H2O + HNO3 + P = NO + NO2 + H3PO4

H_2O water + HNO_3 nitric acid + P red phosphorus ⟶ NO nitric oxide + NO_2 nitrogen dioxide + H_3PO_4 phosphoric acid

Balance the chemical equation algebraically: H_2O + HNO_3 + P ⟶ NO + NO_2 + H_3PO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 HNO_3 + c_3 P ⟶ c_4 NO + c_5 NO_2 + c_6 H_3PO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, N and P: H: | 2 c_1 + c_2 = 3 c_6 O: | c_1 + 3 c_2 = c_4 + 2 c_5 + 4 c_6 N: | c_2 = c_4 + c_5 P: | 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_3 = c_2/3 + 2/3 c_4 = c_2/3 + 5/3 c_5 = (2 c_2)/3 - 5/3 c_6 = c_2/3 + 2/3 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_2 = 7 and solve for the remaining coefficients: c_1 = 1 c_2 = 7 c_3 = 3 c_4 = 4 c_5 = 3 c_6 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + 7 HNO_3 + 3 P ⟶ 4 NO + 3 NO_2 + 3 H_3PO_4

+ + ⟶ + +

water + nitric acid + red phosphorus ⟶ nitric oxide + nitrogen dioxide + phosphoric acid

Construct the equilibrium constant, K, expression for: H_2O + HNO_3 + P ⟶ NO + NO_2 + H_3PO_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: H_2O + 7 HNO_3 + 3 P ⟶ 4 NO + 3 NO_2 + 3 H_3PO_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 | 1 | -1 HNO_3 | 7 | -7 P | 3 | -3 NO | 4 | 4 NO_2 | 3 | 3 H_3PO_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 | 1 | -1 | ([H2O])^(-1) HNO_3 | 7 | -7 | ([HNO3])^(-7) P | 3 | -3 | ([P])^(-3) NO | 4 | 4 | ([NO])^4 NO_2 | 3 | 3 | ([NO2])^3 H_3PO_4 | 3 | 3 | ([H3PO4])^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])^(-1) ([HNO3])^(-7) ([P])^(-3) ([NO])^4 ([NO2])^3 ([H3PO4])^3 = (([NO])^4 ([NO2])^3 ([H3PO4])^3)/([H2O] ([HNO3])^7 ([P])^3)

Construct the rate of reaction expression for: H_2O + HNO_3 + P ⟶ NO + NO_2 + H_3PO_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: H_2O + 7 HNO_3 + 3 P ⟶ 4 NO + 3 NO_2 + 3 H_3PO_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 | 1 | -1 HNO_3 | 7 | -7 P | 3 | -3 NO | 4 | 4 NO_2 | 3 | 3 H_3PO_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 | 1 | -1 | -(Δ[H2O])/(Δt) HNO_3 | 7 | -7 | -1/7 (Δ[HNO3])/(Δt) P | 3 | -3 | -1/3 (Δ[P])/(Δt) NO | 4 | 4 | 1/4 (Δ[NO])/(Δt) NO_2 | 3 | 3 | 1/3 (Δ[NO2])/(Δt) H_3PO_4 | 3 | 3 | 1/3 (Δ[H3PO4])/(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 = -(Δ[H2O])/(Δt) = -1/7 (Δ[HNO3])/(Δt) = -1/3 (Δ[P])/(Δt) = 1/4 (Δ[NO])/(Δt) = 1/3 (Δ[NO2])/(Δt) = 1/3 (Δ[H3PO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | nitric acid | red phosphorus | nitric oxide | nitrogen dioxide | phosphoric acid formula | H_2O | HNO_3 | P | NO | NO_2 | H_3PO_4 Hill formula | H_2O | HNO_3 | P | NO | NO_2 | H_3O_4P name | water | nitric acid | red phosphorus | nitric oxide | nitrogen dioxide | phosphoric acid IUPAC name | water | nitric acid | phosphorus | nitric oxide | Nitrogen dioxide | phosphoric acid

| water | nitric acid | red phosphorus | nitric oxide | nitrogen dioxide | phosphoric acid molar mass | 18.015 g/mol | 63.012 g/mol | 30.973761998 g/mol | 30.006 g/mol | 46.005 g/mol | 97.994 g/mol phase | liquid (at STP) | liquid (at STP) | solid (at STP) | gas (at STP) | gas (at STP) | liquid (at STP) melting point | 0 °C | -41.6 °C | 579.2 °C | -163.6 °C | -11 °C | 42.4 °C boiling point | 99.9839 °C | 83 °C | | -151.7 °C | 21 °C | 158 °C density | 1 g/cm^3 | 1.5129 g/cm^3 | 2.16 g/cm^3 | 0.001226 g/cm^3 (at 25 °C) | 0.00188 g/cm^3 (at 25 °C) | 1.685 g/cm^3 solubility in water | | miscible | insoluble | | reacts | very soluble surface tension | 0.0728 N/m | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | 7.6×10^-4 Pa s (at 25 °C) | 7.6×10^-4 Pa s (at 20.2 °C) | 1.911×10^-5 Pa s (at 25 °C) | 4.02×10^-4 Pa s (at 25 °C) | odor | odorless | | | | | odorless