H2SO4 + MnSO4 + NaBiO3 = H2O + Na2SO4 + NaMnO4 + Bi2(SO4)3

H_2SO_4 sulfuric acid + MnSO_4 manganese(II) sulfate + NaBiO_3 sodium bismuthate ⟶ H_2O water + Na_2SO_4 sodium sulfate + NaMnO_4 sodium permanganate + Bi_2(SO_4)_3 bismuth sulfate

Balance the chemical equation algebraically: H_2SO_4 + MnSO_4 + NaBiO_3 ⟶ H_2O + Na_2SO_4 + NaMnO_4 + Bi_2(SO_4)_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 MnSO_4 + c_3 NaBiO_3 ⟶ c_4 H_2O + c_5 Na_2SO_4 + c_6 NaMnO_4 + c_7 Bi_2(SO_4)_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Mn, Bi and Na: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 + 3 c_3 = c_4 + 4 c_5 + 4 c_6 + 12 c_7 S: | c_1 + c_2 = c_5 + 3 c_7 Mn: | c_2 = c_6 Bi: | c_3 = 2 c_7 Na: | c_3 = 2 c_5 + c_6 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 = 14/3 c_2 = 4/3 c_3 = 10/3 c_4 = 14/3 c_5 = 1 c_6 = 4/3 c_7 = 5/3 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 14 c_2 = 4 c_3 = 10 c_4 = 14 c_5 = 3 c_6 = 4 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 14 H_2SO_4 + 4 MnSO_4 + 10 NaBiO_3 ⟶ 14 H_2O + 3 Na_2SO_4 + 4 NaMnO_4 + 5 Bi_2(SO_4)_3

+ + ⟶ + + +

sulfuric acid + manganese(II) sulfate + sodium bismuthate ⟶ water + sodium sulfate + sodium permanganate + bismuth sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + MnSO_4 + NaBiO_3 ⟶ H_2O + Na_2SO_4 + NaMnO_4 + Bi_2(SO_4)_3 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: 14 H_2SO_4 + 4 MnSO_4 + 10 NaBiO_3 ⟶ 14 H_2O + 3 Na_2SO_4 + 4 NaMnO_4 + 5 Bi_2(SO_4)_3 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 | 14 | -14 MnSO_4 | 4 | -4 NaBiO_3 | 10 | -10 H_2O | 14 | 14 Na_2SO_4 | 3 | 3 NaMnO_4 | 4 | 4 Bi_2(SO_4)_3 | 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 | 14 | -14 | ([H2SO4])^(-14) MnSO_4 | 4 | -4 | ([MnSO4])^(-4) NaBiO_3 | 10 | -10 | ([NaBiO3])^(-10) H_2O | 14 | 14 | ([H2O])^14 Na_2SO_4 | 3 | 3 | ([Na2SO4])^3 NaMnO_4 | 4 | 4 | ([NaMnO4])^4 Bi_2(SO_4)_3 | 5 | 5 | ([Bi2(SO4)3])^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])^(-14) ([MnSO4])^(-4) ([NaBiO3])^(-10) ([H2O])^14 ([Na2SO4])^3 ([NaMnO4])^4 ([Bi2(SO4)3])^5 = (([H2O])^14 ([Na2SO4])^3 ([NaMnO4])^4 ([Bi2(SO4)3])^5)/(([H2SO4])^14 ([MnSO4])^4 ([NaBiO3])^10)

Construct the rate of reaction expression for: H_2SO_4 + MnSO_4 + NaBiO_3 ⟶ H_2O + Na_2SO_4 + NaMnO_4 + Bi_2(SO_4)_3 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: 14 H_2SO_4 + 4 MnSO_4 + 10 NaBiO_3 ⟶ 14 H_2O + 3 Na_2SO_4 + 4 NaMnO_4 + 5 Bi_2(SO_4)_3 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 | 14 | -14 MnSO_4 | 4 | -4 NaBiO_3 | 10 | -10 H_2O | 14 | 14 Na_2SO_4 | 3 | 3 NaMnO_4 | 4 | 4 Bi_2(SO_4)_3 | 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 | 14 | -14 | -1/14 (Δ[H2SO4])/(Δt) MnSO_4 | 4 | -4 | -1/4 (Δ[MnSO4])/(Δt) NaBiO_3 | 10 | -10 | -1/10 (Δ[NaBiO3])/(Δt) H_2O | 14 | 14 | 1/14 (Δ[H2O])/(Δt) Na_2SO_4 | 3 | 3 | 1/3 (Δ[Na2SO4])/(Δt) NaMnO_4 | 4 | 4 | 1/4 (Δ[NaMnO4])/(Δt) Bi_2(SO_4)_3 | 5 | 5 | 1/5 (Δ[Bi2(SO4)3])/(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 = -1/14 (Δ[H2SO4])/(Δt) = -1/4 (Δ[MnSO4])/(Δt) = -1/10 (Δ[NaBiO3])/(Δt) = 1/14 (Δ[H2O])/(Δt) = 1/3 (Δ[Na2SO4])/(Δt) = 1/4 (Δ[NaMnO4])/(Δt) = 1/5 (Δ[Bi2(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | manganese(II) sulfate | sodium bismuthate | water | sodium sulfate | sodium permanganate | bismuth sulfate formula | H_2SO_4 | MnSO_4 | NaBiO_3 | H_2O | Na_2SO_4 | NaMnO_4 | Bi_2(SO_4)_3 Hill formula | H_2O_4S | MnSO_4 | BiNaO_3 | H_2O | Na_2O_4S | MnNaO_4 | Bi_2O_12S_3 name | sulfuric acid | manganese(II) sulfate | sodium bismuthate | water | sodium sulfate | sodium permanganate | bismuth sulfate IUPAC name | sulfuric acid | manganese(+2) cation sulfate | sodium oxido-dioxobismuth | water | disodium sulfate | sodium permanganate | dibismuth trisulfate