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Kinetics_Math_ver1.0

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  • 1. 速度論の数学的基礎 1. 変数分離形積分法 (同時方程式) 2. 定数係数1階線形微分方程式の公式 3. 零次反応 4. 一次反応 5.二次反応 6. 二成分反応 7. 可逆一次反応Ⅰ 8. 可逆一次反応Ⅱ 9. 半減期 Presented by S. Kume 140326 ver 1.0
  • 2. 1. 変数分離形積分法 (同時方程式) ネイピア数 e ≅ 2.718 ex = exp(x) ln(ex) = x ! dy dx = f (x) " g(y) # dy g(y) = f (x)dx dy g(y) $ = f (x)dx$ + C ! dy g(y) " = ln y 例題 ! dy dt = "ay # dy y = "adx dy y $ = "a dx$ + C 変数分離形積分法により ! dy y " = ln y,#a dx" + C = #ax + C ! dy y " = ln y ,#a dx" + C = #ax + C ln y = "ax + C # y = e"ax+C
  • 3. 2. 定数係数1階線形微分方程式の公式 ! dy dx + ay = Q(x) "y = e#ax eax Q(x)dx$ + C{ } 導入方法 ! dy dx + ay = Q(x) " eax dy dx + aeax y = eax Q(x) " eax dy dx + d dx eax # $ % & ' (y = eax Q(x) eax の微分はaeaxとなる性質を利用して ! d dx f (x)g(x) = df (x) dx g(x) + f (x) dg(x) dx 積の微分公式より ! eax dy dx + d dx eax " # $ % & 'y = d dx eax y( ) ! d dx eax y( ) = eax Q(x) つまり 両辺を x で積分すると eax y = eax Q(x)dx" + C # y = e$ax eax Q(x)dx" + C{ }
  • 4. ! A k " #" P 零次反応: 実験的な反応速度が濃度に比例しない ! v = d[P] dt = " d[A] dt = k ! " d[A] dt = k # d[A] = "kdt d[A] =$ " k dt + C #$ [A] = "kt + C t = 0ならば、[A] = [A]0 ! [A] = "kt + C # [A]0 = "k $ 0 + C # C = [A]0 すなわち [A] = [A]0 " kt 3. 零次反応
  • 5. ! A k " #" P 一次反応: 実験的な反応速度が濃度の1乗に比例する ! v = d[P] dt = " d[A] dt = k[A] ! " d[A] dt = k[A] # d[A] = "k[A]dt # d[A] [A] = "kdt ! ln[A] = "kt + C # [A] = e"kt+C 変数分離形積分法により ! d[A] [A] " = #k dt" + C $ ln[A] = #kt + C ! log M = P " M = eP ex+y = ex # ey t = 0ならば、[A] = [A]0 ! [A]0 = e"k#0+C $ [A]0 = eC [A] = e"kt +C # [A] = e"kt eC # [A] = [A]0e"kt # [A] = [A]0 exp "kt( ) すなわち 4. 一次反応
  • 6. 二次反応: 実験的な反応速度が濃度の2乗に比例する ! A k " #" P ! v = d[P] dt = " d[A] dt = k[A]2 " d[A] dt = k[A]2 # " d[A] [A]2 = kdt 変数分離形積分法により ! " d[A] [A]2# = k dt# + C $ " " 1 [A] % & ' ( ) * = kt + C t = 0ならば、[A] = [A]0 すなわち ! " " 1 [A]0 # $ % & ' ( = k ) 0 + C * 1 [A]0 = C " " 1 [A] # $ % & ' ( = kt + C ) 1 [A] = kt + 1 [A]0 ) [A] = 1 kt + 1 [A]0 5. 二次反応
  • 7. 実験的な反応速度が濃度[A]および [B]に比例する ! A + B k " #" P ! v = d[P] dt = " d[A] dt = " d[B] dt = k[A][B] ! dx dt = k[A][B] " dx [A][B] = kdt " dx [A]0 # x( ) [B]0 # x( ) = kdt [A] = [A] 0 – x, [B] = [B]0 – x とおくと ! [A]0 " [B]0 6. 二成分反応 のとき € 1 [A]0 − x( ) [B]0 − x( ) = − 1 [A]0 −[B]0( ) 1 [A]0 − x( ) − 1 [B]0 − x( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − 1 [A]0 −[B]0( ) [B]0 − x( )− [A]0 − x( ) [A]0 − x( ) [B]0 − x( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ € = − 1 [A]0 −[B]0( ) [B]0 −[A]0 [A]0 − x( ) [B]0 − x( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − 1 [A]0 −[B]0( ) −([A]0 −[B]0) [A]0 − x( ) [B]0 − x( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 1 [A]0 ! x( ) [B]0 ! x( )
  • 8. dx [A]0 ! x( ) [B]0 ! x( ) = ! 1 [A]0 ![B]0( ) 1 [A]0 ! x( ) ! 1 [B]0 ! x( ) ! " # # $ % & & dx dx [A]0 ! x( ) [B]0 ! x( ) = kdt 1 [A]0 ! x( ) dx の積分を考える。 ここで 1 [A]0 ! x( ) dx 0 x " # ! 1 z dz [A]0 [A]0!x " [A]0 ! x = z と置くと ! 1 z dz [A]0 [A]0!x " = ! [A]0 [A]0!x ln z#$ %& = ! ln([A]0 ! x)! ln[A]0[ ] = !ln [A]0 ! x [A]0 " # $ % & ' = !ln [A] [A]0 " # $ % & ' したがって の左辺は
  • 9. 1 [A]0 ! x( ) [B]0 ! x( ) dx" = ! 1 [A]0 ![B]0( ) 1 [A]0 ! x( ) ! 1 [B]0 ! x( ) # $ % % & ' ( ( dx" = ! 1 [A]0 ![B]0( ) !ln [A] [A]0 " # $ % & '! !ln [B] [B]0 " # $ % & ' ( ) * + , - . / 0 10 2 3 0 40 = 1 [A]0 ![B]0( ) ln [A] [A]0 " # $ % & '! ln [B] [B]0 " # $ % & ' . / 0 10 2 3 0 40 = 1 [A]0 ![B]0( ) ln [A][B]0 [A]0[B] " # $ % & ' [B]も同様に行い ! [A]0 = [B]0 1 [A] " 1 [A]0 = 1 [B] " 1 [B]0 = kt のとき
  • 10. 実験的な反応速度において逆反応が無視できない場合 ! A ! B ! k1 ! k"1 7. 可逆一次反応Ⅰ ! v = d[B] dt = " d[A] dt = k1[A] " k"1[B] # d[A] dt = k"1[B] " k1[A] # d[A] dt = k"1 [A]0 "[A]( )" k1[A] # d[A] dt = " k1 + k"1( )[A]+ k"1[A]0 # d[A] dt = " k1 + k"1( ) [A] " k"1 k1 + k"1 [A]0 $ % & ' ( ) ! K = [B]eq [A]eq ! [A]0 = [A]eq +[B]eq であるので ! K = [A]0 "[A]eq [A]eq = k1 k"1 ! [A]0 "[A]eq = k1 k"1 [A]eq # [A]0 = k1 k"1 [A]eq +[A]eq # [A]0 = k1 + k"1 k"1 [A]eq # k"1 k1 + k"1 [A]0 = [A]eq 時間 t 後における濃度を [A] とすると " d[A] dt = # k1 + k#1( ) [A] #[A]eq( )" d[A] [A] #[A]eq = # k1 + k#1( )dt
  • 11. ! d[A] [A] "[A]eq # = " k1 + k"1( ) dt# + C $ ln [A] "[A]eq( )= " k1 + k"1( )t + C 変数分離形積分法により t = 0ならば、[A] = [A]0となるので ! C = ln [A]0 "[A]eq( ) つなわち ! ln [A] "[A]eq( )= " k1 + k"1( )t + ln [A]0 "[A]eq( ) # ln [A] "[A]eq( )" ln [A]0 "[A]eq( )= " k1 + k"1( )t # ln [A] "[A]eq [A]0 "[A]eq = " k1 + k"1( )t # [A] "[A]eq [A]0 "[A]eq = e" k1 +k"1( )t # [A] "[A]eq = [A]0 "[A]eq( )e" k1 +k"1( )t # [A] = [A]eq + [A]0 "[A]eq( )e" k1 +k"1( )t kobs = k1 + k"1 とおく および A = [A]0 "[A]eq [A] = [A]eq + Ae"kobs t
  • 12. 実験的な反応速度において逆反応が無視できない場合 ! A ! B ! k1 ! k"1 ! v = d[B] dt = " d[A] dt = k1[A] " k"1[B] 時間 t 後に、[A]が x mol/dm3に変化したとすると ! dx dt = k1 [A] " x( ) " k"1 [B]+ x( ) # dx dt = k1[A] " k"1[B] " x k1 + k"1( ) # dx dt = k1 + k"1( ) k1[A] " k"1[B] k1 + k"1 " x $ % & ' ( ) ! m = k1[A] " k"1[B] k1 + k"1 とおく ! dx dt = k1 + k"1( ) m " x( ) # dx m " x = k1 + k"1( )dt 変数分離形積分法により dx m " x # = k1 + k"1( ) dt# + C $ "ln m " x( ) = k1 + k"1( )t + C 8. 可逆一次反応Ⅱ
  • 13. ! "ln m " x( ) = k1 + k"1( )t + C # "ln m " 0( ) = k1 + k"1( )$ 0 + C # C = "ln m( ) 時間 t = 0ならば、 x = 0 となる ! ln m( ) " ln( ) = k1 + k"1( )t # ln m m " x $ % & ' ( ) = k1 + k"1( )t # m m " x = e k1 +k"1( )t # m " x m = e" k1 +k"1( )t # m " x = me" k1 +k"1( )t # x = m + me" k1 +k"1( )t C = -ln (m)を代入すると ! kobs = k1 + k"1 とおく ! "x = m + me#kobs t $ x = m + m % exp #kobst( )
  • 14. 9. 反応の半減期 [A]=[A0]/2になる時間をt=t1/2とすると € ln [A] [A0] = –kt ⇔ [A] = [A0]exp −kt( ) € [A] = [A0] 2 を代入すると € ln [A0] 2 [A0] = –kt1/ 2 ⇔ ln 1 2 = –kt1/ 2 ⇔ t1/ 2 = − 1 k ln 1 2 = ln2 k ≈ 0.693 k 一次反応における半減期は、初期濃度に依存しない。 一次反応
  • 15. [A]=[A0]/2になる時間をt=t1/2とすると € 1 [A] = kt + 1 [A0] € [A] = [A0] 2 を代入すると 二次反応 1 [A0] 2 = kt1/ 2 + 1 [A0] ⇔ t1/ 2 = 1 [A0]k 一次反応以外の他の反応半減期はすべて濃度に依存する。 零次反応 € [A0] −[A] = kt [A]=[A0]/2になる時間をt=t1/2とすると € [A] = [A0] 2 を代入すると € [A0] − [A0] 2 = kt1/ 2 ⇔ [A0] 2 = kt1/ 2 ⇔ t1/ 2 = [A0] 2k