中和滴定曲線 (4回目)

中和滴定曲線 (4回目)

2026/1/24(土)

中和滴定曲線 (4回目)

(neutralization titration)

3価の酸と1価の強塩基の滴定曲線
c,c',V,V'は元の酸･塩基のモル濃度と体積
Ca = cV/(V+V') , Cb = c'V'/(V+V') は滴定中の酸･塩基のモル濃度
水溶液中に[H₃A],[H₂A^-],[HA^2-],[A^3-],[H⁺],[B⁺],[OH^-]が存在する
Ca = [A^2-] + [HA^-] + [H₂A]… 酸のモル濃度
Cb = [B⁺] + [BOH]… 塩基のモル濃度
Kw = [H⁺][OH^-]                    … 水のイオン積
Ka1 = [H⁺][H₂A^-]/[H₃A]              … 酸の電離定数(第1)
Ka2 = [H⁺][HA^2-]/[H₂A^-]             … 酸の電離定数(第2)
Ka3 = [H⁺][A^3-]/[HA^2-]              … 酸の電離定数(第3)
[B⁺] + [H⁺] = 3[A^3-] + 2[HA^2-] + [H₂A^-] + [OH^-] … 電気的中性
Ka = Ka1Ka2Ka3 = [H⁺]³[A^3-]/[H₃A]

中和滴定曲線 (1回目)
より
[A^n-] = CaKa/Σ_j=0ⁿ([H⁺]^jΠ_i=1^n-jKa_i)
Caα_k = [H_n-_kA^k^-] = [H⁺]^n-k[A^n-]/(Π_i=k+1ⁿKa_i)
x = Σ(k[H_n-kA^k-]) = Σ{k[H⁺]^n-k(Π_i=2^kKa_i)}[A^n-]/(Π_i=2ⁿKa_i)
[B^m+] = CbKb/Σ_j=0^m([OH^-]^jΠ_i=1^m-jKb_i)
Cbβ_k = [B(OH)_m-k^k+] = [OH^-]^m-k[B^m+]/(Π_i=k+1^mKb_i)
y = Σ(k[B(OH)_m-k^k+]) = Σ{k[OH^-]^m-k(Π_i=2^kKb_i)}[B^m+]/(Π_i=2^mKb_i)
[OH^-] = Kw/[H⁺]
[H⁺]² - (x - y)[H⁺] - Kw = 0

n = 3
[A^n-] = CaKa/Σ_j=0ⁿ([H⁺]^jΠ_i=1^n-jKa_i)より
[A^3-] = CaKa/Σ_j=0³([H⁺]^jΠ_i=1^3-jKa_i)
= CaKa/([H⁺]⁰Π_i=1^3-0Ka_i+[H⁺]¹Π_i=1^3-1Ka_i+[H⁺]²Π_i=1^3-2Ka_i+[H⁺]³Π_i=1^3-3Ka_i)
= CaKa/(Ka₁Ka₂Ka₃+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)
= CaKa/(Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)

Caα_k = [H_n-_kA^k^-] = [H⁺]^n-k[A^n-]/(Π_i=k+1ⁿKa_i)より
Caα₁ = [H₂A^-] = [H⁺]^3-1[A^3-]/(Π_i=1+1³Ka_i) = [H⁺]²[A^3-]/(Ka₂Ka₃)
Caα₂ = [HA^2-] = [H⁺]^3-2[A^3-]/(Π_i=2+1³Ka_i) = [H⁺][A^3-]/Ka₃
Caα₃ = [A^3-] = CaKa/(Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)
x = Σ(k[H_n-kA^k-]) = [H₂A^-] + 2[HA^2-] + 3[A^2-]
= [H⁺]²[A^3-]/(Ka₂Ka₃) + 2[H⁺][A^3-]/Ka₃ + 3[A^3-]
= ([H⁺]² + 2[H⁺]Ka₂ + 3Ka₂Ka₃)[A^3-]/(Ka₂Ka₃)

Cbβ = [B⁺] = Cb
y = Σ(k[B(OH)_m-k^k+]) = [B⁺] = Cb

-(x-y)[H⁺] = -{([H⁺]² + 2[H⁺]Ka₂ + 3Ka₂Ka₃)[A^3-]/(Ka₂Ka₃) - Cb}[H⁺]
= [H⁺]Cb - ([H⁺]³ + 2[H⁺]²Ka₂ + 3[H⁺]Ka₂Ka₃)[A^3-]/(Ka₂Ka₃)
を
[H⁺]² - (x - y)[H⁺] - Kw = 0
に代入
[H⁺]² + [H⁺]Cb - ([H⁺]³+2[H⁺]²Ka₂+3[H⁺]Ka₂Ka₃)[A^3-]/(Ka₂Ka₃) - Kw = 0
に
[A^3-]/(Ka₂Ka₃) = Ca{Ka/(Ka₂Ka₃)}/(Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)
= CaKa₁/(Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)
を代入
(Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)[H⁺]²
+ (Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)[H⁺]Cb
- ([H⁺]³ + 2[H⁺]²Ka₂ + 3[H⁺]Ka₂Ka₃)CaKa₁
- (Ka+[H⁺]Ka₁Ka₂+[H⁺]²Ka₁+[H⁺]³)Kw
= [H⁺]²Ka+[H⁺]³Ka₁Ka₂+[H⁺]⁴Ka₁+[H⁺]⁵+[H⁺]KaCb
+ [H⁺]²Ka₁Ka₂Cb+[H⁺]³Ka₁Cb+[H⁺]⁴Cb
- [H⁺]³CaKa₁-2[H⁺]²CaKa₁Ka₂-3[H⁺]CaKa-KaKw
- [H⁺]Ka₁Ka₂Kw-[H⁺]²Ka₁Kw-[H⁺]³Kw
= [H⁺]⁵+[H⁺]⁴Ka₁+[H⁺]⁴Cb+[H⁺]³Ka₁Ka₂+[H⁺]³Ka₁Cb- [H⁺]³CaKa₁+ [H⁺]²Ka+[H⁺]²Ka₁Ka₂Cb-2[H⁺]²CaKa₁Ka₂-[H⁺]²Ka₁Kw
+ [H⁺]KaCb-3[H⁺]CaKa-KaKw-[H⁺]Ka₁Ka₂Kw-[H⁺]³Kw
= [H⁺]⁵+(Ka₁+Cb)[H⁺]⁴+{(Ka₂+Cb-Ca)Ka₁-Kw}[H⁺]³
+ (Ka₂(Ka₃+Cb-2Ca)-Kw)Ka₁[H⁺]²+{(Cb-3Ca)Ka₃-Kw}Ka₁Ka₂[H⁺]-KaKw = 0

[H⁺]⁵ + (Ka1 + Cb)[H⁺]⁴ + {Ka1(Ka2 + Cb - Ca) - Kw}[H⁺]³
+ Ka1{Ka2(Ka3 + Cb - 2Ca) - Kw}[H⁺]²
+ Ka1Ka2{Ka3(Cb - 3Ca) - Kw}[H⁺]
- Ka1Ka2Ka3Kw = 0

Ca = cV/(V+V') , Cb = c'V'/(V+V') は滴定中の酸･塩基のモル濃度
Caα₁ = [H⁺]²[A^3-]/(Ka2Ka3)
Caα₂ = [H⁺][A^3-]/Ka3
Caα₃ = [A^3-] = CaKa/(Ka+[H⁺]Ka1Ka2+[H⁺]²Ka1+[H⁺]³)
Cbβ = [B⁺] = Cb

V'を求める式
[H]⁵ + {Ka1 + c'V'/(V+V')}[H⁺]⁴
+ {Ka1{Ka2 + (c'V'-cV)/(V+V')} - Kw}[H⁺]³
+ Ka1{Ka2{Ka3 + (c'V'-2cV)/(V+V')} - Kw}[H⁺]²
+ Ka1Ka2{Ka3(c'V'-3cV)/(V+V') - Kw}[H⁺] - Ka1Ka2Ka3Kw = 0

(V+V')[H⁺]⁵ + {(V+V')Ka1 + c'V'}[H⁺]⁴
+ {Ka1{(V+V')Ka2 + c'V' - cV} - (V+V')Kw}[H⁺]³
+ Ka1{Ka2{(V+V')Ka3 + c'V' - 2cV} - (V+V')Kw}[H⁺]²
+ Ka1Ka2{Ka3(c'V' - 3cV) - (V+V')Kw}[H⁺]
- (V+V')Ka1Ka2Ka3Kw = 0

A = [H⁺]⁵ + Ka1[H⁺]⁴ + Ka1Ka2[H⁺]³ - Ka1Kw[H⁺]³
+ Ka1Ka2Ka3[H⁺]² - Ka1Kw[H⁺]² - Ka1Ka2Kw[H⁺] - Ka1Ka2Ka3Kw
= [H⁺]{[H⁺]{[H⁺]{[H⁺]([H⁺] + Ka1) + Ka1(Ka2 - Kw)}
+ Ka1(Ka2Ka3 - Kw)} - Ka1Ka2Kw} - Ka1Ka2Ka3Kw
B = -(-Ka1c[H⁺]³ - 2cKa1Ka2[H⁺]² - 3cKa1Ka2Ka3[H⁺] + A)
= cKa1[H⁺]³ + 2cKa1Ka2[H⁺]² + 3cKa1Ka2Ka3[H⁺] - A
= [H⁺]{[H⁺]cKa1([H⁺] + 2Ka2) + 3cKa1Ka2Ka3} - A
C = c'[H⁺]⁴ + c'Ka1[H⁺]³ + c'Ka1Ka2[H⁺]² + c'Ka1Ka2Ka3[H⁺]
= c'[H⁺]{[H⁺]³ + Ka1[H⁺]² + Ka1Ka2[H⁺] + Ka1Ka2Ka3}
= c'[H⁺]{[H⁺]{[H⁺]² + Ka1[H⁺] + Ka1Ka2} + Ka1Ka2Ka3}
= c'[H⁺]{[H⁺]{[H⁺]([H⁺] + Ka1) + Ka1Ka2} + Ka1Ka2Ka3}

A = [H⁺]{[H⁺]{[H⁺]{[H⁺]([H⁺] + Ka1) + Ka1(Ka2 - Kw)}
+ Ka1(Ka2Ka3 - Kw)} - Ka1Ka2Kw} - Ka1Ka2Ka3Kw
B = [H⁺]{[H⁺]cKa1([H⁺] + 2Ka2) + 3cKa1Ka2Ka3} - A
C = c'[H⁺]{[H⁺]{[H⁺]([H⁺] + Ka1) + Ka1Ka2} + Ka1Ka2Ka3} + A
V' = (B/C)V

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