H2SO4 + KMnO4 + H2C2O4 = H2O + CO2 + MnSO4 + KHSO4

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + HO_2CCO_2H oxalic acid ⟶ H_2O water + CO_2 carbon dioxide + MnSO_4 manganese(II) sulfate + KHSO_4 potassium bisulfate

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + HO_2CCO_2H ⟶ H_2O + CO_2 + MnSO_4 + KHSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 HO_2CCO_2H ⟶ c_4 H_2O + c_5 CO_2 + c_6 MnSO_4 + c_7 KHSO_4 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 + 2 c_3 = 2 c_4 + c_7 O: | 4 c_1 + 4 c_2 + 4 c_3 = c_4 + 2 c_5 + 4 c_6 + 4 c_7 S: | c_1 = c_6 + c_7 K: | c_2 = c_7 Mn: | c_2 = c_6 C: | 2 c_3 = c_5 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 2 c_2 = 1 c_3 = 5/2 c_4 = 4 c_5 = 5 c_6 = 1 c_7 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 4 c_2 = 2 c_3 = 5 c_4 = 8 c_5 = 10 c_6 = 2 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2SO_4 + 2 KMnO_4 + 5 HO_2CCO_2H ⟶ 8 H_2O + 10 CO_2 + 2 MnSO_4 + 2 KHSO_4

+ + ⟶ + + +

sulfuric acid + potassium permanganate + oxalic acid ⟶ water + carbon dioxide + manganese(II) sulfate + potassium bisulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + HO_2CCO_2H ⟶ H_2O + CO_2 + MnSO_4 + KHSO_4 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: 4 H_2SO_4 + 2 KMnO_4 + 5 HO_2CCO_2H ⟶ 8 H_2O + 10 CO_2 + 2 MnSO_4 + 2 KHSO_4 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 | 4 | -4 KMnO_4 | 2 | -2 HO_2CCO_2H | 5 | -5 H_2O | 8 | 8 CO_2 | 10 | 10 MnSO_4 | 2 | 2 KHSO_4 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 4 | -4 | ([H2SO4])^(-4) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) HO_2CCO_2H | 5 | -5 | ([HO2CCO2H])^(-5) H_2O | 8 | 8 | ([H2O])^8 CO_2 | 10 | 10 | ([CO2])^10 MnSO_4 | 2 | 2 | ([MnSO4])^2 KHSO_4 | 2 | 2 | ([KHSO4])^2 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])^(-4) ([KMnO4])^(-2) ([HO2CCO2H])^(-5) ([H2O])^8 ([CO2])^10 ([MnSO4])^2 ([KHSO4])^2 = (([H2O])^8 ([CO2])^10 ([MnSO4])^2 ([KHSO4])^2)/(([H2SO4])^4 ([KMnO4])^2 ([HO2CCO2H])^5)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + HO_2CCO_2H ⟶ H_2O + CO_2 + MnSO_4 + KHSO_4 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: 4 H_2SO_4 + 2 KMnO_4 + 5 HO_2CCO_2H ⟶ 8 H_2O + 10 CO_2 + 2 MnSO_4 + 2 KHSO_4 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 | 4 | -4 KMnO_4 | 2 | -2 HO_2CCO_2H | 5 | -5 H_2O | 8 | 8 CO_2 | 10 | 10 MnSO_4 | 2 | 2 KHSO_4 | 2 | 2 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 | 4 | -4 | -1/4 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) HO_2CCO_2H | 5 | -5 | -1/5 (Δ[HO2CCO2H])/(Δt) H_2O | 8 | 8 | 1/8 (Δ[H2O])/(Δt) CO_2 | 10 | 10 | 1/10 (Δ[CO2])/(Δt) MnSO_4 | 2 | 2 | 1/2 (Δ[MnSO4])/(Δt) KHSO_4 | 2 | 2 | 1/2 (Δ[KHSO4])/(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 = -1/4 (Δ[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/5 (Δ[HO2CCO2H])/(Δt) = 1/8 (Δ[H2O])/(Δt) = 1/10 (Δ[CO2])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) = 1/2 (Δ[KHSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | oxalic acid | water | carbon dioxide | manganese(II) sulfate | potassium bisulfate formula | H_2SO_4 | KMnO_4 | HO_2CCO_2H | H_2O | CO_2 | MnSO_4 | KHSO_4 Hill formula | H_2O_4S | KMnO_4 | C_2H_2O_4 | H_2O | CO_2 | MnSO_4 | HKO_4S name | sulfuric acid | potassium permanganate | oxalic acid | water | carbon dioxide | manganese(II) sulfate | potassium bisulfate IUPAC name | sulfuric acid | potassium permanganate | oxalic acid | water | carbon dioxide | manganese(+2) cation sulfate | potassium hydrogen sulfate

| sulfuric acid | potassium permanganate | oxalic acid | water | carbon dioxide | manganese(II) sulfate | potassium bisulfate molar mass | 98.07 g/mol | 158.03 g/mol | 90.03 g/mol | 18.015 g/mol | 44.009 g/mol | 150.99 g/mol | 136.16 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | gas (at STP) | solid (at STP) | solid (at STP) melting point | 10.371 °C | 240 °C | 189.5 °C | 0 °C | -56.56 °C (at triple point) | 710 °C | 214 °C boiling point | 279.6 °C | | 365 °C | 99.9839 °C | -78.5 °C (at sublimation point) | | density | 1.8305 g/cm^3 | 1 g/cm^3 | 1.948 g/cm^3 | 1 g/cm^3 | 0.00184212 g/cm^3 (at 20 °C) | 3.25 g/cm^3 | 2.32 g/cm^3 solubility in water | very soluble | | | | | soluble | surface tension | 0.0735 N/m | | | 0.0728 N/m | | | dynamic viscosity | 0.021 Pa s (at 25 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | 1.491×10^-5 Pa s (at 25 °C) | | odor | odorless | odorless | | odorless | odorless | |