H2O + P4O6 = H3PO4 + P + PH3

H_2O water + O_6P_4 tetraphosphorus(III) hexoxide ⟶ H_3PO_4 phosphoric acid + P red phosphorus + PH_3 phosphine

Balance the chemical equation algebraically: H_2O + O_6P_4 ⟶ H_3PO_4 + P + PH_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 O_6P_4 ⟶ c_3 H_3PO_4 + c_4 P + c_5 PH_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O and P: H: | 2 c_1 = 3 c_3 + 3 c_5 O: | c_1 + 6 c_2 = 4 c_3 P: | 4 c_2 = c_3 + 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_2 = (5 c_1)/18 - 2/3 c_3 = (2 c_1)/3 - 1 c_4 = (4 c_1)/9 - 8/3 c_5 = 1 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 24 and solve for the remaining coefficients: c_1 = 24 c_2 = 6 c_3 = 15 c_4 = 8 c_5 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 24 H_2O + 6 O_6P_4 ⟶ 15 H_3PO_4 + 8 P + PH_3

+ ⟶ + +

water + tetraphosphorus(III) hexoxide ⟶ phosphoric acid + red phosphorus + phosphine

Construct the equilibrium constant, K, expression for: H_2O + O_6P_4 ⟶ H_3PO_4 + P + PH_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: 24 H_2O + 6 O_6P_4 ⟶ 15 H_3PO_4 + 8 P + PH_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 H_2O | 24 | -24 O_6P_4 | 6 | -6 H_3PO_4 | 15 | 15 P | 8 | 8 PH_3 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 24 | -24 | ([H2O])^(-24) O_6P_4 | 6 | -6 | ([O6P4])^(-6) H_3PO_4 | 15 | 15 | ([H3PO4])^15 P | 8 | 8 | ([P])^8 PH_3 | 1 | 1 | [PH3] 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])^(-24) ([O6P4])^(-6) ([H3PO4])^15 ([P])^8 [PH3] = (([H3PO4])^15 ([P])^8 [PH3])/(([H2O])^24 ([O6P4])^6)

Construct the rate of reaction expression for: H_2O + O_6P_4 ⟶ H_3PO_4 + P + PH_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: 24 H_2O + 6 O_6P_4 ⟶ 15 H_3PO_4 + 8 P + PH_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 H_2O | 24 | -24 O_6P_4 | 6 | -6 H_3PO_4 | 15 | 15 P | 8 | 8 PH_3 | 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 H_2O | 24 | -24 | -1/24 (Δ[H2O])/(Δt) O_6P_4 | 6 | -6 | -1/6 (Δ[O6P4])/(Δt) H_3PO_4 | 15 | 15 | 1/15 (Δ[H3PO4])/(Δt) P | 8 | 8 | 1/8 (Δ[P])/(Δt) PH_3 | 1 | 1 | (Δ[PH3])/(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 = -1/24 (Δ[H2O])/(Δt) = -1/6 (Δ[O6P4])/(Δt) = 1/15 (Δ[H3PO4])/(Δt) = 1/8 (Δ[P])/(Δt) = (Δ[PH3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | tetraphosphorus(III) hexoxide | phosphoric acid | red phosphorus | phosphine formula | H_2O | O_6P_4 | H_3PO_4 | P | PH_3 Hill formula | H_2O | O_6P_4 | H_3O_4P | P | H_3P name | water | tetraphosphorus(III) hexoxide | phosphoric acid | red phosphorus | phosphine IUPAC name | water | | phosphoric acid | phosphorus | phosphine