H2O + AgNO3 + P = HNO3 + H3PO4 + Ag

H_2O water + AgNO_3 silver nitrate + P red phosphorus ⟶ HNO_3 nitric acid + H_3PO_4 phosphoric acid + Ag silver

Balance the chemical equation algebraically: H_2O + AgNO_3 + P ⟶ HNO_3 + H_3PO_4 + Ag Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 AgNO_3 + c_3 P ⟶ c_4 HNO_3 + c_5 H_3PO_4 + c_6 Ag Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Ag, N and P: H: | 2 c_1 = c_4 + 3 c_5 O: | c_1 + 3 c_2 = 3 c_4 + 4 c_5 Ag: | c_2 = c_6 N: | c_2 = c_4 P: | c_3 = 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 4 c_2 = 5 c_3 = 1 c_4 = 5 c_5 = 1 c_6 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2O + 5 AgNO_3 + P ⟶ 5 HNO_3 + H_3PO_4 + 5 Ag

+ + ⟶ + +

water + silver nitrate + red phosphorus ⟶ nitric acid + phosphoric acid + silver

Construct the equilibrium constant, K, expression for: H_2O + AgNO_3 + P ⟶ HNO_3 + H_3PO_4 + Ag 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 H_2O + 5 AgNO_3 + P ⟶ 5 HNO_3 + H_3PO_4 + 5 Ag 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 | 4 | -4 AgNO_3 | 5 | -5 P | 1 | -1 HNO_3 | 5 | 5 H_3PO_4 | 1 | 1 Ag | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 4 | -4 | ([H2O])^(-4) AgNO_3 | 5 | -5 | ([AgNO3])^(-5) P | 1 | -1 | ([P])^(-1) HNO_3 | 5 | 5 | ([HNO3])^5 H_3PO_4 | 1 | 1 | [H3PO4] Ag | 5 | 5 | ([Ag])^5 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])^(-4) ([AgNO3])^(-5) ([P])^(-1) ([HNO3])^5 [H3PO4] ([Ag])^5 = (([HNO3])^5 [H3PO4] ([Ag])^5)/(([H2O])^4 ([AgNO3])^5 [P])

Construct the rate of reaction expression for: H_2O + AgNO_3 + P ⟶ HNO_3 + H_3PO_4 + Ag 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 H_2O + 5 AgNO_3 + P ⟶ 5 HNO_3 + H_3PO_4 + 5 Ag 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 | 4 | -4 AgNO_3 | 5 | -5 P | 1 | -1 HNO_3 | 5 | 5 H_3PO_4 | 1 | 1 Ag | 5 | 5 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 | 4 | -4 | -1/4 (Δ[H2O])/(Δt) AgNO_3 | 5 | -5 | -1/5 (Δ[AgNO3])/(Δt) P | 1 | -1 | -(Δ[P])/(Δt) HNO_3 | 5 | 5 | 1/5 (Δ[HNO3])/(Δt) H_3PO_4 | 1 | 1 | (Δ[H3PO4])/(Δt) Ag | 5 | 5 | 1/5 (Δ[Ag])/(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[H2O])/(Δt) = -1/5 (Δ[AgNO3])/(Δt) = -(Δ[P])/(Δt) = 1/5 (Δ[HNO3])/(Δt) = (Δ[H3PO4])/(Δt) = 1/5 (Δ[Ag])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | silver nitrate | red phosphorus | nitric acid | phosphoric acid | silver formula | H_2O | AgNO_3 | P | HNO_3 | H_3PO_4 | Ag Hill formula | H_2O | AgNO_3 | P | HNO_3 | H_3O_4P | Ag name | water | silver nitrate | red phosphorus | nitric acid | phosphoric acid | silver IUPAC name | water | silver nitrate | phosphorus | nitric acid | phosphoric acid | silver