H2O + KMnO4 + VOSO4 = H2SO4 + K2SO4 + MnSO4 + HVO3

H_2O water + KMnO_4 potassium permanganate + VOSO_4·2H_2O vanadyl sulfate ⟶ H_2SO_4 sulfuric acid + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + HVO3

Balance the chemical equation algebraically: H_2O + KMnO_4 + VOSO_4·2H_2O ⟶ H_2SO_4 + K_2SO_4 + MnSO_4 + HVO3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 VOSO_4·2H_2O ⟶ c_4 H_2SO_4 + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 HVO3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn, S and V: H: | 2 c_1 = 2 c_4 + c_7 O: | c_1 + 4 c_2 + 5 c_3 = 4 c_4 + 4 c_5 + 4 c_6 + 3 c_7 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 S: | c_3 = c_4 + c_5 + c_6 V: | 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 = 12 c_2 = 2 c_3 = 10 c_4 = 7 c_5 = 1 c_6 = 2 c_7 = 10 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 12 H_2O + 2 KMnO_4 + 10 VOSO_4·2H_2O ⟶ 7 H_2SO_4 + K_2SO_4 + 2 MnSO_4 + 10 HVO3

+ + ⟶ + + + HVO3

water + potassium permanganate + vanadyl sulfate ⟶ sulfuric acid + potassium sulfate + manganese(II) sulfate + HVO3

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + VOSO_4·2H_2O ⟶ H_2SO_4 + K_2SO_4 + MnSO_4 + HVO3 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: 12 H_2O + 2 KMnO_4 + 10 VOSO_4·2H_2O ⟶ 7 H_2SO_4 + K_2SO_4 + 2 MnSO_4 + 10 HVO3 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_2O | 12 | -12 KMnO_4 | 2 | -2 VOSO_4·2H_2O | 10 | -10 H_2SO_4 | 7 | 7 K_2SO_4 | 1 | 1 MnSO_4 | 2 | 2 HVO3 | 10 | 10 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 12 | -12 | ([H2O])^(-12) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) VOSO_4·2H_2O | 10 | -10 | ([VOSO4·2H2O])^(-10) H_2SO_4 | 7 | 7 | ([H2SO4])^7 K_2SO_4 | 1 | 1 | [K2SO4] MnSO_4 | 2 | 2 | ([MnSO4])^2 HVO3 | 10 | 10 | ([HVO3])^10 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 = ([H2O])^(-12) ([KMnO4])^(-2) ([VOSO4·2H2O])^(-10) ([H2SO4])^7 [K2SO4] ([MnSO4])^2 ([HVO3])^10 = (([H2SO4])^7 [K2SO4] ([MnSO4])^2 ([HVO3])^10)/(([H2O])^12 ([KMnO4])^2 ([VOSO4·2H2O])^10)

Construct the rate of reaction expression for: H_2O + KMnO_4 + VOSO_4·2H_2O ⟶ H_2SO_4 + K_2SO_4 + MnSO_4 + HVO3 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: 12 H_2O + 2 KMnO_4 + 10 VOSO_4·2H_2O ⟶ 7 H_2SO_4 + K_2SO_4 + 2 MnSO_4 + 10 HVO3 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_2O | 12 | -12 KMnO_4 | 2 | -2 VOSO_4·2H_2O | 10 | -10 H_2SO_4 | 7 | 7 K_2SO_4 | 1 | 1 MnSO_4 | 2 | 2 HVO3 | 10 | 10 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_2O | 12 | -12 | -1/12 (Δ[H2O])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) VOSO_4·2H_2O | 10 | -10 | -1/10 (Δ[VOSO4·2H2O])/(Δt) H_2SO_4 | 7 | 7 | 1/7 (Δ[H2SO4])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) MnSO_4 | 2 | 2 | 1/2 (Δ[MnSO4])/(Δt) HVO3 | 10 | 10 | 1/10 (Δ[HVO3])/(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 = -1/12 (Δ[H2O])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/10 (Δ[VOSO4·2H2O])/(Δt) = 1/7 (Δ[H2SO4])/(Δt) = (Δ[K2SO4])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) = 1/10 (Δ[HVO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | vanadyl sulfate | sulfuric acid | potassium sulfate | manganese(II) sulfate | HVO3 formula | H_2O | KMnO_4 | VOSO_4·2H_2O | H_2SO_4 | K_2SO_4 | MnSO_4 | HVO3 Hill formula | H_2O | KMnO_4 | O_5SV | H_2O_4S | K_2O_4S | MnSO_4 | HO3V name | water | potassium permanganate | vanadyl sulfate | sulfuric acid | potassium sulfate | manganese(II) sulfate | IUPAC name | water | potassium permanganate | oxovanadium sulfate | sulfuric acid | dipotassium sulfate | manganese(+2) cation sulfate |

| water | potassium permanganate | vanadyl sulfate | sulfuric acid | potassium sulfate | manganese(II) sulfate | HVO3 molar mass | 18.015 g/mol | 158.03 g/mol | 163 g/mol | 98.07 g/mol | 174.25 g/mol | 150.99 g/mol | 99.947 g/mol phase | liquid (at STP) | solid (at STP) | | liquid (at STP) | | solid (at STP) | melting point | 0 °C | 240 °C | | 10.371 °C | | 710 °C | boiling point | 99.9839 °C | | | 279.6 °C | | | density | 1 g/cm^3 | 1 g/cm^3 | 3 g/cm^3 | 1.8305 g/cm^3 | | 3.25 g/cm^3 | solubility in water | | | soluble | very soluble | soluble | soluble | surface tension | 0.0728 N/m | | | 0.0735 N/m | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.021 Pa s (at 25 °C) | | | odor | odorless | odorless | | odorless | | |