H2O + KMnO4 + P = KOH + MnO2 + KH2PO4 + K2HPO4

H_2O water + KMnO_4 potassium permanganate + P red phosphorus ⟶ KOH potassium hydroxide + MnO_2 manganese dioxide + KH_2PO_4 potassium dihydrogen phosphate + K_2HPO_4 dipotassium hydrogen phosphate

Balance the chemical equation algebraically: H_2O + KMnO_4 + P ⟶ KOH + MnO_2 + KH_2PO_4 + K_2HPO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 P ⟶ c_4 KOH + c_5 MnO_2 + c_6 KH_2PO_4 + c_7 K_2HPO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn and P: H: | 2 c_1 = c_4 + 2 c_6 + c_7 O: | c_1 + 4 c_2 = c_4 + 2 c_5 + 4 c_6 + 4 c_7 K: | c_2 = c_4 + c_6 + 2 c_7 Mn: | c_2 = c_5 P: | c_3 = c_6 + 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_4 = 1 and solve the system of equations for the remaining coefficients: c_2 = (5 c_1)/2 - 5/2 c_3 = (3 c_1)/2 - 3/2 c_4 = 1 c_5 = (5 c_1)/2 - 5/2 c_6 = c_1/2 + 1/2 c_7 = c_1 - 2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 5 and solve for the remaining coefficients: c_1 = 5 c_2 = 10 c_3 = 6 c_4 = 1 c_5 = 10 c_6 = 3 c_7 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 5 H_2O + 10 KMnO_4 + 6 P ⟶ KOH + 10 MnO_2 + 3 KH_2PO_4 + 3 K_2HPO_4

+ + ⟶ + + +

water + potassium permanganate + red phosphorus ⟶ potassium hydroxide + manganese dioxide + potassium dihydrogen phosphate + dipotassium hydrogen phosphate

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + P ⟶ KOH + MnO_2 + KH_2PO_4 + K_2HPO_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: 5 H_2O + 10 KMnO_4 + 6 P ⟶ KOH + 10 MnO_2 + 3 KH_2PO_4 + 3 K_2HPO_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 H_2O | 5 | -5 KMnO_4 | 10 | -10 P | 6 | -6 KOH | 1 | 1 MnO_2 | 10 | 10 KH_2PO_4 | 3 | 3 K_2HPO_4 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 5 | -5 | ([H2O])^(-5) KMnO_4 | 10 | -10 | ([KMnO4])^(-10) P | 6 | -6 | ([P])^(-6) KOH | 1 | 1 | [KOH] MnO_2 | 10 | 10 | ([MnO2])^10 KH_2PO_4 | 3 | 3 | ([KH2PO4])^3 K_2HPO_4 | 3 | 3 | ([K2HPO4])^3 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])^(-5) ([KMnO4])^(-10) ([P])^(-6) [KOH] ([MnO2])^10 ([KH2PO4])^3 ([K2HPO4])^3 = ([KOH] ([MnO2])^10 ([KH2PO4])^3 ([K2HPO4])^3)/(([H2O])^5 ([KMnO4])^10 ([P])^6)

Construct the rate of reaction expression for: H_2O + KMnO_4 + P ⟶ KOH + MnO_2 + KH_2PO_4 + K_2HPO_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: 5 H_2O + 10 KMnO_4 + 6 P ⟶ KOH + 10 MnO_2 + 3 KH_2PO_4 + 3 K_2HPO_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 H_2O | 5 | -5 KMnO_4 | 10 | -10 P | 6 | -6 KOH | 1 | 1 MnO_2 | 10 | 10 KH_2PO_4 | 3 | 3 K_2HPO_4 | 3 | 3 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 | 5 | -5 | -1/5 (Δ[H2O])/(Δt) KMnO_4 | 10 | -10 | -1/10 (Δ[KMnO4])/(Δt) P | 6 | -6 | -1/6 (Δ[P])/(Δt) KOH | 1 | 1 | (Δ[KOH])/(Δt) MnO_2 | 10 | 10 | 1/10 (Δ[MnO2])/(Δt) KH_2PO_4 | 3 | 3 | 1/3 (Δ[KH2PO4])/(Δt) K_2HPO_4 | 3 | 3 | 1/3 (Δ[K2HPO4])/(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 = -1/5 (Δ[H2O])/(Δt) = -1/10 (Δ[KMnO4])/(Δt) = -1/6 (Δ[P])/(Δt) = (Δ[KOH])/(Δt) = 1/10 (Δ[MnO2])/(Δt) = 1/3 (Δ[KH2PO4])/(Δt) = 1/3 (Δ[K2HPO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | red phosphorus | potassium hydroxide | manganese dioxide | potassium dihydrogen phosphate | dipotassium hydrogen phosphate formula | H_2O | KMnO_4 | P | KOH | MnO_2 | KH_2PO_4 | K_2HPO_4 Hill formula | H_2O | KMnO_4 | P | HKO | MnO_2 | H_2KO_4P | HK_2O_4P name | water | potassium permanganate | red phosphorus | potassium hydroxide | manganese dioxide | potassium dihydrogen phosphate | dipotassium hydrogen phosphate IUPAC name | water | potassium permanganate | phosphorus | potassium hydroxide | dioxomanganese | potassium dihydrogen phosphate | dipotassium hydrogen phosphate