Activities and Activity Coefficients

The Definition of the Activity For any real system, the chemical potential for the solute (or solvent) is given by

Activities of Pure Solids/Liquids The chemical potential is essentially invariant with pressure for condensed phases

Pure Solids and Pure Liquids For a pure solid or a pure liquid at standard to moderately high pressures or a J = 1

Activities in Gaseous Systems The chemical potential of a real gas is written in terms of its fugacity

Define the Activity Coefficient The activity coefficient (  J ) relates the activity to the concentration terms of interest. In gaseous systems, we relate the fugacity (or activity) to the ideal pressure of the gas via

Activities in Solutions Two conventions Convention I Raoult’s Law is applied to both solute and solvent Convention II Raoult’s Law is applied to the solvent; Henry’s Law is applied to the solute

Convention I We substitute the activity of the solute and solvent into our expressions for Raoult’s Law

Convention I (cont’d) Vapour pressure above real solutions is related to its liquid phase mole fraction and the activity coefficient Note – as X J  1  J I  1 and P J  P J id

Convention II The solvent is treated in the same manner as for Convention I For the solute, substitute the solute activity into our Henry’s Law expression

Convention II (cont’d) Vapour pressure above real dilute solutions is related to its liquid phase mole fraction and activity coefficient Note – as X J  0  J II  1 and P J  P J id

Convention II - Molalities For the solute, we use the molality as our concentration scale Note – as m J  0  J m  1 and a J (m)  m J

The Gibbs-Duhem Equation The Gibbs-Duhem gives us an interrelationship amongst all partial molar quantities in a mixture

Thermodynamics of Ions in Solutions Electrolyte solutions – deviations from ideal behaviour occur at molalities as low as 0.01 mole/kg. How do we obtain thermodynamic properties of ionic species in solution?

Thermodynamics (II) For the H + (aq) ion, we define  f H  = 0 kJ/mole at all temperatures S  = 0 J/(K mole) at all temperatures  f G  = 0 kJ/mole at all temperatures

Activities in Electrolyte Solutions For the following discussion Solvent “s” Cation “+” Anion “-“ Consider 1 mole of an electrolyte dissociating into + cations and - anions G = n s  s + n  = n s  s + n +  + + n -  - Note – since =   = +   -

The Mean Ionic Chemical Potential We define   =  / We now proceed to define the activities  =  + RT ln a  + =  +  + RT ln a +  - =  -  + RT ln a -   =    + RT ln a 

The Relationship Between a and a  Since   =  /  =  + RT ln a = (    + RT ln a  ) Since    =  / This gives us the relationship between the electrolyte activity and the mean activity (a  ) = a

The Relationship Between a , a - and a + We note that  = +   - and   =  / This gives us the following relationship (    + RT ln a  ) = + (  +  + RT ln a + ) + - (  -  + RT ln a - ) Since    = +  +  + -  -  (a  ) = (a + ) + (a - ) -

Activities in Electrolyte Solutions The activities of various components in an electrolyte solution are defined as follows a + =  + m + a - =  - m - a + =  + m + As with the activities (   ) = (  + ) + (  - ) - (m  ) = (m + ) + (m - ) -

The Chemical Potential Expression This can be factored into two parts The ideal part Deviations from ideal behaviour

KCl CaCl 2 H 2 SO 4 HCl LaCl 3 Activity Coefficients As a Function of Molality Data obtained from Glasstone et al., Introduction to Electrochemistry, Van Nostrand (1942). CRC Handbook of Chemistry and Physics, 63 rd ed.; R.C. Weast Ed.; CRC Press, Boca Raton, Fl (1982).

Determination of Activity Coefficients in Solution Two ways Use the Gibbs-Duhem equation and  for the solvent to estimate  for the solute. Determination of osmotic coefficients from colligative properties or vapour pressure measurements (we will examine later)

Estimates of Activity Coefficients in Electrolyte Solutions A few have been proposed to allow the theoretical estimation of the mean activity coefficients of an electrolyte. Each has a limited range of applicability.

u This is valid in the up to a concentration of molal! The Debye Hűckel Limiting Law Z + = charge of cation; z - = charge of anion

Debye Hűckel Extended Law This equation can reliably estimate the activity coefficients up to a concentration of 0.10 mole/kg. B = 1.00 (kg/mole) 1/2

The Davies Equation This equation can reliably estimate the activity coefficients up to a concentration of 1.00 mole/kg. k = 0.30 (kg/mole)

The Chemical Potential for Real Gases The fugacity (f) represents the chemical potential of a real gas. Define the fugacity coefficient   = f / P For a real gas

Obtaining Fugacity Coefficients Comparing the chemical potential of the real gas to the chemical potential of an ideal gas at the same pressure

Calculating Fugacity Coefficients The fugacity coefficients are obtained from the compression factors (Z) as shown below