H2SO4 + KMnO4 + CH3OH = H2O + K2SO4 + MnSO4 + HCOOH

H_2SO_4 (sulfuric acid) + KMnO_4 (potassium permanganate) + CH_3OH (methanol) ⟶ H_2O (water) + K_2SO_4 (potassium sulfate) + MnSO_4 (manganese(II) sulfate) + HCOOH (formic acid)

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + CH_3OH ⟶ H_2O + K_2SO_4 + MnSO_4 + HCOOH Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 CH_3OH ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 HCOOH Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K, Mn and C: H: | 2 c_1 + 4 c_3 = 2 c_4 + 2 c_7 O: | 4 c_1 + 4 c_2 + c_3 = c_4 + 4 c_5 + 4 c_6 + 2 c_7 S: | c_1 = c_5 + c_6 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 C: | c_3 = c_7 Since the coefficients are relative quantities and underdetermined, choose a coefficient to set arbitrarily. To keep the coefficients small, the arbitrary value is ordinarily one. For instance, set c_5 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 5/2 c_4 = 11/2 c_5 = 1 c_6 = 2 c_7 = 5/2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 6 c_2 = 4 c_3 = 5 c_4 = 11 c_5 = 2 c_6 = 4 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 H_2SO_4 + 4 KMnO_4 + 5 CH_3OH ⟶ 11 H_2O + 2 K_2SO_4 + 4 MnSO_4 + 5 HCOOH

+ + ⟶ + + +

sulfuric acid + potassium permanganate + methanol ⟶ water + potassium sulfate + manganese(II) sulfate + formic acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + CH_3OH ⟶ H_2O + K_2SO_4 + MnSO_4 + HCOOH Plan: • Balance the chemical equation. • Determine the stoichiometric numbers. • Assemble the activity expression for each chemical species. • Use the activity expressions to build the equilibrium constant expression. Write the balanced chemical equation: 6 H_2SO_4 + 4 KMnO_4 + 5 CH_3OH ⟶ 11 H_2O + 2 K_2SO_4 + 4 MnSO_4 + 5 HCOOH Assign stoichiometric numbers, ν_i, using the stoichiometric coefficients, c_i, from the balanced chemical equation in the following manner: ν_i = -c_i for reactants and ν_i = c_i for products: chemical species | c_i | ν_i H_2SO_4 | 6 | -6 KMnO_4 | 4 | -4 CH_3OH | 5 | -5 H_2O | 11 | 11 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 HCOOH | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 6 | -6 | ([H2SO4])^(-6) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) CH_3OH | 5 | -5 | ([CH3OH])^(-5) H_2O | 11 | 11 | ([H2O])^11 K_2SO_4 | 2 | 2 | ([K2SO4])^2 MnSO_4 | 4 | 4 | ([MnSO4])^4 HCOOH | 5 | 5 | ([HCOOH])^5 The equilibrium constant symbol in the concentration basis is: K_c Mulitply the activity expressions to arrive at the K_c expression: Answer: | | K_c = ([H2SO4])^(-6) ([KMnO4])^(-4) ([CH3OH])^(-5) ([H2O])^11 ([K2SO4])^2 ([MnSO4])^4 ([HCOOH])^5 = (([H2O])^11 ([K2SO4])^2 ([MnSO4])^4 ([HCOOH])^5)/(([H2SO4])^6 ([KMnO4])^4 ([CH3OH])^5)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + CH_3OH ⟶ H_2O + K_2SO_4 + MnSO_4 + HCOOH Plan: • Balance the chemical equation. • Determine the stoichiometric numbers. • Assemble the rate term for each chemical species. • Write the rate of reaction expression. Write the balanced chemical equation: 6 H_2SO_4 + 4 KMnO_4 + 5 CH_3OH ⟶ 11 H_2O + 2 K_2SO_4 + 4 MnSO_4 + 5 HCOOH Assign stoichiometric numbers, ν_i, using the stoichiometric coefficients, c_i, from the balanced chemical equation in the following manner: ν_i = -c_i for reactants and ν_i = c_i for products: chemical species | c_i | ν_i H_2SO_4 | 6 | -6 KMnO_4 | 4 | -4 CH_3OH | 5 | -5 H_2O | 11 | 11 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 HCOOH | 5 | 5 The rate term for each chemical species, B_i, is 1/ν_i(Δ[B_i])/(Δt) where [B_i] is the amount concentration and t is time: chemical species | c_i | ν_i | rate term H_2SO_4 | 6 | -6 | -1/6 (Δ[H2SO4])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) CH_3OH | 5 | -5 | -1/5 (Δ[CH3OH])/(Δt) H_2O | 11 | 11 | 1/11 (Δ[H2O])/(Δt) K_2SO_4 | 2 | 2 | 1/2 (Δ[K2SO4])/(Δt) MnSO_4 | 4 | 4 | 1/4 (Δ[MnSO4])/(Δt) HCOOH | 5 | 5 | 1/5 (Δ[HCOOH])/(Δt) (for infinitesimal rate of change, replace Δ with d) Set the rate terms equal to each other to arrive at the rate expression: Answer: | | rate = -1/6 (Δ[H2SO4])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/5 (Δ[CH3OH])/(Δt) = 1/11 (Δ[H2O])/(Δt) = 1/2 (Δ[K2SO4])/(Δt) = 1/4 (Δ[MnSO4])/(Δt) = 1/5 (Δ[HCOOH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | methanol | water | potassium sulfate | manganese(II) sulfate | formic acid formula | H_2SO_4 | KMnO_4 | CH_3OH | H_2O | K_2SO_4 | MnSO_4 | HCOOH Hill formula | H_2O_4S | KMnO_4 | CH_4O | H_2O | K_2O_4S | MnSO_4 | CH_2O_2 name | sulfuric acid | potassium permanganate | methanol | water | potassium sulfate | manganese(II) sulfate | formic acid IUPAC name | sulfuric acid | potassium permanganate | methanol | water | dipotassium sulfate | manganese(+2) cation sulfate | formic acid