H2SO4 + H2Cr2O7 + H2S2O3 = H2O + Cr2(SO4)3

H_2SO_4 sulfuric acid + H_2Cr_2O_7 dichromic acid + H2S2O3 ⟶ H_2O water + Cr_2(SO_4)_3 chromium sulfate

Balance the chemical equation algebraically: H_2SO_4 + H_2Cr_2O_7 + H2S2O3 ⟶ H_2O + Cr_2(SO_4)_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 H_2Cr_2O_7 + c_3 H2S2O3 ⟶ c_4 H_2O + c_5 Cr_2(SO_4)_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S and Cr: H: | 2 c_1 + 2 c_2 + 2 c_3 = 2 c_4 O: | 4 c_1 + 7 c_2 + 3 c_3 = c_4 + 12 c_5 S: | c_1 + 2 c_3 = 3 c_5 Cr: | 2 c_2 = 2 c_5 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_1 = 2 c_2 = 4/3 c_3 = 1 c_4 = 13/3 c_5 = 4/3 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 6 c_2 = 4 c_3 = 3 c_4 = 13 c_5 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 H_2SO_4 + 4 H_2Cr_2O_7 + 3 H2S2O3 ⟶ 13 H_2O + 4 Cr_2(SO_4)_3

+ + H2S2O3 ⟶ +

sulfuric acid + dichromic acid + H2S2O3 ⟶ water + chromium sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + H_2Cr_2O_7 + H2S2O3 ⟶ H_2O + Cr_2(SO_4)_3 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 H_2SO_4 + 4 H_2Cr_2O_7 + 3 H2S2O3 ⟶ 13 H_2O + 4 Cr_2(SO_4)_3 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 | 6 | -6 H_2Cr_2O_7 | 4 | -4 H2S2O3 | 3 | -3 H_2O | 13 | 13 Cr_2(SO_4)_3 | 4 | 4 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 6 | -6 | ([H2SO4])^(-6) H_2Cr_2O_7 | 4 | -4 | ([H2Cr2O7])^(-4) H2S2O3 | 3 | -3 | ([H2S2O3])^(-3) H_2O | 13 | 13 | ([H2O])^13 Cr_2(SO_4)_3 | 4 | 4 | ([Cr2(SO4)3])^4 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])^(-6) ([H2Cr2O7])^(-4) ([H2S2O3])^(-3) ([H2O])^13 ([Cr2(SO4)3])^4 = (([H2O])^13 ([Cr2(SO4)3])^4)/(([H2SO4])^6 ([H2Cr2O7])^4 ([H2S2O3])^3)

Construct the rate of reaction expression for: H_2SO_4 + H_2Cr_2O_7 + H2S2O3 ⟶ H_2O + Cr_2(SO_4)_3 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 H_2SO_4 + 4 H_2Cr_2O_7 + 3 H2S2O3 ⟶ 13 H_2O + 4 Cr_2(SO_4)_3 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 | 6 | -6 H_2Cr_2O_7 | 4 | -4 H2S2O3 | 3 | -3 H_2O | 13 | 13 Cr_2(SO_4)_3 | 4 | 4 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 | 6 | -6 | -1/6 (Δ[H2SO4])/(Δt) H_2Cr_2O_7 | 4 | -4 | -1/4 (Δ[H2Cr2O7])/(Δt) H2S2O3 | 3 | -3 | -1/3 (Δ[H2S2O3])/(Δt) H_2O | 13 | 13 | 1/13 (Δ[H2O])/(Δt) Cr_2(SO_4)_3 | 4 | 4 | 1/4 (Δ[Cr2(SO4)3])/(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[H2SO4])/(Δt) = -1/4 (Δ[H2Cr2O7])/(Δt) = -1/3 (Δ[H2S2O3])/(Δt) = 1/13 (Δ[H2O])/(Δt) = 1/4 (Δ[Cr2(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | dichromic acid | H2S2O3 | water | chromium sulfate formula | H_2SO_4 | H_2Cr_2O_7 | H2S2O3 | H_2O | Cr_2(SO_4)_3 Hill formula | H_2O_4S | Cr_2H_2O_7 | H2O3S2 | H_2O | Cr_2O_12S_3 name | sulfuric acid | dichromic acid | | water | chromium sulfate IUPAC name | sulfuric acid | hydroxy-(hydroxy-dioxo-chromio)oxy-dioxo-chromium | | water | chromium(+3) cation trisulfate

| sulfuric acid | dichromic acid | H2S2O3 | water | chromium sulfate molar mass | 98.07 g/mol | 218 g/mol | 114.1 g/mol | 18.015 g/mol | 392.2 g/mol phase | liquid (at STP) | | | liquid (at STP) | liquid (at STP) melting point | 10.371 °C | | | 0 °C | boiling point | 279.6 °C | | | 99.9839 °C | 330 °C density | 1.8305 g/cm^3 | 1.66 g/cm^3 | | 1 g/cm^3 | 1.84 g/cm^3 solubility in water | very 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 | odorless