HNO3 + KMnO4 + Na2S4O6 = H2O + H2SO4 + KNO3 + NaHSO4 + Mn(NO3)2

HNO_3 nitric acid + KMnO_4 potassium permanganate + Na2S4O6 ⟶ H_2O water + H_2SO_4 sulfuric acid + KNO_3 potassium nitrate + NaHSO_4 sodium bisulfate + Mn(NO_3)_2 manganese(II) nitrate

Balance the chemical equation algebraically: HNO_3 + KMnO_4 + Na2S4O6 ⟶ H_2O + H_2SO_4 + KNO_3 + NaHSO_4 + Mn(NO_3)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 KMnO_4 + c_3 Na2S4O6 ⟶ c_4 H_2O + c_5 H_2SO_4 + c_6 KNO_3 + c_7 NaHSO_4 + c_8 Mn(NO_3)_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, K, Mn, Na and S: H: | c_1 = 2 c_4 + 2 c_5 + c_7 N: | c_1 = c_6 + 2 c_8 O: | 3 c_1 + 4 c_2 + 6 c_3 = c_4 + 4 c_5 + 3 c_6 + 4 c_7 + 6 c_8 K: | c_2 = c_6 Mn: | c_2 = c_8 Na: | 2 c_3 = c_7 S: | 4 c_3 = c_5 + 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 42/5 c_2 = 14/5 c_3 = 1 c_4 = 6/5 c_5 = 2 c_6 = 14/5 c_7 = 2 c_8 = 14/5 Multiply by the least common denominator, 5, to eliminate fractional coefficients: c_1 = 42 c_2 = 14 c_3 = 5 c_4 = 6 c_5 = 10 c_6 = 14 c_7 = 10 c_8 = 14 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 42 HNO_3 + 14 KMnO_4 + 5 Na2S4O6 ⟶ 6 H_2O + 10 H_2SO_4 + 14 KNO_3 + 10 NaHSO_4 + 14 Mn(NO_3)_2

+ + Na2S4O6 ⟶ + + + +

nitric acid + potassium permanganate + Na2S4O6 ⟶ water + sulfuric acid + potassium nitrate + sodium bisulfate + manganese(II) nitrate

Construct the equilibrium constant, K, expression for: HNO_3 + KMnO_4 + Na2S4O6 ⟶ H_2O + H_2SO_4 + KNO_3 + NaHSO_4 + Mn(NO_3)_2 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: 42 HNO_3 + 14 KMnO_4 + 5 Na2S4O6 ⟶ 6 H_2O + 10 H_2SO_4 + 14 KNO_3 + 10 NaHSO_4 + 14 Mn(NO_3)_2 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 HNO_3 | 42 | -42 KMnO_4 | 14 | -14 Na2S4O6 | 5 | -5 H_2O | 6 | 6 H_2SO_4 | 10 | 10 KNO_3 | 14 | 14 NaHSO_4 | 10 | 10 Mn(NO_3)_2 | 14 | 14 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HNO_3 | 42 | -42 | ([HNO3])^(-42) KMnO_4 | 14 | -14 | ([KMnO4])^(-14) Na2S4O6 | 5 | -5 | ([Na2S4O6])^(-5) H_2O | 6 | 6 | ([H2O])^6 H_2SO_4 | 10 | 10 | ([H2SO4])^10 KNO_3 | 14 | 14 | ([KNO3])^14 NaHSO_4 | 10 | 10 | ([NaHSO4])^10 Mn(NO_3)_2 | 14 | 14 | ([Mn(NO3)2])^14 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 = ([HNO3])^(-42) ([KMnO4])^(-14) ([Na2S4O6])^(-5) ([H2O])^6 ([H2SO4])^10 ([KNO3])^14 ([NaHSO4])^10 ([Mn(NO3)2])^14 = (([H2O])^6 ([H2SO4])^10 ([KNO3])^14 ([NaHSO4])^10 ([Mn(NO3)2])^14)/(([HNO3])^42 ([KMnO4])^14 ([Na2S4O6])^5)

Construct the rate of reaction expression for: HNO_3 + KMnO_4 + Na2S4O6 ⟶ H_2O + H_2SO_4 + KNO_3 + NaHSO_4 + Mn(NO_3)_2 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: 42 HNO_3 + 14 KMnO_4 + 5 Na2S4O6 ⟶ 6 H_2O + 10 H_2SO_4 + 14 KNO_3 + 10 NaHSO_4 + 14 Mn(NO_3)_2 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 HNO_3 | 42 | -42 KMnO_4 | 14 | -14 Na2S4O6 | 5 | -5 H_2O | 6 | 6 H_2SO_4 | 10 | 10 KNO_3 | 14 | 14 NaHSO_4 | 10 | 10 Mn(NO_3)_2 | 14 | 14 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 HNO_3 | 42 | -42 | -1/42 (Δ[HNO3])/(Δt) KMnO_4 | 14 | -14 | -1/14 (Δ[KMnO4])/(Δt) Na2S4O6 | 5 | -5 | -1/5 (Δ[Na2S4O6])/(Δt) H_2O | 6 | 6 | 1/6 (Δ[H2O])/(Δt) H_2SO_4 | 10 | 10 | 1/10 (Δ[H2SO4])/(Δt) KNO_3 | 14 | 14 | 1/14 (Δ[KNO3])/(Δt) NaHSO_4 | 10 | 10 | 1/10 (Δ[NaHSO4])/(Δt) Mn(NO_3)_2 | 14 | 14 | 1/14 (Δ[Mn(NO3)2])/(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 = -1/42 (Δ[HNO3])/(Δt) = -1/14 (Δ[KMnO4])/(Δt) = -1/5 (Δ[Na2S4O6])/(Δt) = 1/6 (Δ[H2O])/(Δt) = 1/10 (Δ[H2SO4])/(Δt) = 1/14 (Δ[KNO3])/(Δt) = 1/10 (Δ[NaHSO4])/(Δt) = 1/14 (Δ[Mn(NO3)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | potassium permanganate | Na2S4O6 | water | sulfuric acid | potassium nitrate | sodium bisulfate | manganese(II) nitrate formula | HNO_3 | KMnO_4 | Na2S4O6 | H_2O | H_2SO_4 | KNO_3 | NaHSO_4 | Mn(NO_3)_2 Hill formula | HNO_3 | KMnO_4 | Na2O6S4 | H_2O | H_2O_4S | KNO_3 | HNaO_4S | MnN_2O_6 name | nitric acid | potassium permanganate | | water | sulfuric acid | potassium nitrate | sodium bisulfate | manganese(II) nitrate IUPAC name | nitric acid | potassium permanganate | | water | sulfuric acid | potassium nitrate | | manganese(2+) dinitrate