H2SO4 + K2Cr2O7 + P2S3 = H2O + K2SO4 + SO2 + Cr2(SO4)3 + CrPO4

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + P_2S_3 phosphorus trisulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + SO_2 sulfur dioxide + Cr_2(SO_4)_3 chromium sulfate + O_4PCr chromium(III) phosphate

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + P_2S_3 ⟶ H_2O + K_2SO_4 + SO_2 + Cr_2(SO_4)_3 + O_4PCr Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 P_2S_3 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 SO_2 + c_7 Cr_2(SO_4)_3 + c_8 O_4PCr Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Cr, K and P: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 7 c_2 = c_4 + 4 c_5 + 2 c_6 + 12 c_7 + 4 c_8 S: | c_1 + 3 c_3 = c_5 + c_6 + 3 c_7 Cr: | 2 c_2 = 2 c_7 + c_8 K: | 2 c_2 = 2 c_5 P: | 2 c_3 = c_8 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_2 = c_1 - 8 c_3 = 1 c_4 = c_1 c_5 = c_1 - 8 c_6 = 38 - 3 c_1 c_7 = c_1 - 9 c_8 = 2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 10 and solve for the remaining coefficients: c_1 = 10 c_2 = 2 c_3 = 1 c_4 = 10 c_5 = 2 c_6 = 8 c_7 = 1 c_8 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 10 H_2SO_4 + 2 K_2Cr_2O_7 + P_2S_3 ⟶ 10 H_2O + 2 K_2SO_4 + 8 SO_2 + Cr_2(SO_4)_3 + 2 O_4PCr

+ + ⟶ + + + +

sulfuric acid + potassium dichromate + phosphorus trisulfide ⟶ water + potassium sulfate + sulfur dioxide + chromium sulfate + chromium(III) phosphate

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + P_2S_3 ⟶ H_2O + K_2SO_4 + SO_2 + Cr_2(SO_4)_3 + O_4PCr 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: 10 H_2SO_4 + 2 K_2Cr_2O_7 + P_2S_3 ⟶ 10 H_2O + 2 K_2SO_4 + 8 SO_2 + Cr_2(SO_4)_3 + 2 O_4PCr 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_2SO_4 | 10 | -10 K_2Cr_2O_7 | 2 | -2 P_2S_3 | 1 | -1 H_2O | 10 | 10 K_2SO_4 | 2 | 2 SO_2 | 8 | 8 Cr_2(SO_4)_3 | 1 | 1 O_4PCr | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 10 | -10 | ([H2SO4])^(-10) K_2Cr_2O_7 | 2 | -2 | ([K2Cr2O7])^(-2) P_2S_3 | 1 | -1 | ([P2S3])^(-1) H_2O | 10 | 10 | ([H2O])^10 K_2SO_4 | 2 | 2 | ([K2SO4])^2 SO_2 | 8 | 8 | ([SO2])^8 Cr_2(SO_4)_3 | 1 | 1 | [Cr2(SO4)3] O_4PCr | 2 | 2 | ([O4P1Cr1])^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 = ([H2SO4])^(-10) ([K2Cr2O7])^(-2) ([P2S3])^(-1) ([H2O])^10 ([K2SO4])^2 ([SO2])^8 [Cr2(SO4)3] ([O4P1Cr1])^2 = (([H2O])^10 ([K2SO4])^2 ([SO2])^8 [Cr2(SO4)3] ([O4P1Cr1])^2)/(([H2SO4])^10 ([K2Cr2O7])^2 [P2S3])

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + P_2S_3 ⟶ H_2O + K_2SO_4 + SO_2 + Cr_2(SO_4)_3 + O_4PCr 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: 10 H_2SO_4 + 2 K_2Cr_2O_7 + P_2S_3 ⟶ 10 H_2O + 2 K_2SO_4 + 8 SO_2 + Cr_2(SO_4)_3 + 2 O_4PCr 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_2SO_4 | 10 | -10 K_2Cr_2O_7 | 2 | -2 P_2S_3 | 1 | -1 H_2O | 10 | 10 K_2SO_4 | 2 | 2 SO_2 | 8 | 8 Cr_2(SO_4)_3 | 1 | 1 O_4PCr | 2 | 2 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_2SO_4 | 10 | -10 | -1/10 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 2 | -2 | -1/2 (Δ[K2Cr2O7])/(Δt) P_2S_3 | 1 | -1 | -(Δ[P2S3])/(Δt) H_2O | 10 | 10 | 1/10 (Δ[H2O])/(Δt) K_2SO_4 | 2 | 2 | 1/2 (Δ[K2SO4])/(Δt) SO_2 | 8 | 8 | 1/8 (Δ[SO2])/(Δt) Cr_2(SO_4)_3 | 1 | 1 | (Δ[Cr2(SO4)3])/(Δt) O_4PCr | 2 | 2 | 1/2 (Δ[O4P1Cr1])/(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 = -1/10 (Δ[H2SO4])/(Δt) = -1/2 (Δ[K2Cr2O7])/(Δt) = -(Δ[P2S3])/(Δt) = 1/10 (Δ[H2O])/(Δt) = 1/2 (Δ[K2SO4])/(Δt) = 1/8 (Δ[SO2])/(Δt) = (Δ[Cr2(SO4)3])/(Δt) = 1/2 (Δ[O4P1Cr1])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | phosphorus trisulfide | water | potassium sulfate | sulfur dioxide | chromium sulfate | chromium(III) phosphate formula | H_2SO_4 | K_2Cr_2O_7 | P_2S_3 | H_2O | K_2SO_4 | SO_2 | Cr_2(SO_4)_3 | O_4PCr Hill formula | H_2O_4S | Cr_2K_2O_7 | P_2S_3 | H_2O | K_2O_4S | O_2S | Cr_2O_12S_3 | CrO_4P name | sulfuric acid | potassium dichromate | phosphorus trisulfide | water | potassium sulfate | sulfur dioxide | chromium sulfate | chromium(III) phosphate IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | | water | dipotassium sulfate | sulfur dioxide | chromium(+3) cation trisulfate | chromium(3+) phosphate