H2O + KOH + Al + KNO3 = NH3 + KAlO2

H_2O water + KOH potassium hydroxide + Al aluminum + KNO_3 potassium nitrate ⟶ NH_3 ammonia + KAlO2

Balance the chemical equation algebraically: H_2O + KOH + Al + KNO_3 ⟶ NH_3 + KAlO2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KOH + c_3 Al + c_4 KNO_3 ⟶ c_5 NH_3 + c_6 KAlO2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Al and N: H: | 2 c_1 + c_2 = 3 c_5 O: | c_1 + c_2 + 3 c_4 = 2 c_6 K: | c_2 + c_4 = c_6 Al: | c_3 = c_6 N: | c_4 = 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 5/2 c_3 = 4 c_4 = 3/2 c_5 = 3/2 c_6 = 4 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 2 c_2 = 5 c_3 = 8 c_4 = 3 c_5 = 3 c_6 = 8 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + 5 KOH + 8 Al + 3 KNO_3 ⟶ 3 NH_3 + 8 KAlO2

+ + + ⟶ + KAlO2

water + potassium hydroxide + aluminum + potassium nitrate ⟶ ammonia + KAlO2

Construct the equilibrium constant, K, expression for: H_2O + KOH + Al + KNO_3 ⟶ NH_3 + KAlO2 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: 2 H_2O + 5 KOH + 8 Al + 3 KNO_3 ⟶ 3 NH_3 + 8 KAlO2 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 | 2 | -2 KOH | 5 | -5 Al | 8 | -8 KNO_3 | 3 | -3 NH_3 | 3 | 3 KAlO2 | 8 | 8 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 2 | -2 | ([H2O])^(-2) KOH | 5 | -5 | ([KOH])^(-5) Al | 8 | -8 | ([Al])^(-8) KNO_3 | 3 | -3 | ([KNO3])^(-3) NH_3 | 3 | 3 | ([NH3])^3 KAlO2 | 8 | 8 | ([KAlO2])^8 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])^(-2) ([KOH])^(-5) ([Al])^(-8) ([KNO3])^(-3) ([NH3])^3 ([KAlO2])^8 = (([NH3])^3 ([KAlO2])^8)/(([H2O])^2 ([KOH])^5 ([Al])^8 ([KNO3])^3)

Construct the rate of reaction expression for: H_2O + KOH + Al + KNO_3 ⟶ NH_3 + KAlO2 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: 2 H_2O + 5 KOH + 8 Al + 3 KNO_3 ⟶ 3 NH_3 + 8 KAlO2 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 | 2 | -2 KOH | 5 | -5 Al | 8 | -8 KNO_3 | 3 | -3 NH_3 | 3 | 3 KAlO2 | 8 | 8 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 | 2 | -2 | -1/2 (Δ[H2O])/(Δt) KOH | 5 | -5 | -1/5 (Δ[KOH])/(Δt) Al | 8 | -8 | -1/8 (Δ[Al])/(Δt) KNO_3 | 3 | -3 | -1/3 (Δ[KNO3])/(Δt) NH_3 | 3 | 3 | 1/3 (Δ[NH3])/(Δt) KAlO2 | 8 | 8 | 1/8 (Δ[KAlO2])/(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 = -1/2 (Δ[H2O])/(Δt) = -1/5 (Δ[KOH])/(Δt) = -1/8 (Δ[Al])/(Δt) = -1/3 (Δ[KNO3])/(Δt) = 1/3 (Δ[NH3])/(Δt) = 1/8 (Δ[KAlO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium hydroxide | aluminum | potassium nitrate | ammonia | KAlO2 formula | H_2O | KOH | Al | KNO_3 | NH_3 | KAlO2 Hill formula | H_2O | HKO | Al | KNO_3 | H_3N | AlKO2 name | water | potassium hydroxide | aluminum | potassium nitrate | ammonia |