers may be disk-shaped with flexible tails on their perimeter. In water the disk-like cores tend to form stacks. It also is possible to apply this idea to dipolar molecules forming chains due to dipole-dipole inter- action.). The right side of Eq. (3.91) is bounded, becausex ≤1. In particular this requiresx1eα <1, because the sum∞
s=msqsdiverges atq=1 (geomet- ric series!). Putting in some numbers we find∞
s=50sqs ≈4×10−3ifq=0.8 and∞
s=50sqs ≈3 ifq =0.9, i.e. forx1<0.8e−αvirtually all ofxis due to free monomers. Addition of monomers at this point leads to their assembly into
Fig. 3.12 Sketch illustrating the reversible assembly of amphiphilic molecules into micelles
s-aggregate xs
μs
monomer s = 1
x1
μ1
aggregates. Figure3.13illustrates this for different combinations of assumed values formandα(Notice the change of scale in the third panel.). Because of the sharpness of the “transition” in the typical case of largemthe threshold concentration
xC MC ≈e−α (3.92)
is called critical aggregate concentration or, in the case of micelles, critical micelle concentration (CMC). While the sharpness is governed bym, the am- phiphile concentration at which the change of behavior occurs is determined byα.
We note that the existence of a CMC is not tied to the specific form of Eq. (3.89). For instance, assuming that monomers may form minimum aggre- gates only, i.e. only thes =m-term in the sum in Eq. (3.91) is present, still yields a CMC. The true size distribution,xs, in fact is a complicated function of molecular interactions as well as thermodynamic conditions. One interest- ing and quite general ingredient ignored here is the aggregate dimensionality, which is discussed in the following.
More information on molecular assemblies (micelles, membranes, etc.) can be found in Israelachvili (1992) or in Evans and Wennerstrửm (1994).
Surface Effects in Condensation
The assumption underlying Eq. (3.90) is that all monomers inside an aggregate are equivalent. For a spherical micelle this is in accord with intuition. But what if we study droplets containing monomers completely embedded in their interior and monomers on their surface? These two are certainly different and Eq. (3.90) no longer holds.