H3PO4 + Ra(OH)2 = H2O + Ra3(PO4)2

H_3PO_4 phosphoric acid + Ra(OH)2 ⟶ H_2O water + Ra3(PO4)2

Balance the chemical equation algebraically: H_3PO_4 + Ra(OH)2 ⟶ H_2O + Ra3(PO4)2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_3PO_4 + c_2 Ra(OH)2 ⟶ c_3 H_2O + c_4 Ra3(PO4)2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, P and Ra: H: | 3 c_1 + 2 c_2 = 2 c_3 O: | 4 c_1 + 2 c_2 = c_3 + 8 c_4 P: | c_1 = 2 c_4 Ra: | c_2 = 3 c_4 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_1 = 2 c_2 = 3 c_3 = 6 c_4 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_3PO_4 + 3 Ra(OH)2 ⟶ 6 H_2O + Ra3(PO4)2

+ Ra(OH)2 ⟶ + Ra3(PO4)2

phosphoric acid + Ra(OH)2 ⟶ water + Ra3(PO4)2

Construct the equilibrium constant, K, expression for: H_3PO_4 + Ra(OH)2 ⟶ H_2O + Ra3(PO4)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_3PO_4 + 3 Ra(OH)2 ⟶ 6 H_2O + Ra3(PO4)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_3PO_4 | 2 | -2 Ra(OH)2 | 3 | -3 H_2O | 6 | 6 Ra3(PO4)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_3PO_4 | 2 | -2 | ([H3PO4])^(-2) Ra(OH)2 | 3 | -3 | ([Ra(OH)2])^(-3) H_2O | 6 | 6 | ([H2O])^6 Ra3(PO4)2 | 1 | 1 | [Ra3(PO4)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 = ([H3PO4])^(-2) ([Ra(OH)2])^(-3) ([H2O])^6 [Ra3(PO4)2] = (([H2O])^6 [Ra3(PO4)2])/(([H3PO4])^2 ([Ra(OH)2])^3)

Construct the rate of reaction expression for: H_3PO_4 + Ra(OH)2 ⟶ H_2O + Ra3(PO4)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_3PO_4 + 3 Ra(OH)2 ⟶ 6 H_2O + Ra3(PO4)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_3PO_4 | 2 | -2 Ra(OH)2 | 3 | -3 H_2O | 6 | 6 Ra3(PO4)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_3PO_4 | 2 | -2 | -1/2 (Δ[H3PO4])/(Δt) Ra(OH)2 | 3 | -3 | -1/3 (Δ[Ra(OH)2])/(Δt) H_2O | 6 | 6 | 1/6 (Δ[H2O])/(Δt) Ra3(PO4)2 | 1 | 1 | (Δ[Ra3(PO4)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[H3PO4])/(Δt) = -1/3 (Δ[Ra(OH)2])/(Δt) = 1/6 (Δ[H2O])/(Δt) = (Δ[Ra3(PO4)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| phosphoric acid | Ra(OH)2 | water | Ra3(PO4)2 formula | H_3PO_4 | Ra(OH)2 | H_2O | Ra3(PO4)2 Hill formula | H_3O_4P | H2O2Ra | H_2O | O8P2Ra3 name | phosphoric acid | | water |

| phosphoric acid | Ra(OH)2 | water | Ra3(PO4)2 molar mass | 97.994 g/mol | 260 g/mol | 18.015 g/mol | 868 g/mol phase | liquid (at STP) | | liquid (at STP) | melting point | 42.4 °C | | 0 °C | boiling point | 158 °C | | 99.9839 °C | density | 1.685 g/cm^3 | | 1 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) | odor | odorless | | odorless |