H2SO4 + KMnO4 + NaNO2 = H2O + K2SO4 + Na2SO4 + NO2 + MnSO4

H_2SO_4 (sulfuric acid) + KMnO_4 (potassium permanganate) + NaNO_2 (sodium nitrite) ⟶ H_2O (water) + K_2SO_4 (potassium sulfate) + Na_2SO_4 (sodium sulfate) + NO_2 (nitrogen dioxide) + MnSO_4 (manganese(II) sulfate)

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + NaNO_2 ⟶ H_2O + K_2SO_4 + Na_2SO_4 + NO_2 + MnSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 NaNO_2 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 Na_2SO_4 + c_7 NO_2 + c_8 MnSO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K, Mn, N and Na: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 + 2 c_3 = c_4 + 4 c_5 + 4 c_6 + 2 c_7 + 4 c_8 S: | c_1 = c_5 + c_6 + c_8 K: | c_2 = 2 c_5 Mn: | c_2 = c_8 N: | c_3 = c_7 Na: | c_3 = 2 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 = 8 c_2 = 2 c_3 = 10 c_4 = 8 c_5 = 1 c_6 = 5 c_7 = 10 c_8 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2SO_4 + 2 KMnO_4 + 10 NaNO_2 ⟶ 8 H_2O + K_2SO_4 + 5 Na_2SO_4 + 10 NO_2 + 2 MnSO_4

+ + ⟶ + + + +

sulfuric acid + potassium permanganate + sodium nitrite ⟶ water + potassium sulfate + sodium sulfate + nitrogen dioxide + manganese(II) sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + NaNO_2 ⟶ H_2O + K_2SO_4 + Na_2SO_4 + NO_2 + MnSO_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: 8 H_2SO_4 + 2 KMnO_4 + 10 NaNO_2 ⟶ 8 H_2O + K_2SO_4 + 5 Na_2SO_4 + 10 NO_2 + 2 MnSO_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 | 8 | -8 KMnO_4 | 2 | -2 NaNO_2 | 10 | -10 H_2O | 8 | 8 K_2SO_4 | 1 | 1 Na_2SO_4 | 5 | 5 NO_2 | 10 | 10 MnSO_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 | 8 | -8 | ([H2SO4])^(-8) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) NaNO_2 | 10 | -10 | ([NaNO2])^(-10) H_2O | 8 | 8 | ([H2O])^8 K_2SO_4 | 1 | 1 | [K2SO4] Na_2SO_4 | 5 | 5 | ([Na2SO4])^5 NO_2 | 10 | 10 | ([NO2])^10 MnSO_4 | 2 | 2 | ([MnSO4])^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])^(-8) ([KMnO4])^(-2) ([NaNO2])^(-10) ([H2O])^8 [K2SO4] ([Na2SO4])^5 ([NO2])^10 ([MnSO4])^2 = (([H2O])^8 [K2SO4] ([Na2SO4])^5 ([NO2])^10 ([MnSO4])^2)/(([H2SO4])^8 ([KMnO4])^2 ([NaNO2])^10)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + NaNO_2 ⟶ H_2O + K_2SO_4 + Na_2SO_4 + NO_2 + MnSO_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: 8 H_2SO_4 + 2 KMnO_4 + 10 NaNO_2 ⟶ 8 H_2O + K_2SO_4 + 5 Na_2SO_4 + 10 NO_2 + 2 MnSO_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 | 8 | -8 KMnO_4 | 2 | -2 NaNO_2 | 10 | -10 H_2O | 8 | 8 K_2SO_4 | 1 | 1 Na_2SO_4 | 5 | 5 NO_2 | 10 | 10 MnSO_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 | 8 | -8 | -1/8 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) NaNO_2 | 10 | -10 | -1/10 (Δ[NaNO2])/(Δt) H_2O | 8 | 8 | 1/8 (Δ[H2O])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) Na_2SO_4 | 5 | 5 | 1/5 (Δ[Na2SO4])/(Δt) NO_2 | 10 | 10 | 1/10 (Δ[NO2])/(Δt) MnSO_4 | 2 | 2 | 1/2 (Δ[MnSO4])/(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 = -1/8 (Δ[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/10 (Δ[NaNO2])/(Δt) = 1/8 (Δ[H2O])/(Δt) = (Δ[K2SO4])/(Δt) = 1/5 (Δ[Na2SO4])/(Δt) = 1/10 (Δ[NO2])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | sodium nitrite | water | potassium sulfate | sodium sulfate | nitrogen dioxide | manganese(II) sulfate formula | H_2SO_4 | KMnO_4 | NaNO_2 | H_2O | K_2SO_4 | Na_2SO_4 | NO_2 | MnSO_4 Hill formula | H_2O_4S | KMnO_4 | NNaO_2 | H_2O | K_2O_4S | Na_2O_4S | NO_2 | MnSO_4 name | sulfuric acid | potassium permanganate | sodium nitrite | water | potassium sulfate | sodium sulfate | nitrogen dioxide | manganese(II) sulfate IUPAC name | sulfuric acid | potassium permanganate | sodium nitrite | water | dipotassium sulfate | disodium sulfate | Nitrogen dioxide | manganese(+2) cation sulfate