H2SO4 + Cr2(SO4)3 + NaBiO3 = H2O + Na2SO4 + H2Cr2O7 + Bi2(SO4)3

H_2SO_4 sulfuric acid + Cr_2(SO_4)_3 chromium sulfate + NaBiO_3 sodium bismuthate ⟶ H_2O water + Na_2SO_4 sodium sulfate + H_2Cr_2O_7 dichromic acid + Bi_2(SO_4)_3 bismuth sulfate

Balance the chemical equation algebraically: H_2SO_4 + Cr_2(SO_4)_3 + NaBiO_3 ⟶ H_2O + Na_2SO_4 + H_2Cr_2O_7 + Bi_2(SO_4)_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 Cr_2(SO_4)_3 + c_3 NaBiO_3 ⟶ c_4 H_2O + c_5 Na_2SO_4 + c_6 H_2Cr_2O_7 + c_7 Bi_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, Cr, Bi and Na: H: | 2 c_1 = 2 c_4 + 2 c_6 O: | 4 c_1 + 12 c_2 + 3 c_3 = c_4 + 4 c_5 + 7 c_6 + 12 c_7 S: | c_1 + 3 c_2 = c_5 + 3 c_7 Cr: | 2 c_2 = 2 c_6 Bi: | c_3 = 2 c_7 Na: | c_3 = 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 1 c_3 = 3 c_4 = 2 c_5 = 3/2 c_6 = 1 c_7 = 3/2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 6 c_2 = 2 c_3 = 6 c_4 = 4 c_5 = 3 c_6 = 2 c_7 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 H_2SO_4 + 2 Cr_2(SO_4)_3 + 6 NaBiO_3 ⟶ 4 H_2O + 3 Na_2SO_4 + 2 H_2Cr_2O_7 + 3 Bi_2(SO_4)_3

+ + ⟶ + + +

sulfuric acid + chromium sulfate + sodium bismuthate ⟶ water + sodium sulfate + dichromic acid + bismuth sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + Cr_2(SO_4)_3 + NaBiO_3 ⟶ H_2O + Na_2SO_4 + H_2Cr_2O_7 + Bi_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 + 2 Cr_2(SO_4)_3 + 6 NaBiO_3 ⟶ 4 H_2O + 3 Na_2SO_4 + 2 H_2Cr_2O_7 + 3 Bi_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 Cr_2(SO_4)_3 | 2 | -2 NaBiO_3 | 6 | -6 H_2O | 4 | 4 Na_2SO_4 | 3 | 3 H_2Cr_2O_7 | 2 | 2 Bi_2(SO_4)_3 | 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 | 6 | -6 | ([H2SO4])^(-6) Cr_2(SO_4)_3 | 2 | -2 | ([Cr2(SO4)3])^(-2) NaBiO_3 | 6 | -6 | ([NaBiO3])^(-6) H_2O | 4 | 4 | ([H2O])^4 Na_2SO_4 | 3 | 3 | ([Na2SO4])^3 H_2Cr_2O_7 | 2 | 2 | ([H2Cr2O7])^2 Bi_2(SO_4)_3 | 3 | 3 | ([Bi2(SO4)3])^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])^(-6) ([Cr2(SO4)3])^(-2) ([NaBiO3])^(-6) ([H2O])^4 ([Na2SO4])^3 ([H2Cr2O7])^2 ([Bi2(SO4)3])^3 = (([H2O])^4 ([Na2SO4])^3 ([H2Cr2O7])^2 ([Bi2(SO4)3])^3)/(([H2SO4])^6 ([Cr2(SO4)3])^2 ([NaBiO3])^6)

Construct the rate of reaction expression for: H_2SO_4 + Cr_2(SO_4)_3 + NaBiO_3 ⟶ H_2O + Na_2SO_4 + H_2Cr_2O_7 + Bi_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 + 2 Cr_2(SO_4)_3 + 6 NaBiO_3 ⟶ 4 H_2O + 3 Na_2SO_4 + 2 H_2Cr_2O_7 + 3 Bi_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 Cr_2(SO_4)_3 | 2 | -2 NaBiO_3 | 6 | -6 H_2O | 4 | 4 Na_2SO_4 | 3 | 3 H_2Cr_2O_7 | 2 | 2 Bi_2(SO_4)_3 | 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 | 6 | -6 | -1/6 (Δ[H2SO4])/(Δt) Cr_2(SO_4)_3 | 2 | -2 | -1/2 (Δ[Cr2(SO4)3])/(Δt) NaBiO_3 | 6 | -6 | -1/6 (Δ[NaBiO3])/(Δt) H_2O | 4 | 4 | 1/4 (Δ[H2O])/(Δt) Na_2SO_4 | 3 | 3 | 1/3 (Δ[Na2SO4])/(Δt) H_2Cr_2O_7 | 2 | 2 | 1/2 (Δ[H2Cr2O7])/(Δt) Bi_2(SO_4)_3 | 3 | 3 | 1/3 (Δ[Bi2(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) = -1/2 (Δ[Cr2(SO4)3])/(Δt) = -1/6 (Δ[NaBiO3])/(Δt) = 1/4 (Δ[H2O])/(Δt) = 1/3 (Δ[Na2SO4])/(Δt) = 1/2 (Δ[H2Cr2O7])/(Δt) = 1/3 (Δ[Bi2(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | chromium sulfate | sodium bismuthate | water | sodium sulfate | dichromic acid | bismuth sulfate formula | H_2SO_4 | Cr_2(SO_4)_3 | NaBiO_3 | H_2O | Na_2SO_4 | H_2Cr_2O_7 | Bi_2(SO_4)_3 Hill formula | H_2O_4S | Cr_2O_12S_3 | BiNaO_3 | H_2O | Na_2O_4S | Cr_2H_2O_7 | Bi_2O_12S_3 name | sulfuric acid | chromium sulfate | sodium bismuthate | water | sodium sulfate | dichromic acid | bismuth sulfate IUPAC name | sulfuric acid | chromium(+3) cation trisulfate | sodium oxido-dioxobismuth | water | disodium sulfate | hydroxy-(hydroxy-dioxo-chromio)oxy-dioxo-chromium | dibismuth trisulfate

| sulfuric acid | chromium sulfate | sodium bismuthate | water | sodium sulfate | dichromic acid | bismuth sulfate molar mass | 98.07 g/mol | 392.2 g/mol | 279.967 g/mol | 18.015 g/mol | 142.04 g/mol | 218 g/mol | 706.1 g/mol phase | liquid (at STP) | liquid (at STP) | | liquid (at STP) | solid (at STP) | | melting point | 10.371 °C | | | 0 °C | 884 °C | | boiling point | 279.6 °C | 330 °C | | 99.9839 °C | 1429 °C | | density | 1.8305 g/cm^3 | 1.84 g/cm^3 | | 1 g/cm^3 | 2.68 g/cm^3 | 1.66 g/cm^3 | solubility in water | very soluble | | insoluble | | 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 | | |