A. Lainez and G. Tardajos Universidad Complutense, Madrid 28040. Spain As we have often verified, students find i t difficult to grasp the meanine of standard states defined for real solutions. ~ h e r m o d ~ n & n ior c physical chemistry texts deal with the situation in varied ways, going from the mere definitions to extended considerations which are either cumbersome or rigorous. In any case, the student hardly understands the need for more than one standard state and the physical meaning of some of the hypothesized states. We believe the presentation of the subject developed here allows an easy confumation that two standard states with easy physical interpretation are possible in the case of real solutions. For a given composition and temperature, the activity a; of component i in solution is given by
Therefore, in both cases & follows a linear relation with respect to In xi, with a slope R T and ordinate values a t the origin p; =ILP+RTlnp:
(7)
pf=py+RTInK;
(8)
A representation of pi as a function of In x; for a real solution might look something like Figure 2. Since the deviation from Raoult's law is negative we have Ki < py and therefore p t < pf, as shown in Figure 2. In the region between the vertical dotted lines, where Raoult's and Henry's law are not oheyed, the exact habit of the pi representation is irrelevant. Since this is of no consequence for our
.
where p: is the chemical potential of the standard state (a; = \
11.
For an ideal gas mixture i t is easily shown' that a; = pi. In this case eqn. ( I ) reduces to p; = p?(T) + RTinpi
(2)
where p:(T) is the chemical potential of the pure i gas at 1atm pressure and the given temperature, assuming ideal behavior. For ideal solutions2 ai = xi and the chemical potential of component i is Now pp is the chemical potential of the pure i liquid at the solution temperature and pressure. I t is usually referred as PI Obviously, if we represent the chemical-potentials versus In pi or In x i for the ideal gas mixture or the ideal solution respectively, we get a straight line whose slope is R T and where pp and p; are the ordinate values a t the origin. Real Solutions In an idea?solution Raoult's law is valid a t any concentration, that is,
0
1 Xi
Figure 1. V a p pressure of tha hh component tor a liquid mimre with negative deviation of Raoult's law.
where py is the vapor pressure of pure i liquid a t T . However, this is not the case for a real solution, and in Figure 1we have reoresented the vanor of comnonent i against mole . nressure . fraction for a real solution with n negative devintion of Raoult's law. We ohserve in the fieure that at x. -* 1 or x. 0 the vawr pressure may he linearii approaima;ed. The dotted and ;he continuous lines corres~ondto Kaoult's law and Henrv's law (pi = K;xj), respectivefy. The constant Kj depends on component i and on the nature of the other solution components, while pp is a characteristic of component i. For a real solution in equilibrium with its vapor, the chemical potential for component i is the same in the liquid and vapor phase. If we assume ideal behavior for the vapor, we ohtain.from (2) and Raoult's and Henry's laws pi=(pP+RTlnpp)+RTinr
'
ifri-1
(5)
Moore. W. J.. "Physical Chemistry." 5th ed., Longman Group LM.. London. 1972, p. 237. Footnote 1, p. 239.
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Figure 2. Variation of the chemical potential wnh respect to in x, for a real sclution. In lhis and Figure 3 the dotted lines llmit the regions of ideal behavior.
*i
xi-1
I" X i Flgure 3. Relation betweenlhe chemical potential faa real solution and those lor the ideal behavior given by Raouit's and Henry's laws.
following considerations, we have assumed the shape drawn in the figure. We therefore visualize the possibility of measuring the chemical potential pi with respect to either of the two straight reference lines, the (a) line if we start from Raoult's law and the (b) line if we do it from Henry's. In other words, we have two different reference conventions. Convention I Let us look a t point C in Figure 3, which corresponds to the mole fraction xi. Its chemical potential measured with reference to the straight line (a) is:
-
D C = ~ + =
(9)
Figure 4. Variation of chemical potential with respect lo In xr for a partially miscible solute.
Now formally using expression (11) (hut keeping in mind that the arti\,itv coefficient value isdifferentLand renlacinnu .. pib by its value; eqn. (15) becomes pi=p?+RTInxi+RTinyi
(16)
As the xi value decreases so does the corresponding segment value. We have now
-
limBC=O 1,-0
limlny,=O l i m y i = l z,-0
r,-0
Consequently, adopting convention I1 where pid. is the deviation from the ideal behavior represented by line (a). If we make
Consequently, adopting convention I, the chemical potential is given by pi =p;+RTlnoi (13) 7;-1 ifxi-1 and the standard chemical potential is that of the pure i liquid a t the solution pressure and temperature.
and the standard state is an extrapolation a t xi = 1of the hehavior a t infinite dilution. Since the requirement for the standard state is that yi = 1and xi = 1simultaneously (so that. a; = I), it is not, therefore, a real state for i, as our convention requires yi = 1when xi = 0. Some textbooks3 define p? as the chemical potential of component i (solute) in a hypothetical ideal solution a t xi = 1. On the other hand, this so-called standard state depends not only on component i, but on the type of solution, as seen from eqn. (8). Why define two standard states instead of being satisfied with convention I, which has a physical meaning that is easy to grasp? The answer appears to he a simple one. If the components are soluble in the whole range, either of the two conventions could he applied. However, let us look a t the case in which they are only partially miscihle, as in Figure 4, which gives a possible plot of the solute chemical potential against In xi. Here x, stands for the mole fraction a t saturation, which sets a maximum for xi. I t would he rather puzzling to talk of pi for xi >I., and i t would not be advisable to use p f as standard state. Therefore, convention I is applicable only to the component whose mole fraction may reach the value of 1 (solvent). Convention I1 should be applies to the other components (solutes).
Convention 11 Consider as before the point C in Figure 3. Referring t o straight line (b) we get
Castellan. G. W., "Physical Chemistry," 2nd ed., Addison-Wesley Publishing Co., 1971, p. 320.
and replace piab y its value, eqn. (10) gives pi=p;+RTlnxi+RTInyi
(12)
Here y; stands for the activity coefficient of component i, and is a measure of the deviation from ideal hehavior. Comparison of eqn. (12) with eqn. (1) gives
xi
As the value inueases, the value of the corresponding segments (A'C', A"C", etc.) decreases until i t vanishes. Therefore, -
limAC=O xi-,
limlny;=O xi-,
limyi=l ri-l
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879
It is obvious that the standard state for solute is defined in an analogous way if concentrations are given in terms of mo-
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Journal of Chemical Education
lality or molarity instead of mole fraction. We simply replace 1 by c, = 1 mol.1-' or mi = 1 mol.kg-'.
xi =
