From WolfWikis

LECTURE 15

Activities

As seen before activities are a way to account for deviation from ideal behavior while still keeping the formulism for the ideal case intact. For example in a ideal solution we have:

μ^sln = μ* + RTlnx_i

is replaced by

μ^sln = μ* + RTlna_i

The relationship between a_i and x_i is often written using an activity coefficient γ:

a_i= γ_ix_i

Raoult versus Henry

Implicitly we have made use of Raoult's law here because we originally used

x_i = P_i/P*_i

In the case of a solvent this makes sense because Raoult's law is still valid in the limiting case, but for the solute it would make more sense to use Henry's law as a basis for the definition of activity:

a_solute,H ≡ P_solute/K_x,H

This does mean that the μ* now becomes a μ*^Henry because the extrapolation of the Henry law all the way to the other side of the diagram where x_solute=1 points to a point that is not the equilibrium vapor pressure of this component. In fact it represents a virtual state of the system that cannot be realized. This however does not affect the usefulness of the convention.

Various concentration unit

I have added a subscript X to the K value because we are still using mole fractions. However Henry's law is often used with other concentration measures. The most important are:

• molarity
• molality
• mole fraction

Both the numerical values and the dimensions of K will differ depending on which concentration measure is used. In addition the pressure units can differ. For example for oxygen in water we have:

K_x,H= 4.259 10⁴ atm

K_cp,H= 1.3 10^-3 mol/lit.atm

K_pc,H= 769.23 lit.atm/mol

As you can see K_cp,H is simply 1/K_pc,H, both conventions are used...

Note that in this case a choice based on Raoult is really not feasible. At room temperature we are far above the critical point of oxygen which make the equilibrium vapor pressure a non-existent entity.

Returning to activities we could use each of the versions of K as a basis for the activity definition. This means that when using activities it must be specified what scale we are using.

Activities and Henry coefficients of dissolved gases in water (both fresh and salt) are quite important in geochemistry, environmental chemistry etc.

Non-volatile solutes

A special case arises if the vapor pressure of a solute is negligible. For example if we dissolve sucrose in water.

In that case we can still use the Henry based definition

a_solute,H ≡ P_solute/K_x,H

Even though both K and P will be exceedingly small their ratio is still finite. However how do we determine either?

The answer lies in the solvent. Even if the vapor pressure of sucrose is immeasurably small, the water vapor pressure above the solution can be measured. The Gibbs-Duhem equation can then be used to translate one into the other.

We can use Raoult to define the activity of the solvent:

a₁ = P₁/P*₁

We can measure the pressures as a function of the solute concentrations

At low concentrations ln a₁ ln x₁ ≈ -x₂

At higher concentrations we will get deviations, we can write:

lnP₁/P*₁=lna₁ ≈ -x₂φ

The 'fudge factor' φ is known as the osmotic coefficient and can thus be determined as a function of the solute concentration from the pressure data.

What we are really interested in is a₂, not a₁:

a₂= γ₂x₂

Using Gibbs-Duhem we can convert φ into γ₂ (see book)

Usually this is done in terms of molalities rather than mole fractions and it leads to this integral:

lnγ_2,m = φ – 1 + ∫ [ φ – 1 ]/m' dm' (from m'=0 to a certain m)

Colligative properties

Colligative properties are properties that depend on the number of particles rather than their total mass. This implies that these properties can be used to measure molar mass.

Colligative properties include:

• melting point depression
• boiling point elevation
• osmotic pressure

Melting point depression

When we freeze a dilute solution the resulting frozen solvent is often quite a bit purer than the original solution. Let us consider this problem and make the following rather opposite assumptions about the solvent component:

• in the liquid state it can be considered to follow Raoult over a sufficient concentration range
• in the solid state the solubility is nil

Under these assumptions we can consider the thermodynamic potentials of the solvent component (1) at the freezing point. They should be equal once equilibrium has been reached:

μ₁^s = μ₁^sln

μ₁^s = μ₁^liq* + RTlna₁

[μ₁^s - μ₁^liq*] / RT=lna₁

[-Δμ₁] / RT=lna₁

We can now apply Gibbs-Helmholtz by differentiation with respect to temperature :

∂ [Δμ₁/T]/∂T = - Δ_fusH_molar,1/T²

∂ [lna₁]/∂T = Δ_fusH_molar,1/RT²

This means that we can actually use the quantity Δ_fusH_molar,1/RT² to determine activities by integration, but usually Raoult is assumed valid:

ln a₁ ln x₁ ≈ -x₂

If we integrate Δ_fusH_molar,1/RT² from the melting point of the pure solvent T_m* to the actual melting point of the solution T_m we get:

-x₂ = Δ_fusH_molar,1/R [ 1/T_m*- 1/ T_m] = Δ_fusH_molar,1/R [ {T_m- T_m*}/T_m* T_m]≈ Δ_fusH_molar,1/R [ -ΔT/T_m*²]

This is often rewritten in terms of molality as:

ΔT = K_f.m

If K_f is known for the solvent we can add a number of grams of an unknown compound to the solvent measure the temperature depression, this tells us the molality. From molality and weight we can then calculate molar mass.

Boiling point elevation is quite similar and we will skip it.

Determining molar masses nowadays

Both melting point depression and boiling point elevation only facilitate the determination of relatively small molar weights. The need for such measurements is no longer felt because we now have good techniques to determine the structure of most small to medium size molecules. For polymers this is a different matter. They usually have a molecular weight (mass) distribution and determining it is an important topic of polymer science

Osmometry -see below- is still of some practical usefulness. It is also colligative and able to measure up to about 8000 daltons. Many polymers are much bigger than that. Their mass distribution is usually determined by different means. The polymers is dissolved and led over a chromatographic column usually based on size-exclusion. The effluent is then probed as function of the elution time by a combination of techniques:

1. UV absorption (determine the monomer concentration)
2. Low Angle Laser Light Scattering (LALLS) and/or Viscometry

The latter two provide information on the molar mass distribution but they give a different moment of that distribution. The combination of techniques gives an idea not only of how much material there is of a given molar mass but also of the linearity or degree of branching of the chains.

Purity analysis

Nevertheless melting point depression is still used in a somewhat different application. When a slightly impure solid is melted its melting point in depressed. Also the melting process is not sudden but takes place over the whole trajectory from typically a lower eutectic temperature up to the depressed melting point (the liquidus line in the phase diagram). In organic synthesis the melting behavior is often used as a first convenient indication of purity. In a DSC experiment the melting peak becomes progressively skewed towards lower temperatures at higher impurity levels. The shape of the curve can be modeled with a modified version of the melting point depression expression. This yields a value for the total impurity level in the solid. This technique is used in the pharmaceutical industry for quality control purposes.

Osmosis

Semipermeable membranes

Some membrane materials are permeable for some molecules but not for others. Often this is a matter of size of the molecule but it can also be a question of solubility of the molecule in the barrier material. Many biological membranes have semipermeable properties and osmosis is therefore an important biological process.

If solvent molecules can pass but solute molecules (or ions) cannot and at the opposite side of the membrane there is pure solvent, the solute molecules get a chance to increase the size of the box they are trapped in. Solvent molecules will spontaneously migrate across the membrane to increase the solution's volume and thus reduce its concentration.

If the solution is ideal this process is in many ways analogous to the spontaneous increase in volume of a gas allowed to expand against vacuum. Of course the volume of the 'solute-gas' is limited by the availability of solvent and -if done under gravity in a U-shaped tube by the build up of hydrostatic pressure. This pressure is known as the osmotic pressure Π

At equilibrium we can write:

μ*(T,P) = μ^sln(T,P+ Π,a₁)

μ*(T,P) = μ*(T,P+ Π)+ RTlna₁

From dG in its natural variables we know that

∂G/∂P |_T,x = V

Taking the partial versus x₁ we get:

∂μ*/∂P |_T,x = V*_bar,1

This means we can integrate over the molar volume to convert μ*(T,P+ Π) to a different pressure

μ*(T,P+ Π) = μ*(T,P) + ∫V*_bar,1 dP (from P to P+ Π )

Thus we get

μ*(T,P) = μ*(T,P+ Π)+ RTlna₁

μ*(T,P) = μ*(T,P)+ ΠV*_bar,1+ RTlna₁

Once again using the ideal approximation

ln a₁ ln x₁ ≈ -x₂

we get:

RTx₂ = ΠV*_bar,1

x₂ = n₂/[n₁+n₂]≈n₂/[n₁]

The combination gives an expression involving the molarity.

Π=RTc

Osmosis can also be used in reverse, if we apply about 30 bar to sea water we can obtain fresh water on the other side of a suitable membrane. This process is used in some places but better membranes would be desirable. Also they easily get clogged. The resulting water is not completely salt-free and this means that if used for agriculture the salt may accumulate on the field over time.