HNO3 + K2MnO4 = H2O + KMnO4 + MnO2 + KNO3

HNO_3 nitric acid + K_2MnO_4 potassium manganate ⟶ H_2O water + KMnO_4 potassium permanganate + MnO_2 manganese dioxide + KNO_3 potassium nitrate

Balance the chemical equation algebraically: HNO_3 + K_2MnO_4 ⟶ H_2O + KMnO_4 + MnO_2 + KNO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 K_2MnO_4 ⟶ c_3 H_2O + c_4 KMnO_4 + c_5 MnO_2 + c_6 KNO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, K and Mn: H: | c_1 = 2 c_3 N: | c_1 = c_6 O: | 3 c_1 + 4 c_2 = c_3 + 4 c_4 + 2 c_5 + 3 c_6 K: | 2 c_2 = c_4 + c_6 Mn: | c_2 = c_4 + c_5 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 = 4 c_2 = 3 c_3 = 2 c_4 = 2 c_5 = 1 c_6 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 HNO_3 + 3 K_2MnO_4 ⟶ 2 H_2O + 2 KMnO_4 + MnO_2 + 4 KNO_3

+ ⟶ + + +

nitric acid + potassium manganate ⟶ water + potassium permanganate + manganese dioxide + potassium nitrate

Construct the equilibrium constant, K, expression for: HNO_3 + K_2MnO_4 ⟶ H_2O + KMnO_4 + MnO_2 + KNO_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: 4 HNO_3 + 3 K_2MnO_4 ⟶ 2 H_2O + 2 KMnO_4 + MnO_2 + 4 KNO_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 HNO_3 | 4 | -4 K_2MnO_4 | 3 | -3 H_2O | 2 | 2 KMnO_4 | 2 | 2 MnO_2 | 1 | 1 KNO_3 | 4 | 4 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HNO_3 | 4 | -4 | ([HNO3])^(-4) K_2MnO_4 | 3 | -3 | ([K2MnO4])^(-3) H_2O | 2 | 2 | ([H2O])^2 KMnO_4 | 2 | 2 | ([KMnO4])^2 MnO_2 | 1 | 1 | [MnO2] KNO_3 | 4 | 4 | ([KNO3])^4 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])^(-4) ([K2MnO4])^(-3) ([H2O])^2 ([KMnO4])^2 [MnO2] ([KNO3])^4 = (([H2O])^2 ([KMnO4])^2 [MnO2] ([KNO3])^4)/(([HNO3])^4 ([K2MnO4])^3)

Construct the rate of reaction expression for: HNO_3 + K_2MnO_4 ⟶ H_2O + KMnO_4 + MnO_2 + KNO_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: 4 HNO_3 + 3 K_2MnO_4 ⟶ 2 H_2O + 2 KMnO_4 + MnO_2 + 4 KNO_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 HNO_3 | 4 | -4 K_2MnO_4 | 3 | -3 H_2O | 2 | 2 KMnO_4 | 2 | 2 MnO_2 | 1 | 1 KNO_3 | 4 | 4 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 | 4 | -4 | -1/4 (Δ[HNO3])/(Δt) K_2MnO_4 | 3 | -3 | -1/3 (Δ[K2MnO4])/(Δt) H_2O | 2 | 2 | 1/2 (Δ[H2O])/(Δt) KMnO_4 | 2 | 2 | 1/2 (Δ[KMnO4])/(Δt) MnO_2 | 1 | 1 | (Δ[MnO2])/(Δt) KNO_3 | 4 | 4 | 1/4 (Δ[KNO3])/(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 = -1/4 (Δ[HNO3])/(Δt) = -1/3 (Δ[K2MnO4])/(Δt) = 1/2 (Δ[H2O])/(Δt) = 1/2 (Δ[KMnO4])/(Δt) = (Δ[MnO2])/(Δt) = 1/4 (Δ[KNO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | potassium manganate | water | potassium permanganate | manganese dioxide | potassium nitrate formula | HNO_3 | K_2MnO_4 | H_2O | KMnO_4 | MnO_2 | KNO_3 name | nitric acid | potassium manganate | water | potassium permanganate | manganese dioxide | potassium nitrate IUPAC name | nitric acid | dipotassium dioxido-dioxomanganese | water | potassium permanganate | dioxomanganese | potassium nitrate

| nitric acid | potassium manganate | water | potassium permanganate | manganese dioxide | potassium nitrate molar mass | 63.012 g/mol | 197.13 g/mol | 18.015 g/mol | 158.03 g/mol | 86.936 g/mol | 101.1 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) | solid (at STP) melting point | -41.6 °C | 190 °C | 0 °C | 240 °C | 535 °C | 334 °C boiling point | 83 °C | | 99.9839 °C | | | density | 1.5129 g/cm^3 | | 1 g/cm^3 | 1 g/cm^3 | 5.03 g/cm^3 | solubility in water | miscible | decomposes | | | insoluble | soluble surface tension | | | 0.0728 N/m | | | dynamic viscosity | 7.6×10^-4 Pa s (at 25 °C) | | 8.9×10^-4 Pa s (at 25 °C) | | | odor | | | odorless | odorless | | odorless