Activity Standard states Gases Fugacity of real gases

Activity Standard states

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

• 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.

• 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

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

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

• 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 ‡ c. RT for non ideal gases γc ‡ γp

• 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

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

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 • Letf 1. be fugacity of pure solvent • Pure solvents under these conditions is the standard state so f 1. = f 1 o • On adding solute, if fugacity of solvent becomes f 1, f 1<f 1 o

• Therefore activity in this case is a 1 = f 1/f 1 o For an ideal solution • f 1 = f 1 o X 1 = f 1. X 1 • So a 1 = f 1 o. X 1/f 1 o = X 1 ie a 1 = X 1 • ie activity of solvent in an ideal solution at 1 atm is equal to its mole fraction

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

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

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

• Activity coefficient a 2/X 2 approaches one as X 1 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

Rational system • If mole fraction of solute is X 2 • Henry’s law is applicable to solute , f 2 =k. X 2 • 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 a 2/X 2 tends to one as X 2 tends to zero (1) Figure

• For very dilute solution as X 2 tends to zero actual cure merges with Henry’s law line • Since a 2 = f 2/f 2 o (2) • So (1) can be written as • Lim X 2→ 0 a 2/X 2 = Lim X 2 → 0 f 2/f 2 o. X 2 = 1 (3)

• Since Henry’s law is also applicable to solute • In very dilute solution Lim X 2→ 0 f 2/X 2 for solid line = limiting slope = k • For Henry’s line (dotted) X 2→ 0 Lim X 2→ 0 f 21/X 2 = k (4) Therefore Lim X 2→ 0 f 2/X 2 = Lim X 2→ 0 f 21/X 2 = k (5) Since f 21 = k. X 2 (5) becomes Lim X 2→ 0 f 2/k. X 2 =1 (6)

• If (3) & (6)should hold good simultaneously f 2 o = k • Form figure , this state can be found by extrapolating the dotted line to a concentration X 2 =1 • From Henry’s law f 21 = k X 2, when X 2 = 1, f 21 = k • This fugacity is the standard fugacity for the solute • Standard fugacity f 2 o is a hypothetical quantity and is not equal to the fugacity f 2. of the pure solute.

• 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 X 2 = 0 to 1 Then a 2 = f 2/f 2 o = f 2/k =k. X 2/k =X 2 (7)

• Thus as X 2 → 1 a 2 becomes unity and the activity at any other concentration will be equal to X 2 • The activity of the pure solute a 2. is different from a 2 o • 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

• 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 X 2 → 0 • Thus possible to choose the infinitely dilute solution as the reference state, Such that as X 2 → 0, γX → 1 or a 2 → X 2

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 f 2 = km 2 • If f 2 is plotted vs m 2, • k can be obtained from the limiting slope of the curve • figure

• The choice of standard fugacity should be as m 2→ 0 , a 2/m 2 → 1 or Lim m 2 → 0 a 2/m 2 = Lim m 2 → 0 f 2/f 2 om 2 = 1 Under such limiting conditions, Henry’s law is valid ie Lim m 2 → 0 f 2/f 2 om 2 = 0

• 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

• A similar cure can be obtained by plotting a 2 vs m 2 • Since the mole fraction scale has limits of 0 to 1 – choice of X 2 =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

• The choice of standard state m 2 o =1 mole/kg is arbitrary • The standard state is the hypothetical 1 molal solution obtained by extrapolating Henry’s law line to m 2= 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 c 2 = 1 mol/L

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.

• By the equation ai =fi/f 1 o , the activity of the solvent is equivalent to f 1/f 1 o • 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.

• 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.