C + Al2O3 = CO2 + Al

C activated charcoal + Al_2O_3 aluminum oxide ⟶ CO_2 carbon dioxide + Al aluminum

Balance the chemical equation algebraically: C + Al_2O_3 ⟶ CO_2 + Al Add stoichiometric coefficients, c_i, to the reactants and products: c_1 C + c_2 Al_2O_3 ⟶ c_3 CO_2 + c_4 Al Set the number of atoms in the reactants equal to the number of atoms in the products for C, Al and O: C: | c_1 = c_3 Al: | 2 c_2 = c_4 O: | 3 c_2 = 2 c_3 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 = 3/2 c_2 = 1 c_3 = 3/2 c_4 = 2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 3 c_2 = 2 c_3 = 3 c_4 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 C + 2 Al_2O_3 ⟶ 3 CO_2 + 4 Al

+ ⟶ +

activated charcoal + aluminum oxide ⟶ carbon dioxide + aluminum

Construct the equilibrium constant, K, expression for: C + Al_2O_3 ⟶ CO_2 + Al 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: 3 C + 2 Al_2O_3 ⟶ 3 CO_2 + 4 Al 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 C | 3 | -3 Al_2O_3 | 2 | -2 CO_2 | 3 | 3 Al | 4 | 4 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression C | 3 | -3 | ([C])^(-3) Al_2O_3 | 2 | -2 | ([Al2O3])^(-2) CO_2 | 3 | 3 | ([CO2])^3 Al | 4 | 4 | ([Al])^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 = ([C])^(-3) ([Al2O3])^(-2) ([CO2])^3 ([Al])^4 = (([CO2])^3 ([Al])^4)/(([C])^3 ([Al2O3])^2)

Construct the rate of reaction expression for: C + Al_2O_3 ⟶ CO_2 + Al 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: 3 C + 2 Al_2O_3 ⟶ 3 CO_2 + 4 Al 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 C | 3 | -3 Al_2O_3 | 2 | -2 CO_2 | 3 | 3 Al | 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 C | 3 | -3 | -1/3 (Δ[C])/(Δt) Al_2O_3 | 2 | -2 | -1/2 (Δ[Al2O3])/(Δt) CO_2 | 3 | 3 | 1/3 (Δ[CO2])/(Δt) Al | 4 | 4 | 1/4 (Δ[Al])/(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 = -1/3 (Δ[C])/(Δt) = -1/2 (Δ[Al2O3])/(Δt) = 1/3 (Δ[CO2])/(Δt) = 1/4 (Δ[Al])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| activated charcoal | aluminum oxide | carbon dioxide | aluminum formula | C | Al_2O_3 | CO_2 | Al name | activated charcoal | aluminum oxide | carbon dioxide | aluminum IUPAC name | carbon | dialuminum;oxygen(2-) | carbon dioxide | aluminum

| activated charcoal | aluminum oxide | carbon dioxide | aluminum molar mass | 12.011 g/mol | 101.96 g/mol | 44.009 g/mol | 26.9815385 g/mol phase | solid (at STP) | solid (at STP) | gas (at STP) | solid (at STP) melting point | 3550 °C | 2040 °C | -56.56 °C (at triple point) | 660.4 °C boiling point | 4027 °C | | -78.5 °C (at sublimation point) | 2460 °C density | 2.26 g/cm^3 | | 0.00184212 g/cm^3 (at 20 °C) | 2.7 g/cm^3 solubility in water | insoluble | | | insoluble surface tension | | | | 0.817 N/m dynamic viscosity | | | 1.491×10^-5 Pa s (at 25 °C) | 1.5×10^-4 Pa s (at 760 °C) odor | | odorless | odorless | odorless