1 Gases
2 Microscopic Energies
3 First Law
4 2nd & 3rd Law
5 Phase transitions
6 Mixtures
7 Phase Diagrams
8 Equilibrium
9 Molecular Interactions
Ideal Solutions
If we consider only ideal solutions, we can calculate the Gibbs energy of mixing the same way as we did for a gas, hence,
Similarly, the entropy of mixing of liquids, when forming an ideal solution is the same as that for ideal gas.
and, since all intermolecular forces are unchanged in creating an ideal solution, the enthalpy of mixing is zero.
Clearly then, for ideal solutions, the driving force of mixing is purely entropy, just as in gases.
Real Solutions
Real solutions have intermolecular forces A-A, B-B and A-B that are all different from each other. There will be a contribution to the various thermodynamic parameters for mixing due to these differences. We talk of the excess functions, for example, the excess entropy is
| SE = ΔmixS - ΔmixS ideal, |
In this case, ΔmixS ideal, comes from equation 6.24 and the value of ΔmixS comes from the real solution. There would be a non-zero enthalpy of mixing because of the differing energies of interaction (intermolecular forces) of a non-ideal solution, thus, HE ≠ 0. The excess functions give us the measure of the amount of deviation from ideality. We will name a "Regular Solution", one which deviates from ideality in enthalpy but which has zero excess entropy, HE ≠ 0 and SE = 0. In other words, the randomness of the distribution of the molecules in solution is the same as if it were ideal but there are differing intermolecular forces, hence the non-zero excess enthalpy.
| HE = nβRTχAχB |
|
Here, β is a dimensionless parameter that gives us a measure of the difference between the A-B forces versus the A-A and B-B forces. This function clearly has a maximum (or minimum) at when χA = cB = 0.5.
In this equation, if β is positive then there is an increase in the enthalpy of the solution compared to the individual liquids, in other words, the intermolecular forces (of attraction) for A-B are weaker than for A-A or B-B.
On the other hand, if β is negative it means there is a decrease in the enthalpy of the solution compared to the individual liquids. The intermolecular forces of the solution are stronger than in the individual liquids.
|
The Gibbs energy of mixing can be calculated quite easily for a regular solution by simply adding in the enthalpy term:
Clearly, we see that there are possible conditions when we will not
get complete mixing. If the final value of Gibbs energy of
mixing is positive then we will see two phases as the liquids become
immiscible at those particular conditions. In our diagram, we
see that if b
is high enough then the liquids may be miscible in dilute
concentrations (A in B) or (B in A) only but when comparable amounts
of A and B are present, we do not get complete mixing.
Activities
Previously, we saw a term called fugacity that was meant to account for deviation of gas from ideal behavior while keeping the same general form of the equations for the ideal gases. We now develop the concept of activity as a way to account for non-ideal behavior of solutions in similar manner.
The Solvent Activity
We saw in equation 6.22 that for an ideal solution, we could relate the chemical potential to the concentration of the solvent (A) in an ideal solution. The mole fraction of the solute was actually the ratio pA/pA*, which may not be linearly related to concentration in non-ideal solutions. We will replace mole fraction with parameter called "activity", a, such that the same form of the equation is valid for real solutions but the activity now contains the information about the deviation of the solution from non-ideality.
Hence, we see the activity in this case is like an "effective" mole fraction in the same way that fugacity is the "effective" pressure for non-ideal gases. Hence, even if the solution is non-ideal, we can determine it's activity by measuring the vapour pressures via the following:
Example: The measured vapour pressure of a solution containing 0.500 M KNO3(aq) is 99.95 kPa at 100ºC. What is the activity of the water?
Note that the vapour pressure of water at 100ºC is 1 atm = 101.325kPa, as defined by the normal boiling point.
From our previous discussion of Roult's Law, we remember that all solvents obey Raoult's law as the solute (B) concentration approaches zero. hence,
| aA → χA as χA → 1. |
We often use a parameter to show this. The activity coefficient, g, is used to describe the deviation from ideality.
| aA = gA χA and hence gA → 1 as χA → 1. |
In other words, we can only really use concentrations in our thermodynamic equations for infinitely dilute solutions. for all others, we should use activities or at least the activity coefficient. So now, our equation equation 6.22 becomes
We can see the activity coefficient now as a correction term showing how much the solvent deviates from ideality. Clearly, when the χA = 1, the solvent (at 1 bar) attains standard state.
The Solute activity
Solute (B) behavior also approaches a linear relationship in the infinite dilute conditions, however, it's Henry's Law, pB = KBχB, rather than Raoult's Law that solutes approach. Hence, we can write
because KB and pB* are related only to the solute B. so if we take the first two terms only of the equation, we will get the standard chemical potential of the solute.
We can use this to simplify the relation for the chemical potential of the solute in it's infinite dilute solution.
If the solution is ideal then
KB = pB*
and mB0
= mB*.
For less dilute solutions, we deviate from this linear (Henry's Law) behavior so we introduce the activity in place of mole fraction again to account for the deviation from "ideal-dilute" behavior.
This time, the activity is given by
and hence, we can also define the activity coefficient as we did in equation 6.31, but this time we have
| aA → χA and gA → 1 as χA → 0. |
So the solvent approaches ideal behavior as it's mole fraction approached 1 but the solute approached ideal-dilute behavior as it's mole fraction approaches zero. (both of these are really the same thing, infinite dilution)
Activities using molality
In chemistry, we often use methods of measuring concentration that are more convenient than mole fraction, for example, Molarity or molality. We can work with any such concentration measure and define a standard state for those measures.
Let's discuss a solution with solute B having molality b. Our equation for chemical potential is now written as
In this case, we are defining the standard molality bº as 1 mol/kg. Hence, it's not in the denominator of the ln function above. Our activity and activity coefficient is again introduced as follows
This time, we see that γB → 0 as bB → 0. In other words, the solution approaches ideal behavior in the infinite dilute limit.
We can now write our equation
Which gives us a way to calculate the chemical potential of a real solute at any molality.
new definition of standard state
Normal standard state using activities can be defined quite simply now. The standard state of a substance occurs when its activity is 1. Thus, if we are using molals to determine activities then standard state of a solute occurs at activity = 1 (which would correspond to molality = 1 only in an ideal solution) This is called the thermodynamic standard state. While it may not always be possible to actually make a solution with solute concentrations of 1 m, we can at least use it as a defined standard state. the choice of standard state is largely done for convenience. Thus, activity = 1 is not always chosen in science. In biochemistry, this standard is very inconvenient for processes involving acids and bases. If the activity of H+ = 1, for example, that would be a pH = 0. Generally, biological processes occur in neutral or near neutral solutions, pH = 7. Thus, for biological chemistry, organic chemists, biochemists, the standard state for the hydrogen ion is often accepted to be 1×10–7. Any thermodynamic functions determined with this standard state use the superscript o-plus to indicate that the biological standard state is being used for example, .
To convert from the thermodynamic to the biological chemical potential for the hydrogen ion, we use the equation
Activities of ions in solutions
When dealing with ionic solutions, we cannot separate the contribution of the positive and the negative ions since they much both exist together to maintain neutrality. For constant T and p conditions, we can write the overall Gibbs energy of an ideal solution from the individual chemical potentials as follows
For a real solution MX forming M+ and X-, we would need to incorporate the activity coefficients of each ion,
Since we cannot determine an individual activity coefficient, we use a mean activity coefficient
Thus, the individual chemical potentials can now be approximated as
In cases where the mole ratio of the metal M to non metal X, we need to reformulate to account for the stoichiometry. Suppose our salt is MpXq. the p and q need to be incorporated into the equation for G as follows
Finally, we reformulate our mean activity coefficient as
So now, the equation 6.44 works for all ionic compounds now both ions share the non-ideality.
One further complication to ionic solutions is the fact that the ions don't tend to distribute themselves fully randomly. The positive ions tend to surround themselves by negative ions and vice versa. This type of effect increases as the numbers of ions increases. The measure of this effect we call the ionic strength, I.
For dilute solutions, we can calculate the activity coefficient using the Debye-Hückel Limiting law. Limiting in that it is only perfectly accurate at infinite dilution.
| Log γ+ = -|z+z-|AI1/2 |
were z is the charge number of the ions and A = 0.509 for aqueous solution at 25ºC and
|
I = 1/2(b+z+2
+ b-z-2)/b |
Example: what is the mean activity coefficient of a 5.0×10-3 mol/kg KCl (aq) solution at 25ºC?
In this case, z+
= z- = 1 so
and b+ =
b- = 5.0×10-3
and finally
b0
= 1 so, we substitute into equation 6.49
|
I = 1/2(b+z+2
+ b-z-2)/b |
|
Now, using equation 6.48, we have
| Log γ+ = –0.509 (5.0×10-3 )1/2 = –0.036 |
|
 |
As the name "limiting Law" implies, this only gives accurate values in the limit of infinite dilution. To get values for solutions of any concentration, we use the extended Debye-Hückel Law
parameters A, B and C are dimensionless adjustable parameters that depend on the particular mixture in question. By fitting experimental data to this equation, we can derive values A, B and C that allow the Extended Law to predict the mean activity coefficient at a wide range of concentrations.
