P + SO3 = S + P2O5

P red phosphorus + SO_3 sulfur trioxide ⟶ S mixed sulfur + P2O5

Balance the chemical equation algebraically: P + SO_3 ⟶ S + P2O5 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 P + c_2 SO_3 ⟶ c_3 S + c_4 P2O5 Set the number of atoms in the reactants equal to the number of atoms in the products for P, O and S: P: | c_1 = 2 c_4 O: | 3 c_2 = 5 c_4 S: | c_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_4 = 1 and solve the system of equations for the remaining coefficients: c_1 = 2 c_2 = 5/3 c_3 = 5/3 c_4 = 1 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 6 c_2 = 5 c_3 = 5 c_4 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 P + 5 SO_3 ⟶ 5 S + 3 P2O5

+ ⟶ + P2O5

red phosphorus + sulfur trioxide ⟶ mixed sulfur + P2O5

Construct the equilibrium constant, K, expression for: P + SO_3 ⟶ S + P2O5 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: 6 P + 5 SO_3 ⟶ 5 S + 3 P2O5 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 P | 6 | -6 SO_3 | 5 | -5 S | 5 | 5 P2O5 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression P | 6 | -6 | ([P])^(-6) SO_3 | 5 | -5 | ([SO3])^(-5) S | 5 | 5 | ([S])^5 P2O5 | 3 | 3 | ([P2O5])^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 = ([P])^(-6) ([SO3])^(-5) ([S])^5 ([P2O5])^3 = (([S])^5 ([P2O5])^3)/(([P])^6 ([SO3])^5)

Construct the rate of reaction expression for: P + SO_3 ⟶ S + P2O5 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: 6 P + 5 SO_3 ⟶ 5 S + 3 P2O5 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 P | 6 | -6 SO_3 | 5 | -5 S | 5 | 5 P2O5 | 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 P | 6 | -6 | -1/6 (Δ[P])/(Δt) SO_3 | 5 | -5 | -1/5 (Δ[SO3])/(Δt) S | 5 | 5 | 1/5 (Δ[S])/(Δt) P2O5 | 3 | 3 | 1/3 (Δ[P2O5])/(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 = -1/6 (Δ[P])/(Δt) = -1/5 (Δ[SO3])/(Δt) = 1/5 (Δ[S])/(Δt) = 1/3 (Δ[P2O5])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| red phosphorus | sulfur trioxide | mixed sulfur | P2O5 formula | P | SO_3 | S | P2O5 Hill formula | P | O_3S | S | O5P2 name | red phosphorus | sulfur trioxide | mixed sulfur | IUPAC name | phosphorus | sulfur trioxide | sulfur |

| red phosphorus | sulfur trioxide | mixed sulfur | P2O5 molar mass | 30.973761998 g/mol | 80.06 g/mol | 32.06 g/mol | 141.94 g/mol phase | solid (at STP) | liquid (at STP) | solid (at STP) | melting point | 579.2 °C | 16.8 °C | 112.8 °C | boiling point | | 44.7 °C | 444.7 °C | density | 2.16 g/cm^3 | 1.97 g/cm^3 | 2.07 g/cm^3 | solubility in water | insoluble | reacts | | dynamic viscosity | 7.6×10^-4 Pa s (at 20.2 °C) | 0.00159 Pa s (at 30 °C) | |