KOH + KMnO4 + Cr2O3 = H2O + MnO2 + K2CrO4

KOH potassium hydroxide + KMnO_4 potassium permanganate + Cr_2O_3 chromium(III) oxide ⟶ H_2O water + MnO_2 manganese dioxide + K_2CrO_4 potassium chromate

Balance the chemical equation algebraically: KOH + KMnO_4 + Cr_2O_3 ⟶ H_2O + MnO_2 + K_2CrO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KOH + c_2 KMnO_4 + c_3 Cr_2O_3 ⟶ c_4 H_2O + c_5 MnO_2 + c_6 K_2CrO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, K, O, Mn and Cr: H: | c_1 = 2 c_4 K: | c_1 + c_2 = 2 c_6 O: | c_1 + 4 c_2 + 3 c_3 = c_4 + 2 c_5 + 4 c_6 Mn: | c_2 = c_5 Cr: | 2 c_3 = 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 = 2 c_2 = 2 c_3 = 1 c_4 = 1 c_5 = 2 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 KOH + 2 KMnO_4 + Cr_2O_3 ⟶ H_2O + 2 MnO_2 + 2 K_2CrO_4

+ + ⟶ + +

potassium hydroxide + potassium permanganate + chromium(III) oxide ⟶ water + manganese dioxide + potassium chromate

Construct the equilibrium constant, K, expression for: KOH + KMnO_4 + Cr_2O_3 ⟶ H_2O + MnO_2 + K_2CrO_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: 2 KOH + 2 KMnO_4 + Cr_2O_3 ⟶ H_2O + 2 MnO_2 + 2 K_2CrO_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 KOH | 2 | -2 KMnO_4 | 2 | -2 Cr_2O_3 | 1 | -1 H_2O | 1 | 1 MnO_2 | 2 | 2 K_2CrO_4 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KOH | 2 | -2 | ([KOH])^(-2) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) Cr_2O_3 | 1 | -1 | ([Cr2O3])^(-1) H_2O | 1 | 1 | [H2O] MnO_2 | 2 | 2 | ([MnO2])^2 K_2CrO_4 | 2 | 2 | ([K2CrO4])^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 = ([KOH])^(-2) ([KMnO4])^(-2) ([Cr2O3])^(-1) [H2O] ([MnO2])^2 ([K2CrO4])^2 = ([H2O] ([MnO2])^2 ([K2CrO4])^2)/(([KOH])^2 ([KMnO4])^2 [Cr2O3])

Construct the rate of reaction expression for: KOH + KMnO_4 + Cr_2O_3 ⟶ H_2O + MnO_2 + K_2CrO_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: 2 KOH + 2 KMnO_4 + Cr_2O_3 ⟶ H_2O + 2 MnO_2 + 2 K_2CrO_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 KOH | 2 | -2 KMnO_4 | 2 | -2 Cr_2O_3 | 1 | -1 H_2O | 1 | 1 MnO_2 | 2 | 2 K_2CrO_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 KOH | 2 | -2 | -1/2 (Δ[KOH])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) Cr_2O_3 | 1 | -1 | -(Δ[Cr2O3])/(Δt) H_2O | 1 | 1 | (Δ[H2O])/(Δt) MnO_2 | 2 | 2 | 1/2 (Δ[MnO2])/(Δt) K_2CrO_4 | 2 | 2 | 1/2 (Δ[K2CrO4])/(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 = -1/2 (Δ[KOH])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -(Δ[Cr2O3])/(Δt) = (Δ[H2O])/(Δt) = 1/2 (Δ[MnO2])/(Δt) = 1/2 (Δ[K2CrO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium hydroxide | potassium permanganate | chromium(III) oxide | water | manganese dioxide | potassium chromate formula | KOH | KMnO_4 | Cr_2O_3 | H_2O | MnO_2 | K_2CrO_4 Hill formula | HKO | KMnO_4 | Cr_2O_3 | H_2O | MnO_2 | CrK_2O_4 name | potassium hydroxide | potassium permanganate | chromium(III) oxide | water | manganese dioxide | potassium chromate IUPAC name | potassium hydroxide | potassium permanganate | | water | dioxomanganese | dipotassium dioxido-dioxochromium

| potassium hydroxide | potassium permanganate | chromium(III) oxide | water | manganese dioxide | potassium chromate molar mass | 56.105 g/mol | 158.03 g/mol | 151.99 g/mol | 18.015 g/mol | 86.936 g/mol | 194.19 g/mol phase | solid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) melting point | 406 °C | 240 °C | 2435 °C | 0 °C | 535 °C | 971 °C boiling point | 1327 °C | | 4000 °C | 99.9839 °C | | density | 2.044 g/cm^3 | 1 g/cm^3 | 4.8 g/cm^3 | 1 g/cm^3 | 5.03 g/cm^3 | 2.73 g/cm^3 solubility in water | soluble | | insoluble | | insoluble | soluble surface tension | | | | 0.0728 N/m | | dynamic viscosity | 0.001 Pa s (at 550 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | | odor | | odorless | | odorless | | odorless