Debye-Hückel theory: a way to calculate ionic activity coefficients for dilute electrolyte solutions.

Thermodynamic behavior → solution thermodynamics → constituent characteristic behavior

Activity:

(1)ideal solution a_i=C_i; concentrated constituent solvent if aqueous Raoultian, H2O=Xᵢ pure s.s. (2)solute dilute solutio, C= 1M s.s., aᵢ=Cᵢ Herian

(3)activity coeff.

A. γᵢ=aᵢ/Xᵢ or aᵢ/Cᵢ<1 比較喜歡變成水溶液

B. γᵢ>1活性強, 彼此不喜歡在一起, 只是因ΔS而在一起

ex. desulfurization: 加入某種物質,使γ_sulfur>>1讓S活性增加,與其他元素作用, 從Fe中去掉

Chemical potential, μᵢ=μᵢº+RTlnaᵢ i.e. aᵢ=Cᵢγᵢ

Electrochemical potential, η μᵢ=ηᵢ= μᵢº+RTlnaᵢ+zᵢΔϕ

⸫μ_(elec)=ηᵢ-μᵢ_(chem)= μᵢ-μᵢ_(chem)= μᵢ-( μᵢº+RTlnCᵢ) i.e. aᵢ=Cᵢ if ideal

⸪ μᵢ=ηᵢ= μᵢº+RTlnaᵢ=μᵢº+RTlnCᵢ+RTlnγᵢ_(elec)

→μ_(elec)=RTlnγᵢ_(elec)=∫ ϕ_atmdQ, ϕ: surrounding potential(ion atmosphere)

ϕ(r)=(aϕₐ/r)∙exp[(a-r)/X_A], i.e. ϕₐ=σa/ [ϵ ϵ₀(1+a/X_A)] & σ=Q/4πa² a=particle size

→ϕ(r)=(Q/4πrϵ ϵ₀)∙{exp[(a-r)/X_A]/(1+a/X_A)}

ϕ(r)=ϕ_central+ϕ_atm, ϕ_atm=ϕ(r)-ϕ_central=(Q/4πrϵ ϵ₀)∙{exp[(a-r)/X_A]/(1+a/X_A)-1}, ϕ_central=Q/4πrϵ ϵ₀

at r=a, ϕ_atm=(Q/4πaϵ ϵ₀)∙[ X_A/( X_A+a)-1]=-(Q/4πϵ ϵ₀)[1/( X_A+a)]

→μ_(elec)=∫ ϕ_atmdQ=-(1/4πϵ ϵ₀)[1/( X_A+a)]∫QdQ= -(zᵢ²e²/8πϵ ϵ₀)[1/( X_A+a)]

lnγᵢ_(elec)= -(zᵢ²e²/8πRTϵ ϵ₀)[1/( X_A+a)]

ex. 電解質溶液, NaCl solution → 獲得Na⁺, Cl⁻平均mole 值ν₊和ν₋, ν=ν₊+ν₋

平均化學勢 μ_±=(ν₊μ₊+ν₋μ₋)/ν

平均activity, a_± μ_±=μ_±º+RTlna_±=(ν₊/ν)(μ₊º+RTlna₊)+(ν₋/ν)(μ₋º+RTlna₋)=

(ν₊μ₊º+ν₋μ₋º)/ν+RT[lna₊^ν₊/ν+lna₋^ν₋/ν]=(ν₊μ₊º+ν₋μ₋º)/ν+RTln(a₊^ν₊/ν∙a₋^ν₋/ν)

⸫ μ_±º=(ν₊μ₊º+ν₋μ₋º)/ν, a_±= a₊^ν₊/ν∙a₋^ν₋/ν=C_±γ_±

a_±= a₊ ^ν₊/ν∙a₋^ν₋/ν=C_±γ_± → a_±^ν= a₊ ^ν₊∙a₋^ν₋=C_±^νγ_±^ν → (C₊γ₊)^ν₊(C₋γ₋)^ν₋=C_±^νγ_±^ν ⸫C_±^ν=C₊^ν₊C₋^ν₋, γ_±^ν=γ₊^ν₊γ₋^ν₋

⸪C₊=ν₊C & C₋=ν₋C → C_±^ν=(ν₊C)^ν₊ (ν₋C)^ν₋=(ν₊^ν₊∙ν₋^ν₋)C^ν → C_±=C(ν₊^ν₊∙ν₋^ν₋)^1/ν

γ_±^ν=γ₊^ν₊γ₋^ν₋ → 取ln, νlnγ_±=ν₊lnγ₊+ν₋lnγ₋ ⸫lnγ_±=(ν₊/ν)lnγ₊+(ν₋/ν)lnγ₋

lnγ_±=(ν₊/ν)lnγ₊+(ν₋/ν)lnγ₋= -(e²/8πRTϵ ϵ₀)[1/( X_A+a)][(ν₊z₊²+ν₋z₋²)/ν]

由於電荷平衡, |ν₊z₊|=|ν₋z₋| and ν=ν₊+ν₋

lnγ_±= -(e²/8πRTϵ ϵ₀)[1/( X_A+a)][(ν₊z₊²+ν₋z₋²)/(ν₊+ν₋)]= -(e²/8πRTϵ ϵ₀)[|z₊z₋|/( X_A+a)]

pf: (ν₊z₊²+ν₋z₋²)/(ν₊+ν₋)=|ν₊z₊|(z₊+z₋)/(ν₊+ν₋)=|ν₊z₊|(z₊+z₋)/(ν₊+ν₋)∙|z₊z₋|/|z₊z₋|

=[|ν₊z₊|(z₊+z₋)/(ν₊|z₊z₋|+ν₋|z₊z₋|)]∙|z₊z₋|=[|ν₊z₊|(z₊+z₋)/|ν₊z₊|(z₋+z₊)]∙|z₊z₋|=|z₊z₋|

簡化 lnγ_±= -A|z₊z₋|I^½/(1+BaI^½)= -0.05092|z₊z₋|I^½/(1+3.29aI^½) i.e. I=½Σzᵢ²Cᵢ