H2SO4 + Na2S + Na2Cr2O7 = H2O + S + Na2SO4 + Cr2(SO4)2

H_2SO_4 sulfuric acid + Na_2S sodium sulfide + Na_2Cr_2O_7 sodium bichromate ⟶ H_2O water + S mixed sulfur + Na_2SO_4 sodium sulfate + Cr2(SO4)2

Balance the chemical equation algebraically: H_2SO_4 + Na_2S + Na_2Cr_2O_7 ⟶ H_2O + S + Na_2SO_4 + Cr2(SO4)2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 Na_2S + c_3 Na_2Cr_2O_7 ⟶ c_4 H_2O + c_5 S + c_6 Na_2SO_4 + c_7 Cr2(SO4)2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Na and Cr: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 7 c_3 = c_4 + 4 c_6 + 8 c_7 S: | c_1 + c_2 = c_5 + c_6 + 2 c_7 Na: | 2 c_2 + 2 c_3 = 2 c_6 Cr: | 2 c_3 = 2 c_7 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 = (3 c_1)/4 - 5/4 c_3 = 1 c_4 = c_1 c_5 = c_1 - 3 c_6 = (3 c_1)/4 - 1/4 c_7 = 1 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_2 = (3 c_1)/4 - 15/4 c_3 = 3 c_4 = c_1 c_5 = c_1 - 9 c_6 = (3 c_1)/4 - 3/4 c_7 = 3 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 13 and solve for the remaining coefficients: c_1 = 13 c_2 = 6 c_3 = 3 c_4 = 13 c_5 = 4 c_6 = 9 c_7 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 13 H_2SO_4 + 6 Na_2S + 3 Na_2Cr_2O_7 ⟶ 13 H_2O + 4 S + 9 Na_2SO_4 + 3 Cr2(SO4)2

+ + ⟶ + + + Cr2(SO4)2

sulfuric acid + sodium sulfide + sodium bichromate ⟶ water + mixed sulfur + sodium sulfate + Cr2(SO4)2

Construct the equilibrium constant, K, expression for: H_2SO_4 + Na_2S + Na_2Cr_2O_7 ⟶ H_2O + S + Na_2SO_4 + Cr2(SO4)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: 13 H_2SO_4 + 6 Na_2S + 3 Na_2Cr_2O_7 ⟶ 13 H_2O + 4 S + 9 Na_2SO_4 + 3 Cr2(SO4)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_2SO_4 | 13 | -13 Na_2S | 6 | -6 Na_2Cr_2O_7 | 3 | -3 H_2O | 13 | 13 S | 4 | 4 Na_2SO_4 | 9 | 9 Cr2(SO4)2 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 13 | -13 | ([H2SO4])^(-13) Na_2S | 6 | -6 | ([Na2S])^(-6) Na_2Cr_2O_7 | 3 | -3 | ([Na2Cr2O7])^(-3) H_2O | 13 | 13 | ([H2O])^13 S | 4 | 4 | ([S])^4 Na_2SO_4 | 9 | 9 | ([Na2SO4])^9 Cr2(SO4)2 | 3 | 3 | ([Cr2(SO4)2])^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 = ([H2SO4])^(-13) ([Na2S])^(-6) ([Na2Cr2O7])^(-3) ([H2O])^13 ([S])^4 ([Na2SO4])^9 ([Cr2(SO4)2])^3 = (([H2O])^13 ([S])^4 ([Na2SO4])^9 ([Cr2(SO4)2])^3)/(([H2SO4])^13 ([Na2S])^6 ([Na2Cr2O7])^3)

Construct the rate of reaction expression for: H_2SO_4 + Na_2S + Na_2Cr_2O_7 ⟶ H_2O + S + Na_2SO_4 + Cr2(SO4)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: 13 H_2SO_4 + 6 Na_2S + 3 Na_2Cr_2O_7 ⟶ 13 H_2O + 4 S + 9 Na_2SO_4 + 3 Cr2(SO4)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_2SO_4 | 13 | -13 Na_2S | 6 | -6 Na_2Cr_2O_7 | 3 | -3 H_2O | 13 | 13 S | 4 | 4 Na_2SO_4 | 9 | 9 Cr2(SO4)2 | 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_2SO_4 | 13 | -13 | -1/13 (Δ[H2SO4])/(Δt) Na_2S | 6 | -6 | -1/6 (Δ[Na2S])/(Δt) Na_2Cr_2O_7 | 3 | -3 | -1/3 (Δ[Na2Cr2O7])/(Δt) H_2O | 13 | 13 | 1/13 (Δ[H2O])/(Δt) S | 4 | 4 | 1/4 (Δ[S])/(Δt) Na_2SO_4 | 9 | 9 | 1/9 (Δ[Na2SO4])/(Δt) Cr2(SO4)2 | 3 | 3 | 1/3 (Δ[Cr2(SO4)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 = -1/13 (Δ[H2SO4])/(Δt) = -1/6 (Δ[Na2S])/(Δt) = -1/3 (Δ[Na2Cr2O7])/(Δt) = 1/13 (Δ[H2O])/(Δt) = 1/4 (Δ[S])/(Δt) = 1/9 (Δ[Na2SO4])/(Δt) = 1/3 (Δ[Cr2(SO4)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | sodium sulfide | sodium bichromate | water | mixed sulfur | sodium sulfate | Cr2(SO4)2 formula | H_2SO_4 | Na_2S | Na_2Cr_2O_7 | H_2O | S | Na_2SO_4 | Cr2(SO4)2 Hill formula | H_2O_4S | Na_2S_1 | Cr_2Na_2O_7 | H_2O | S | Na_2O_4S | Cr2O8S2 name | sulfuric acid | sodium sulfide | sodium bichromate | water | mixed sulfur | sodium sulfate | IUPAC name | sulfuric acid | | disodium oxido-(oxido-dioxo-chromio)oxy-dioxo-chromium | water | sulfur | disodium sulfate |

| sulfuric acid | sodium sulfide | sodium bichromate | water | mixed sulfur | sodium sulfate | Cr2(SO4)2 molar mass | 98.07 g/mol | 78.04 g/mol | 261.96 g/mol | 18.015 g/mol | 32.06 g/mol | 142.04 g/mol | 296.1 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) | melting point | 10.371 °C | 1172 °C | 356.7 °C | 0 °C | 112.8 °C | 884 °C | boiling point | 279.6 °C | | | 99.9839 °C | 444.7 °C | 1429 °C | density | 1.8305 g/cm^3 | 1.856 g/cm^3 | 2.35 g/cm^3 | 1 g/cm^3 | 2.07 g/cm^3 | 2.68 g/cm^3 | solubility in water | very soluble | | | | | soluble | surface tension | 0.0735 N/m | | | 0.0728 N/m | | | dynamic viscosity | 0.021 Pa s (at 25 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | | | odor | odorless | | | odorless | | |