ν + and ν − are stoichiometric coefficients whereas z+ and z- are the cationc and anionic charges, respectively. For charge neutrality in solution we have ν + z+ = ν − z−

• •

•

If the solution is a very dilute, aqueous solution, the water solvent obeys the usual rules in the dilute limit… µ1 = µ1• + RT ln a1 ≈ µ1• + RT ln χ1 The cationic and anionic solutes obey the equation: µ2 = ν + a+ +ν − a− = v+ ( µ+D + RT ln a+ ) + ν − ( µ−D + RT ln a− )

This is not a useful definition of the chemical potential of an electrolyte solution because we cannot physically separate the cationic and anionic fractions. Thus the components of the chemical potential are not measureable. We must formulate instead a mean chemical potential ν µ +ν µ ν ν µ± = + + − − = + ( µ+D + RT ln a+ ) + − ( µ−D + RT ln a− )

ν

=

•

(ν ν 1

ν

µ+D +ν − µ−D ) + RT ln ( aν+ aν− +

−

)

1/ν

= µ±D + RT ln a±

where ν = ν + + ν − and a± is called the mean activity. • The mean chemical potential and mean activity are required in order to obtain a mean standard state that is realistic. A mean standard state in turn is necessary because a pure cationic and/or anionic standard state cannot be physically realized. Only a mean activity is measureable. We take

a± = ( a+ a−

)

ν + ν − 1/ν

•

+

ν

1/ν

⎛ ⎛ γ m ⎞ν ⎛ γ m ⎞ν ⎞ = ⎜⎜ + D+ ⎟ ⎜ − D− ⎟ ⎟ ⎜⎝ m ⎠ ⎝ m ⎠ ⎟ ⎝ ⎠ +

−

(

ν+ ν−

= γ+ γ−

)

1/ν

b g bγ g

γ ± is the mean activity coefficient γ ν± = γ +

ν+

ν + /ν

⎛ m+ ⎞ ⎜ D⎟ ⎝m ⎠ ν−

ν − /ν

⎛ m− ⎞ ⎜ D⎟ ⎝m ⎠

. It is the only activity coefficient that is measureable from an electrolyte solution. −

ν + /ν

⎛m ⎞ = γ ± ⎜ +D ⎟ ⎝m ⎠

ν − /ν

⎛ m− ⎞ ⎜ D⎟ ⎝m ⎠

•

We define the molalities m± of the ions in terms of the molality m of the original

salt: m± = ν ± m2 , and the mean molaltity m± = ( mν++ mν−− ) . Then the mean acitivity 1/ν

is: ν + /ν

⎛m ⎞ a± = γ ± ⎜ +D ⎟ ⎝m ⎠

ν − /ν

⎛ m− ⎞ ⎜ D⎟ ⎝m ⎠

⎛m = γ ± ⎜ ±D ⎝m

ν + /ν

⎞ ⎛ν +m ⎞ ⎟ = γ± ⎜ D ⎟ ⎠ ⎝ m ⎠

ν − /ν

⎛ν −m ⎞ ⎜ D ⎟ ⎝ m ⎠

+

= γ ±ν ν+ /νν ν−

−

/ν

⎛ m2 ⎞ ⎜ D⎟ ⎝m ⎠

•

The mean chemical potential is now: + − µ ⎡ ⎛ m ⎞⎤ µ± = 2 = µ±D + RT ln a± = µ±D + RT ln ⎢γ ±ν ν+ /νν ν− /ν ⎜ 2D ⎟ ⎥ ν ⎝ m ⎠⎦ ⎣ + − ⎡ ⎛ m ⎞⎤ ∴ µ2 = νµ ±D +ν RT ln ⎢γ ±ν ν+ /νν ν− /ν ⎜ 2D ⎟ ⎥ ⎝ m ⎠⎦ ⎣ ⎡ν ν+ /νν ν− = νµ +ν RT ln ⎢ D ⎣⎢ m +

D ±

−

/ν

⎤ ⎥ + ν RT ln [γ ± m2 ] ⎦⎥

B. Debye-Huckel Theory •

Debye-Huckel theory treats the properties of electrolyte solutions. This theory views a the non-ideality of an electrolyte solution as arising from Coulomb forces between ions in a continuous solution with dielectric constant εr of the form: Q2 f (r ) = where Q=ze 4πε 0ε r r 2 Q and where the electrical potential is ψ ( r ) = 4πε 0ε r r • Theprimary assumption of D-H theory:

The dissolved electrolyte is completely dissociated; it is a strong electrolyte . Ions are spherical and are not polarized by the surrounding electric field . The solvent plays no role other than providing a medium of constant dielectric constant d) Individual ions surrounding a "central" ion can be represented by a statistically averaged cloud of continuous charge density, with a minimum distance of closest approach. This model is shown at right . a) b) c)

•

•

The average effect of surrounding ions on a given ion is represented by a continuous charge distribution or “ion atmosphere”. If the electrical potential at point is ψ the energy of E of a charge Q=ze at that point is E = Qψ . The charge density ρ around an ion is assumed to be

ρ = ∑ Ci Qi e− E / k T = ∑ Ci Qi e− Q ψ / k T i

B

i

B

i

e 2ψ Ci zi2 ∑ k BT i i Note: the last step is only possible if zi eψ k BT . This is a major limitation of DH theory because it states the inter-0ionic energy is small compared to the thermal energy, which limits D-h theory to very dilute solutions. With this expression for the charge density the electrical potential can be obtained by solving the Poisson-Boltzmann equation: ρ (r ) d 2ψ e 2ψ = − ≈ Ci zi2 = κ 2ψ ∑ 2 ε 0ε r ε 0ε r k B T i dr = ∑ Ci Qi e − zi eψ / kBT ≈ −

•

•

•

•

•

•

e2 Ci zi2 . The quantity κ −1 is called the Debye length ∑ ε 0 ε r k BT i and measures the thickness of the ion atmosphere or the range over which the electric field of an ion extends. It is possible to solve the Poisson-Boltzmann equation and obtain an expression for the average electrical potential ψ DH in the dilute limit ze Q ψ DH ≈ − κ =− κ 4πε 0ε r 4πε 0ε r The contribution to the free energy from the charge is ze κ z 2e2 ∆G = ∫ ψ DH dQ = − 8πε 0ε r 0

The parameter κ 2 =

Because this contribution to the free energy is strictly non-deal and arises from Coulombic forces, it can be treated as an activity coefficient. For the ith ion with charge Qi = zi e we express the electrical contribution to ∆G as an activity coefficient which on a per mole basis is : RT ln γ i = −

•

•

•

κ zi2e 2 . 8πε 0ε r

On a per ion basis the electrical contribution to the non-ideality is κ zi2e2 ln γ i = − 8πε 0ε r k BT For a binary salt the mean activity coefficient is 1/ν ⎛ ⎞ e 2κ ln γ ± = ln ( γ ν+ + γ ν− − ) = − z+ z− ⎜ ⎟ ⎝ 8πε 0ε r k BT ⎠ For aqueous solutions at T=298K ln γ ± = −1.173 z+ z− I , which is called the Debye-Huckel limiting law •

The ionic strength is Ι is defined as 1 I = ∑ ( Ci + zi2+ + Ci − zi2− ) 2 i

• • •

• •

C±i is the concentration of the ith cationic (+) or anionic (-) species with corresponding charge z±i. It is common to see the D-H limiting law written in terms of common logarithms: log γ ± = −0.5091 z+ z− I The DH limiting law is valid only for dilute solutions and for which concentrations are