H2O + HNO3 + KMnO4 + TeO2 = KNO3 + H6TeO6 + Mn(NO3)

H_2O water + HNO_3 nitric acid + KMnO_4 potassium permanganate + TeO_2 tellurium dioxide ⟶ KNO_3 potassium nitrate + H_6TeO_6 telluric(VI) acid + Mn(NO_3)_2 manganese(II) nitrate

Balance the chemical equation algebraically: H_2O + HNO_3 + KMnO_4 + TeO_2 ⟶ KNO_3 + H_6TeO_6 + Mn(NO_3)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 HNO_3 + c_3 KMnO_4 + c_4 TeO_2 ⟶ c_5 KNO_3 + c_6 H_6TeO_6 + c_7 Mn(NO_3)_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, N, K, Mn and Te: H: | 2 c_1 + c_2 = 6 c_6 O: | c_1 + 3 c_2 + 4 c_3 + 2 c_4 = 3 c_5 + 6 c_6 + 6 c_7 N: | c_2 = c_5 + 2 c_7 K: | c_3 = c_5 Mn: | c_3 = c_7 Te: | c_4 = 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 6 c_2 = 3 c_3 = 1 c_4 = 5/2 c_5 = 1 c_6 = 5/2 c_7 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 12 c_2 = 6 c_3 = 2 c_4 = 5 c_5 = 2 c_6 = 5 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 12 H_2O + 6 HNO_3 + 2 KMnO_4 + 5 TeO_2 ⟶ 2 KNO_3 + 5 H_6TeO_6 + 2 Mn(NO_3)_2

+ + + ⟶ + +

water + nitric acid + potassium permanganate + tellurium dioxide ⟶ potassium nitrate + telluric(VI) acid + manganese(II) nitrate

Construct the equilibrium constant, K, expression for: H_2O + HNO_3 + KMnO_4 + TeO_2 ⟶ KNO_3 + H_6TeO_6 + 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: 12 H_2O + 6 HNO_3 + 2 KMnO_4 + 5 TeO_2 ⟶ 2 KNO_3 + 5 H_6TeO_6 + 2 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 H_2O | 12 | -12 HNO_3 | 6 | -6 KMnO_4 | 2 | -2 TeO_2 | 5 | -5 KNO_3 | 2 | 2 H_6TeO_6 | 5 | 5 Mn(NO_3)_2 | 2 | 2 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) HNO_3 | 6 | -6 | ([HNO3])^(-6) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) TeO_2 | 5 | -5 | ([TeO2])^(-5) KNO_3 | 2 | 2 | ([KNO3])^2 H_6TeO_6 | 5 | 5 | ([H6TeO6])^5 Mn(NO_3)_2 | 2 | 2 | ([Mn(NO3)2])^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 = ([H2O])^(-12) ([HNO3])^(-6) ([KMnO4])^(-2) ([TeO2])^(-5) ([KNO3])^2 ([H6TeO6])^5 ([Mn(NO3)2])^2 = (([KNO3])^2 ([H6TeO6])^5 ([Mn(NO3)2])^2)/(([H2O])^12 ([HNO3])^6 ([KMnO4])^2 ([TeO2])^5)

Construct the rate of reaction expression for: H_2O + HNO_3 + KMnO_4 + TeO_2 ⟶ KNO_3 + H_6TeO_6 + 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: 12 H_2O + 6 HNO_3 + 2 KMnO_4 + 5 TeO_2 ⟶ 2 KNO_3 + 5 H_6TeO_6 + 2 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 H_2O | 12 | -12 HNO_3 | 6 | -6 KMnO_4 | 2 | -2 TeO_2 | 5 | -5 KNO_3 | 2 | 2 H_6TeO_6 | 5 | 5 Mn(NO_3)_2 | 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_2O | 12 | -12 | -1/12 (Δ[H2O])/(Δt) HNO_3 | 6 | -6 | -1/6 (Δ[HNO3])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) TeO_2 | 5 | -5 | -1/5 (Δ[TeO2])/(Δt) KNO_3 | 2 | 2 | 1/2 (Δ[KNO3])/(Δt) H_6TeO_6 | 5 | 5 | 1/5 (Δ[H6TeO6])/(Δt) Mn(NO_3)_2 | 2 | 2 | 1/2 (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 = -1/12 (Δ[H2O])/(Δt) = -1/6 (Δ[HNO3])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/5 (Δ[TeO2])/(Δt) = 1/2 (Δ[KNO3])/(Δt) = 1/5 (Δ[H6TeO6])/(Δt) = 1/2 (Δ[Mn(NO3)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | nitric acid | potassium permanganate | tellurium dioxide | potassium nitrate | telluric(VI) acid | manganese(II) nitrate formula | H_2O | HNO_3 | KMnO_4 | TeO_2 | KNO_3 | H_6TeO_6 | Mn(NO_3)_2 Hill formula | H_2O | HNO_3 | KMnO_4 | O_2Te | KNO_3 | H_6O_6Te | MnN_2O_6 name | water | nitric acid | potassium permanganate | tellurium dioxide | potassium nitrate | telluric(VI) acid | manganese(II) nitrate IUPAC name | water | nitric acid | potassium permanganate | tellurium dioxide | potassium nitrate | | manganese(2+) dinitrate