H2SO4 + K2Cr2O7 + SmSO4 = H2O + K2SO4 + Cr2(SO4)3 + Sm(SO4)3

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + SmSO4 ⟶ H_2O water + K_2SO_4 potassium sulfate + Cr_2(SO_4)_3 chromium sulfate + Sm(SO4)3

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + SmSO4 ⟶ H_2O + K_2SO_4 + Cr_2(SO_4)_3 + Sm(SO4)3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 SmSO4 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 Cr_2(SO_4)_3 + c_7 Sm(SO4)3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Cr, K and Sm: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 7 c_2 + 4 c_3 = c_4 + 4 c_5 + 12 c_6 + 12 c_7 S: | c_1 + c_3 = c_5 + 3 c_6 + 3 c_7 Cr: | 2 c_2 = 2 c_6 K: | 2 c_2 = 2 c_5 Sm: | c_3 = 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 7 c_2 = 1 c_3 = 3/2 c_4 = 7 c_5 = 1 c_6 = 1 c_7 = 3/2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 14 c_2 = 2 c_3 = 3 c_4 = 14 c_5 = 2 c_6 = 2 c_7 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 14 H_2SO_4 + 2 K_2Cr_2O_7 + 3 SmSO4 ⟶ 14 H_2O + 2 K_2SO_4 + 2 Cr_2(SO_4)_3 + 3 Sm(SO4)3

+ + SmSO4 ⟶ + + + Sm(SO4)3

sulfuric acid + potassium dichromate + SmSO4 ⟶ water + potassium sulfate + chromium sulfate + Sm(SO4)3

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + SmSO4 ⟶ H_2O + K_2SO_4 + Cr_2(SO_4)_3 + Sm(SO4)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: 14 H_2SO_4 + 2 K_2Cr_2O_7 + 3 SmSO4 ⟶ 14 H_2O + 2 K_2SO_4 + 2 Cr_2(SO_4)_3 + 3 Sm(SO4)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 | 14 | -14 K_2Cr_2O_7 | 2 | -2 SmSO4 | 3 | -3 H_2O | 14 | 14 K_2SO_4 | 2 | 2 Cr_2(SO_4)_3 | 2 | 2 Sm(SO4)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 | 14 | -14 | ([H2SO4])^(-14) K_2Cr_2O_7 | 2 | -2 | ([K2Cr2O7])^(-2) SmSO4 | 3 | -3 | ([SmSO4])^(-3) H_2O | 14 | 14 | ([H2O])^14 K_2SO_4 | 2 | 2 | ([K2SO4])^2 Cr_2(SO_4)_3 | 2 | 2 | ([Cr2(SO4)3])^2 Sm(SO4)3 | 3 | 3 | ([Sm(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])^(-14) ([K2Cr2O7])^(-2) ([SmSO4])^(-3) ([H2O])^14 ([K2SO4])^2 ([Cr2(SO4)3])^2 ([Sm(SO4)3])^3 = (([H2O])^14 ([K2SO4])^2 ([Cr2(SO4)3])^2 ([Sm(SO4)3])^3)/(([H2SO4])^14 ([K2Cr2O7])^2 ([SmSO4])^3)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + SmSO4 ⟶ H_2O + K_2SO_4 + Cr_2(SO_4)_3 + Sm(SO4)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: 14 H_2SO_4 + 2 K_2Cr_2O_7 + 3 SmSO4 ⟶ 14 H_2O + 2 K_2SO_4 + 2 Cr_2(SO_4)_3 + 3 Sm(SO4)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 | 14 | -14 K_2Cr_2O_7 | 2 | -2 SmSO4 | 3 | -3 H_2O | 14 | 14 K_2SO_4 | 2 | 2 Cr_2(SO_4)_3 | 2 | 2 Sm(SO4)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 | 14 | -14 | -1/14 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 2 | -2 | -1/2 (Δ[K2Cr2O7])/(Δt) SmSO4 | 3 | -3 | -1/3 (Δ[SmSO4])/(Δt) H_2O | 14 | 14 | 1/14 (Δ[H2O])/(Δt) K_2SO_4 | 2 | 2 | 1/2 (Δ[K2SO4])/(Δt) Cr_2(SO_4)_3 | 2 | 2 | 1/2 (Δ[Cr2(SO4)3])/(Δt) Sm(SO4)3 | 3 | 3 | 1/3 (Δ[Sm(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 = -1/14 (Δ[H2SO4])/(Δt) = -1/2 (Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[SmSO4])/(Δt) = 1/14 (Δ[H2O])/(Δt) = 1/2 (Δ[K2SO4])/(Δt) = 1/2 (Δ[Cr2(SO4)3])/(Δt) = 1/3 (Δ[Sm(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | SmSO4 | water | potassium sulfate | chromium sulfate | Sm(SO4)3 formula | H_2SO_4 | K_2Cr_2O_7 | SmSO4 | H_2O | K_2SO_4 | Cr_2(SO_4)_3 | Sm(SO4)3 Hill formula | H_2O_4S | Cr_2K_2O_7 | O4SSm | H_2O | K_2O_4S | Cr_2O_12S_3 | O12S3Sm name | sulfuric acid | potassium dichromate | | water | potassium sulfate | chromium sulfate | IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | | water | dipotassium sulfate | chromium(+3) cation trisulfate |

| sulfuric acid | potassium dichromate | SmSO4 | water | potassium sulfate | chromium sulfate | Sm(SO4)3 molar mass | 98.07 g/mol | 294.18 g/mol | 246.42 g/mol | 18.015 g/mol | 174.25 g/mol | 392.2 g/mol | 438.5 g/mol phase | liquid (at STP) | solid (at STP) | | liquid (at STP) | | liquid (at STP) | melting point | 10.371 °C | 398 °C | | 0 °C | | | boiling point | 279.6 °C | | | 99.9839 °C | | 330 °C | density | 1.8305 g/cm^3 | 2.67 g/cm^3 | | 1 g/cm^3 | | 1.84 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 | | odorless | | odorless |