H2SO4 + K2Cr2O7 + Na2S = H2O + K2SO4 + Na2SO4 + Cr2S

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + Na_2S sodium sulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + Na_2SO_4 sodium sulfate + Cr2S

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + Na_2S ⟶ H_2O + K_2SO_4 + Na_2SO_4 + Cr2S Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 Na_2S ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 Na_2SO_4 + c_7 Cr2S Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Cr, K and Na: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 7 c_2 = c_4 + 4 c_5 + 4 c_6 S: | c_1 + c_3 = c_5 + c_6 + c_7 Cr: | 2 c_2 = 2 c_7 K: | 2 c_2 = 2 c_5 Na: | 2 c_3 = 2 c_6 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 = 2 c_2 = 1 c_3 = 9/4 c_4 = 2 c_5 = 1 c_6 = 9/4 c_7 = 1 Multiply by the least common denominator, 4, to eliminate fractional coefficients: c_1 = 8 c_2 = 4 c_3 = 9 c_4 = 8 c_5 = 4 c_6 = 9 c_7 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2SO_4 + 4 K_2Cr_2O_7 + 9 Na_2S ⟶ 8 H_2O + 4 K_2SO_4 + 9 Na_2SO_4 + 4 Cr2S

+ + ⟶ + + + Cr2S

sulfuric acid + potassium dichromate + sodium sulfide ⟶ water + potassium sulfate + sodium sulfate + Cr2S

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + Na_2S ⟶ H_2O + K_2SO_4 + Na_2SO_4 + Cr2S 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: 8 H_2SO_4 + 4 K_2Cr_2O_7 + 9 Na_2S ⟶ 8 H_2O + 4 K_2SO_4 + 9 Na_2SO_4 + 4 Cr2S 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 | 8 | -8 K_2Cr_2O_7 | 4 | -4 Na_2S | 9 | -9 H_2O | 8 | 8 K_2SO_4 | 4 | 4 Na_2SO_4 | 9 | 9 Cr2S | 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 | 8 | -8 | ([H2SO4])^(-8) K_2Cr_2O_7 | 4 | -4 | ([K2Cr2O7])^(-4) Na_2S | 9 | -9 | ([Na2S])^(-9) H_2O | 8 | 8 | ([H2O])^8 K_2SO_4 | 4 | 4 | ([K2SO4])^4 Na_2SO_4 | 9 | 9 | ([Na2SO4])^9 Cr2S | 4 | 4 | ([Cr2S])^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])^(-8) ([K2Cr2O7])^(-4) ([Na2S])^(-9) ([H2O])^8 ([K2SO4])^4 ([Na2SO4])^9 ([Cr2S])^4 = (([H2O])^8 ([K2SO4])^4 ([Na2SO4])^9 ([Cr2S])^4)/(([H2SO4])^8 ([K2Cr2O7])^4 ([Na2S])^9)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + Na_2S ⟶ H_2O + K_2SO_4 + Na_2SO_4 + Cr2S 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: 8 H_2SO_4 + 4 K_2Cr_2O_7 + 9 Na_2S ⟶ 8 H_2O + 4 K_2SO_4 + 9 Na_2SO_4 + 4 Cr2S 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 | 8 | -8 K_2Cr_2O_7 | 4 | -4 Na_2S | 9 | -9 H_2O | 8 | 8 K_2SO_4 | 4 | 4 Na_2SO_4 | 9 | 9 Cr2S | 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 | 8 | -8 | -1/8 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 4 | -4 | -1/4 (Δ[K2Cr2O7])/(Δt) Na_2S | 9 | -9 | -1/9 (Δ[Na2S])/(Δt) H_2O | 8 | 8 | 1/8 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) Na_2SO_4 | 9 | 9 | 1/9 (Δ[Na2SO4])/(Δt) Cr2S | 4 | 4 | 1/4 (Δ[Cr2S])/(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 = -1/8 (Δ[H2SO4])/(Δt) = -1/4 (Δ[K2Cr2O7])/(Δt) = -1/9 (Δ[Na2S])/(Δt) = 1/8 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/9 (Δ[Na2SO4])/(Δt) = 1/4 (Δ[Cr2S])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | sodium sulfide | water | potassium sulfate | sodium sulfate | Cr2S formula | H_2SO_4 | K_2Cr_2O_7 | Na_2S | H_2O | K_2SO_4 | Na_2SO_4 | Cr2S Hill formula | H_2O_4S | Cr_2K_2O_7 | Na_2S_1 | H_2O | K_2O_4S | Na_2O_4S | Cr2S name | sulfuric acid | potassium dichromate | sodium sulfide | water | potassium sulfate | sodium sulfate | IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | | water | dipotassium sulfate | disodium sulfate |

| sulfuric acid | potassium dichromate | sodium sulfide | water | potassium sulfate | sodium sulfate | Cr2S molar mass | 98.07 g/mol | 294.18 g/mol | 78.04 g/mol | 18.015 g/mol | 174.25 g/mol | 142.04 g/mol | 136.05 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | | solid (at STP) | melting point | 10.371 °C | 398 °C | 1172 °C | 0 °C | | 884 °C | boiling point | 279.6 °C | | | 99.9839 °C | | 1429 °C | density | 1.8305 g/cm^3 | 2.67 g/cm^3 | 1.856 g/cm^3 | 1 g/cm^3 | | 2.68 g/cm^3 | solubility in water | very soluble | | | | 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 | | |