KMnO4 + (NH4)2SO4 = H2O + K2SO4 + MnO2 + N2

KMnO_4 potassium permanganate + (NH_4)_2SO_4 ammonium sulfate ⟶ H_2O water + K_2SO_4 potassium sulfate + MnO_2 manganese dioxide + N_2 nitrogen

Balance the chemical equation algebraically: KMnO_4 + (NH_4)_2SO_4 ⟶ H_2O + K_2SO_4 + MnO_2 + N_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KMnO_4 + c_2 (NH_4)_2SO_4 ⟶ c_3 H_2O + c_4 K_2SO_4 + c_5 MnO_2 + c_6 N_2 Set the number of atoms in the reactants equal to the number of atoms in the products for K, Mn, O, H, N and S: K: | c_1 = 2 c_4 Mn: | c_1 = c_5 O: | 4 c_1 + 4 c_2 = c_3 + 4 c_4 + 2 c_5 H: | 8 c_2 = 2 c_3 N: | 2 c_2 = 2 c_6 S: | c_2 = c_4 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 2 c_2 = 1 c_3 = 4 c_4 = 1 c_5 = 2 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 KMnO_4 + (NH_4)_2SO_4 ⟶ 4 H_2O + K_2SO_4 + 2 MnO_2 + N_2

+ ⟶ + + +

potassium permanganate + ammonium sulfate ⟶ water + potassium sulfate + manganese dioxide + nitrogen

Construct the equilibrium constant, K, expression for: KMnO_4 + (NH_4)_2SO_4 ⟶ H_2O + K_2SO_4 + MnO_2 + N_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: 2 KMnO_4 + (NH_4)_2SO_4 ⟶ 4 H_2O + K_2SO_4 + 2 MnO_2 + N_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 KMnO_4 | 2 | -2 (NH_4)_2SO_4 | 1 | -1 H_2O | 4 | 4 K_2SO_4 | 1 | 1 MnO_2 | 2 | 2 N_2 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KMnO_4 | 2 | -2 | ([KMnO4])^(-2) (NH_4)_2SO_4 | 1 | -1 | ([(NH4)2SO4])^(-1) H_2O | 4 | 4 | ([H2O])^4 K_2SO_4 | 1 | 1 | [K2SO4] MnO_2 | 2 | 2 | ([MnO2])^2 N_2 | 1 | 1 | [N2] 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 = ([KMnO4])^(-2) ([(NH4)2SO4])^(-1) ([H2O])^4 [K2SO4] ([MnO2])^2 [N2] = (([H2O])^4 [K2SO4] ([MnO2])^2 [N2])/(([KMnO4])^2 [(NH4)2SO4])

Construct the rate of reaction expression for: KMnO_4 + (NH_4)_2SO_4 ⟶ H_2O + K_2SO_4 + MnO_2 + N_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: 2 KMnO_4 + (NH_4)_2SO_4 ⟶ 4 H_2O + K_2SO_4 + 2 MnO_2 + N_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 KMnO_4 | 2 | -2 (NH_4)_2SO_4 | 1 | -1 H_2O | 4 | 4 K_2SO_4 | 1 | 1 MnO_2 | 2 | 2 N_2 | 1 | 1 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 KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) (NH_4)_2SO_4 | 1 | -1 | -(Δ[(NH4)2SO4])/(Δt) H_2O | 4 | 4 | 1/4 (Δ[H2O])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) MnO_2 | 2 | 2 | 1/2 (Δ[MnO2])/(Δt) N_2 | 1 | 1 | (Δ[N2])/(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[KMnO4])/(Δt) = -(Δ[(NH4)2SO4])/(Δt) = 1/4 (Δ[H2O])/(Δt) = (Δ[K2SO4])/(Δt) = 1/2 (Δ[MnO2])/(Δt) = (Δ[N2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium permanganate | ammonium sulfate | water | potassium sulfate | manganese dioxide | nitrogen formula | KMnO_4 | (NH_4)_2SO_4 | H_2O | K_2SO_4 | MnO_2 | N_2 Hill formula | KMnO_4 | H_8N_2O_4S | H_2O | K_2O_4S | MnO_2 | N_2 name | potassium permanganate | ammonium sulfate | water | potassium sulfate | manganese dioxide | nitrogen IUPAC name | potassium permanganate | | water | dipotassium sulfate | dioxomanganese | molecular nitrogen

| potassium permanganate | ammonium sulfate | water | potassium sulfate | manganese dioxide | nitrogen molar mass | 158.03 g/mol | 132.1 g/mol | 18.015 g/mol | 174.25 g/mol | 86.936 g/mol | 28.014 g/mol phase | solid (at STP) | solid (at STP) | liquid (at STP) | | solid (at STP) | gas (at STP) melting point | 240 °C | 280 °C | 0 °C | | 535 °C | -210 °C boiling point | | | 99.9839 °C | | | -195.79 °C density | 1 g/cm^3 | 1.77 g/cm^3 | 1 g/cm^3 | | 5.03 g/cm^3 | 0.001251 g/cm^3 (at 0 °C) solubility in water | | | | soluble | insoluble | insoluble surface tension | | | 0.0728 N/m | | | 0.0066 N/m dynamic viscosity | | | 8.9×10^-4 Pa s (at 25 °C) | | | 1.78×10^-5 Pa s (at 25 °C) odor | odorless | odorless | odorless | | | odorless