H2O + K2Cr2O7 + (NH4)2S = S + KOH + NH3 + Cr(OH)3

H_2O water + K_2Cr_2O_7 potassium dichromate + (NH_4)_2S diammonium sulfide ⟶ S mixed sulfur + KOH potassium hydroxide + NH_3 ammonia + Cr(OH)3

Balance the chemical equation algebraically: H_2O + K_2Cr_2O_7 + (NH_4)_2S ⟶ S + KOH + NH_3 + Cr(OH)3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 K_2Cr_2O_7 + c_3 (NH_4)_2S ⟶ c_4 S + c_5 KOH + c_6 NH_3 + c_7 Cr(OH)3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Cr, K, N and S: H: | 2 c_1 + 8 c_3 = c_5 + 3 c_6 + 3 c_7 O: | c_1 + 7 c_2 = c_5 + 3 c_7 Cr: | 2 c_2 = c_7 K: | 2 c_2 = c_5 N: | 2 c_3 = c_6 S: | c_3 = c_4 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 1 c_3 = 3 c_4 = 3 c_5 = 2 c_6 = 6 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 3 S + 2 KOH + 6 NH_3 + 2 Cr(OH)3

+ + ⟶ + + + Cr(OH)3

water + potassium dichromate + diammonium sulfide ⟶ mixed sulfur + potassium hydroxide + ammonia + Cr(OH)3

Construct the equilibrium constant, K, expression for: H_2O + K_2Cr_2O_7 + (NH_4)_2S ⟶ S + KOH + NH_3 + Cr(OH)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: H_2O + K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 3 S + 2 KOH + 6 NH_3 + 2 Cr(OH)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_2O | 1 | -1 K_2Cr_2O_7 | 1 | -1 (NH_4)_2S | 3 | -3 S | 3 | 3 KOH | 2 | 2 NH_3 | 6 | 6 Cr(OH)3 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 1 | -1 | ([H2O])^(-1) K_2Cr_2O_7 | 1 | -1 | ([K2Cr2O7])^(-1) (NH_4)_2S | 3 | -3 | ([(NH4)2S])^(-3) S | 3 | 3 | ([S])^3 KOH | 2 | 2 | ([KOH])^2 NH_3 | 6 | 6 | ([NH3])^6 Cr(OH)3 | 2 | 2 | ([Cr(OH)3])^2 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 = ([H2O])^(-1) ([K2Cr2O7])^(-1) ([(NH4)2S])^(-3) ([S])^3 ([KOH])^2 ([NH3])^6 ([Cr(OH)3])^2 = (([S])^3 ([KOH])^2 ([NH3])^6 ([Cr(OH)3])^2)/([H2O] [K2Cr2O7] ([(NH4)2S])^3)

Construct the rate of reaction expression for: H_2O + K_2Cr_2O_7 + (NH_4)_2S ⟶ S + KOH + NH_3 + Cr(OH)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: H_2O + K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 3 S + 2 KOH + 6 NH_3 + 2 Cr(OH)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_2O | 1 | -1 K_2Cr_2O_7 | 1 | -1 (NH_4)_2S | 3 | -3 S | 3 | 3 KOH | 2 | 2 NH_3 | 6 | 6 Cr(OH)3 | 2 | 2 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_2O | 1 | -1 | -(Δ[H2O])/(Δt) K_2Cr_2O_7 | 1 | -1 | -(Δ[K2Cr2O7])/(Δt) (NH_4)_2S | 3 | -3 | -1/3 (Δ[(NH4)2S])/(Δt) S | 3 | 3 | 1/3 (Δ[S])/(Δt) KOH | 2 | 2 | 1/2 (Δ[KOH])/(Δt) NH_3 | 6 | 6 | 1/6 (Δ[NH3])/(Δt) Cr(OH)3 | 2 | 2 | 1/2 (Δ[Cr(OH)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 = -(Δ[H2O])/(Δt) = -(Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[(NH4)2S])/(Δt) = 1/3 (Δ[S])/(Δt) = 1/2 (Δ[KOH])/(Δt) = 1/6 (Δ[NH3])/(Δt) = 1/2 (Δ[Cr(OH)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium dichromate | diammonium sulfide | mixed sulfur | potassium hydroxide | ammonia | Cr(OH)3 formula | H_2O | K_2Cr_2O_7 | (NH_4)_2S | S | KOH | NH_3 | Cr(OH)3 Hill formula | H_2O | Cr_2K_2O_7 | H_8N_2S | S | HKO | H_3N | H3CrO3 name | water | potassium dichromate | diammonium sulfide | mixed sulfur | potassium hydroxide | ammonia | IUPAC name | water | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | diammonium sulfide | sulfur | potassium hydroxide | ammonia |

| water | potassium dichromate | diammonium sulfide | mixed sulfur | potassium hydroxide | ammonia | Cr(OH)3 molar mass | 18.015 g/mol | 294.18 g/mol | 68.14 g/mol | 32.06 g/mol | 56.105 g/mol | 17.031 g/mol | 103.02 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) | gas (at STP) | melting point | 0 °C | 398 °C | -18 °C | 112.8 °C | 406 °C | -77.73 °C | boiling point | 99.9839 °C | | | 444.7 °C | 1327 °C | -33.33 °C | density | 1 g/cm^3 | 2.67 g/cm^3 | 0.997 g/cm^3 | 2.07 g/cm^3 | 2.044 g/cm^3 | 6.96×10^-4 g/cm^3 (at 25 °C) | solubility in water | | | very soluble | | soluble | | surface tension | 0.0728 N/m | | | | | 0.0234 N/m | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | | 0.001 Pa s (at 550 °C) | 1.009×10^-5 Pa s (at 25 °C) | odor | odorless | odorless | | | | |