H2O + HNO3 + KMnO4 + TcO2 = KNO3 + Mn(NO3)2 + H6TcO6

H_2O water + HNO_3 nitric acid + KMnO_4 potassium permanganate + TcO2 ⟶ KNO_3 potassium nitrate + Mn(NO_3)_2 manganese(II) nitrate + H6TcO6

Balance the chemical equation algebraically: H_2O + HNO_3 + KMnO_4 + TcO2 ⟶ KNO_3 + Mn(NO_3)_2 + H6TcO6 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 HNO_3 + c_3 KMnO_4 + c_4 TcO2 ⟶ c_5 KNO_3 + c_6 Mn(NO_3)_2 + c_7 H6TcO6 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, N, K, Mn and Tc: H: | 2 c_1 + c_2 = 6 c_7 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_6 K: | c_3 = c_5 Mn: | c_3 = c_6 Tc: | c_4 = 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 = 6 c_2 = 3 c_3 = 1 c_4 = 5/2 c_5 = 1 c_6 = 1 c_7 = 5/2 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 = 2 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 12 H_2O + 6 HNO_3 + 2 KMnO_4 + 5 TcO2 ⟶ 2 KNO_3 + 2 Mn(NO_3)_2 + 5 H6TcO6

+ + + TcO2 ⟶ + + H6TcO6

water + nitric acid + potassium permanganate + TcO2 ⟶ potassium nitrate + manganese(II) nitrate + H6TcO6

Construct the equilibrium constant, K, expression for: H_2O + HNO_3 + KMnO_4 + TcO2 ⟶ KNO_3 + Mn(NO_3)_2 + H6TcO6 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 TcO2 ⟶ 2 KNO_3 + 2 Mn(NO_3)_2 + 5 H6TcO6 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 TcO2 | 5 | -5 KNO_3 | 2 | 2 Mn(NO_3)_2 | 2 | 2 H6TcO6 | 5 | 5 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) TcO2 | 5 | -5 | ([TcO2])^(-5) KNO_3 | 2 | 2 | ([KNO3])^2 Mn(NO_3)_2 | 2 | 2 | ([Mn(NO3)2])^2 H6TcO6 | 5 | 5 | ([H6TcO6])^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 = ([H2O])^(-12) ([HNO3])^(-6) ([KMnO4])^(-2) ([TcO2])^(-5) ([KNO3])^2 ([Mn(NO3)2])^2 ([H6TcO6])^5 = (([KNO3])^2 ([Mn(NO3)2])^2 ([H6TcO6])^5)/(([H2O])^12 ([HNO3])^6 ([KMnO4])^2 ([TcO2])^5)

Construct the rate of reaction expression for: H_2O + HNO_3 + KMnO_4 + TcO2 ⟶ KNO_3 + Mn(NO_3)_2 + H6TcO6 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 TcO2 ⟶ 2 KNO_3 + 2 Mn(NO_3)_2 + 5 H6TcO6 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 TcO2 | 5 | -5 KNO_3 | 2 | 2 Mn(NO_3)_2 | 2 | 2 H6TcO6 | 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_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) TcO2 | 5 | -5 | -1/5 (Δ[TcO2])/(Δt) KNO_3 | 2 | 2 | 1/2 (Δ[KNO3])/(Δt) Mn(NO_3)_2 | 2 | 2 | 1/2 (Δ[Mn(NO3)2])/(Δt) H6TcO6 | 5 | 5 | 1/5 (Δ[H6TcO6])/(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[TcO2])/(Δt) = 1/2 (Δ[KNO3])/(Δt) = 1/2 (Δ[Mn(NO3)2])/(Δt) = 1/5 (Δ[H6TcO6])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | nitric acid | potassium permanganate | TcO2 | potassium nitrate | manganese(II) nitrate | H6TcO6 formula | H_2O | HNO_3 | KMnO_4 | TcO2 | KNO_3 | Mn(NO_3)_2 | H6TcO6 Hill formula | H_2O | HNO_3 | KMnO_4 | O2Tc | KNO_3 | MnN_2O_6 | H6O6Tc name | water | nitric acid | potassium permanganate | | potassium nitrate | manganese(II) nitrate | IUPAC name | water | nitric acid | potassium permanganate | | potassium nitrate | manganese(2+) dinitrate |

| water | nitric acid | potassium permanganate | TcO2 | potassium nitrate | manganese(II) nitrate | H6TcO6 molar mass | 18.015 g/mol | 63.012 g/mol | 158.03 g/mol | 130 g/mol | 101.1 g/mol | 178.95 g/mol | 200 g/mol phase | liquid (at STP) | liquid (at STP) | solid (at STP) | | solid (at STP) | | melting point | 0 °C | -41.6 °C | 240 °C | | 334 °C | | boiling point | 99.9839 °C | 83 °C | | | | | density | 1 g/cm^3 | 1.5129 g/cm^3 | 1 g/cm^3 | | | 1.536 g/cm^3 | solubility in water | | miscible | | | soluble | | surface tension | 0.0728 N/m | | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | 7.6×10^-4 Pa s (at 25 °C) | | | | | odor | odorless | | odorless | | odorless | |