H2O + P2O5 = H4P2O7

H_2O water + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_2 pyrophosphoric acid

Balance the chemical equation algebraically: H_2O + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 P2O5 ⟶ c_3 (HO)_2P(O)OP(O)(OH)_2 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 = 4 c_3 O: | c_1 + 5 c_2 = 7 c_3 P: | 2 c_2 = 2 c_3 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 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_2

+ P2O5 ⟶

water + P2O5 ⟶ pyrophosphoric acid

Construct the equilibrium constant, K, expression for: H_2O + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_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 H_2O + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_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 H_2O | 2 | -2 P2O5 | 1 | -1 (HO)_2P(O)OP(O)(OH)_2 | 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 | 2 | -2 | ([H2O])^(-2) P2O5 | 1 | -1 | ([P2O5])^(-1) (HO)_2P(O)OP(O)(OH)_2 | 1 | 1 | [(HO)2P(O)OP(O)(OH)2] 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])^(-2) ([P2O5])^(-1) [(HO)2P(O)OP(O)(OH)2] = ([(HO)2P(O)OP(O)(OH)2])/(([H2O])^2 [P2O5])

Construct the rate of reaction expression for: H_2O + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_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 H_2O + P2O5 ⟶ (HO)_2P(O)OP(O)(OH)_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 H_2O | 2 | -2 P2O5 | 1 | -1 (HO)_2P(O)OP(O)(OH)_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 H_2O | 2 | -2 | -1/2 (Δ[H2O])/(Δt) P2O5 | 1 | -1 | -(Δ[P2O5])/(Δt) (HO)_2P(O)OP(O)(OH)_2 | 1 | 1 | (Δ[(HO)2P(O)OP(O)(OH)2])/(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[H2O])/(Δt) = -(Δ[P2O5])/(Δt) = (Δ[(HO)2P(O)OP(O)(OH)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | P2O5 | pyrophosphoric acid formula | H_2O | P2O5 | (HO)_2P(O)OP(O)(OH)_2 Hill formula | H_2O | O5P2 | H_4O_7P_2 name | water | | pyrophosphoric acid IUPAC name | water | | phosphono dihydrogen phosphate

| water | P2O5 | pyrophosphoric acid molar mass | 18.015 g/mol | 141.94 g/mol | 177.97 g/mol phase | liquid (at STP) | | solid (at STP) melting point | 0 °C | | 61 °C boiling point | 99.9839 °C | | density | 1 g/cm^3 | | 1.75 g/cm^3 solubility in water | | | very soluble surface tension | 0.0728 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | 0.62 Pa s (at 25 °C) odor | odorless | |