Activity Gets Promoted

Before grappling with the chemical potentials of ionic solutions, we took a moment to update the colligative property formulas for ionic solutes. Since these properties depend not on what is dissolved in solution but how much, the fact that salts break into ions needs to be addressed, which is accomplished by i, the van't Hoff factor:

• freezing point depression: ∆T_f=iK_fm₂
• boiling point elevation: ∆T_b=iK_bm₂
• osmotic pressure: Π=iM₂RT

where i = (actual number of free particles)/(number before dissociation).

For very dilute solutions, i is approximately equal to the number of ions in the formula (i=2 for KCl, i=3 for Na₂SO₄, etc.). As the concentration increases, the actual van't Hoff factor will be less (sometimes very much so) than these stoichiometrically derived values. We also saw that we could relate i to the degree of dissociation α by the formula α=(i-1)/(ν-1).

Although earlier we defined the chemical potential of a solution in terms of the Raoult's standard state [purity and mole fraction], it is more convenient to use a standard state based on molality:

For a solute in an ideal solution: μ_i = μ°_i + RT lnm_i
For a solute in a real solution: μ_i = μ°_i + RT lna_i

Henceforth the symbol ° will mean 1 bar for gases and 1 m for solutions. Since we now have six different chemical potential expressions [ideal gas, real gas, ideal solution, real solution, liquid, solid] it is customary to promote the idea of activity so that it covers all possibilities. We will then use a general expression μ_i = μ°_i + RT lna_i with the following mappings:

• ideal gas: a_i = P_i [partial pressure]
  
• real gas: a_i = f_i [fugacity]
  
• ideal soln: a_i = m_i [molality]
  
• real soln: a_i = a_i [activity]
  
• liquid: a_i = 1 [assuming normal pressure]
  
• solid: a_i = 1 [assuming normal pressure]

Now, for any reversible chemical reaction aA( )+bB( ) ↔ cC( )+dD( ) we can construct reaction quotient and equilibrium constant expressions immediately in terms of the relevant activities.

Tomorrow we will go through the powerful Debye-Hückel theory, which gives considerable insight into ionic solutions, including a method to calculate activity coefficients. Also, next hw set..