1. Activity
Standard states

2. Gases Fugacity of real gases can be measured
So standard states fixed in terms of fugacity
At any fixed temperature, standard state can be defined as one in which the gas has a fugacity of 1 atm
Therefore a= f/fo =f since fo = 1 atm
ie activity and fugacity are same
figure

3. From figure
Can be seen that standard state is a hypothetical state , at 1 atm pressure the gas behaves ideally, so that f=p and equal to unity
Activity of an ideal gas is numerically equal to its pressure since f=p
For gases activity thus has the same meaning of fugacity, since standard state for both of them are same.

4. Like fj/pj, aj/pj is called activity coefficient(γp), which is a measure of deviation of the real gas from ideal behaviour.
Therefore μj = μjo + RTln γppj (1)
Instead of choosing the ideal gas at 1 atm as a standard state, unit molar concentration (c) can be chosen as the standard state

5. Therefore μj = (μjo )c+ RTln(aj)c (2)
For an ideal gas ac = c
We know that μj = μjo + RTln pj (3)
Since pj = cjRT
μj = μjo + RTln RT + RT lncj (4)
For component ‘j’ which behaves ideally ,its molar concentration cj is cj = (aj)c
therefore

6. μj = μjo + RTln RT + RT ln(aj)c (5)
μj - μjo = RTln fj (6)
= RTln(aj)p
= RTlnRT + RTln (aj)c (7)
= RTln (aj)c RT
From (7) & (6)
(aj)c = (aj)p/RT = fi/RT (8)

7. This equation gives the relation between fugacity and activity
The activity coefficient in terms of concentration (γc) is
γc = ac/c = f/RTc (9)
Here γp = f/p
Since p ‡ cRT for non ideal gases γc ‡ γp

8. But as pressure is lowered ,gases behave ideally , so γc ‡ γp
For an ideal gas γc = γp even though ac ‡ ap
Difference in ac and ap arises from the difference in the choice of standard sate.
Not usual to activity in terms of concentration for gases since activity in terms of pressure is same as fugacity
Fugacity is used only for gases

9. In solution
For pure liquids and solids, standard states are taken to be the pure condensed phase at a total pressure of 1 atm
Thus activity at 1 atm is taken as one
But cannot be true for solutions containing liquid or solid
Activities of solute and solvents has to be considered separately.
on increasing the dilution , a solvent in solution approaches ideal behaviour given by Raoult’s law
Solute approaches ideal behaviour specified by Henry’s law

10. Solvent
Standard state is chosen that the pure liquid solvent at the given temperature and at a total pressure of 1 atm, has unit activity
Letf1. be fugacity of pure solvent
Pure solvents under these conditions is the standard state so f1. = f1o
On adding solute, if fugacity of solvent becomes f1, f1<f1o

11. Therefore activity in this case is a1 = f1/f1o
For an ideal solution
f1 = f1oX1 = f1.X1
So a1 = f1oX1/f1o = X1 ie a1 = X1
ie activity of solvent in an ideal solution at 1 atm is equal to its mole fraction

12. For real solution,
Raoult’s law not obeyed
So a1/X1 differs from unity
Activity coefficient(γX) is a measure of the extent of deviation
When γX > 1 (a1>X1) - system shows positive deviation
When γX <1 - negative deviation

13. Activity coefficient of ‘j’ , γj =aj/Xj is called rational activity coefficient of ‘j’

14. Solutes
Several different standard states are chosen depending circumstances.
If solute and solvent are completely miscible in all proportions,
The standard state of the solute is chosen as the pure liquid at atmospheric pressure
This is the same standard state as for the solvent

15. Activity coefficient a2/X2 approaches one as X1 tends to one
If the solute has a limited solubility,
different standard states chosen based on the concentration unit used to express the composition of the solution
If choice is mole fraction it is referred to as rational system
If molality/molarity - practical system

16. Rational system If mole fraction of solute is X2
Henry’s law is applicable to solute , f2 =kX2
So standard state for solute is chosen in such a way that in a dilute solution the activity becomes equal to mole fraction of the solute
Thus a2/X2 tends to one
as X2 tends to zero (1)
Figure

17. For very dilute solution as X2 tends to zero actual cure merges with Henry’s law line
Since a2 = f2/f2o (2)
So (1) can be written as
Lim X2→ 0 a2/X2 = Lim X2 → 0 f2/f2oX2 = (3)

18. Since Henry’s law is also applicable to solute
In very dilute solution Lim X2→ 0 f2/X2 for solid line = limiting slope = k
For Henry’s line (dotted) X2→ 0
Lim X2→ 0 f21/X2 = k (4)
Therefore
Lim X2→ 0 f2/X2 = Lim X2→ 0 f21/X2 = k (5)
Since f21 = kX2 (5) becomes
Lim X2→ 0 f2/kX2 =1 (6) Rv

19. If (3) & (6)should hold good simultaneously f2o = k
Form figure , this state can be found by extrapolating the dotted line to a concentration X2 =1
From Henry’s law f21 = k X2, when X2 = 1, f21 = k
This fugacity is the standard fugacity for the solute
Standard fugacity f2o is a hypothetical quantity and is not equal to the fugacity f2.of the pure solute.

20. Standard state for the solute is chosen as the hypothetical liquid solution at the given temperature and 1 atm total pressure – mole fraction of solute is unity and behaves ideally obeying Henry’s law
If this law is obeyed over entire range of composition X2 = 0 to 1
Then
a2 = f2/f2o = f2/k =kX2/k =X2 (7)

21. Thus as X2 → 1 a2 becomes unity and the activity at any other concentration will be equal to X2
The activity of the pure solute a2. is different from a2o
For any mole fraction Xj,γX is aj/Xj
For a solution behaving ideally over the whole range of concentration the activity will be equal to its mole fraction

22. For non – ideal solution the standard state has no reality and it is preferable define the standard state in terms of reference state.
The activity coefficient becomes equal to unity as X2 → 0
Thus possible to choose the infinitely dilute solution as the reference state, Such that as
X2 → 0, γX → 1 or a2 → X2

23. Practical system
Molality is widely used to express concentration than mole fraction
In very dilute solution molality is proportional to mole fraction
Henry’s law is valid under these conditions
ie f2 = km2
If f2 is plotted vs m2,
k can be obtained from the limiting slope of the curve
figure

24. The choice of standard fugacity should be as
m2→0 , a2/m2 → 1 or
Lim m2 → 0 a2/m2 = Lim m2 → 0 f2/f2om2 = 1
Under such limiting conditions, Henry’s law is valid ie Lim m2 → 0 f2/f2om2 = 0

25. The standard state of the solute is the state, which at the fugacity that the solute of unit molality would have , Henry’s law is obeyed at this concentration
With increasing dilution – solute approaches ideal behaviour

26. A similar cure can be obtained by plotting a2 vs m2
Since the mole fraction scale has limits of 0 to 1 – choice of X2 =1 as standard state is quite natural
Theoretically molality has no upper limit, but in practice the upper limit is the solubility of the substance

27. The choice of standard state m2o =1 mole/kg is arbitrary
The standard state is the hypothetical 1 molal solution obtained by extrapolating Henry’s law line to m2= 1
If the concentration of solute is expressed in molarity(c) the standard state is chosen as the hypothetical state obtained when Henry’s law plot is extrapolated to c2 = 1 mol/L

28. Solids
The activity of pure liquid or pure solid solvent , at atmospheric pressure, is taken as unity at each temperature
The corresponding reference state in the pure liquid or solid at 1 atm. Pressure, the activity coefficient is equal to unity.
With increasing dilution of the solution the mole fraction of solute tends to zero and that of the solvent to unity.

29. By the equation ai =fi/f1o , the activity of the solvent is equivalent to f1/f1o
Therefore from fi = Nifio, for ideal solution the activity of the solvent should always be equal to its mole fraction at 1 atm pressure.
For non-ideal solution- the deviation of ai/Ni from unity at 1 atm.

30. Pressure may be taken as a measure of the departure from ideal behaviour.
Since activities of liquids are not greatly affected by pressure , this conclusion is generally applicable provided pressure is not too high.