C + TiO2 + B4C = CO + TiB2

C activated charcoal + TiO_2 titanium dioxide + B_4C boron carbide ⟶ CO carbon monoxide + TiB_2 titanium boride

Balance the chemical equation algebraically: C + TiO_2 + B_4C ⟶ CO + TiB_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 C + c_2 TiO_2 + c_3 B_4C ⟶ c_4 CO + c_5 TiB_2 Set the number of atoms in the reactants equal to the number of atoms in the products for C, O, Ti and B: C: | c_1 + c_3 = c_4 O: | 2 c_2 = c_4 Ti: | c_2 = c_5 B: | 4 c_3 = 2 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 1 c_4 = 4 c_5 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 C + 2 TiO_2 + B_4C ⟶ 4 CO + 2 TiB_2

+ + ⟶ + TiB_2

activated charcoal + titanium dioxide + boron carbide ⟶ carbon monoxide + titanium boride

Construct the equilibrium constant, K, expression for: C + TiO_2 + B_4C ⟶ CO + TiB_2 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 TiO_2 + B_4C ⟶ 4 CO + 2 TiB_2 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 TiO_2 | 2 | -2 B_4C | 1 | -1 CO | 4 | 4 TiB_2 | 2 | 2 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) TiO_2 | 2 | -2 | ([TiO2])^(-2) B_4C | 1 | -1 | ([B4C])^(-1) CO | 4 | 4 | ([CO])^4 TiB_2 | 2 | 2 | ([TiB2])^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 = ([C])^(-3) ([TiO2])^(-2) ([B4C])^(-1) ([CO])^4 ([TiB2])^2 = (([CO])^4 ([TiB2])^2)/(([C])^3 ([TiO2])^2 [B4C])

Construct the rate of reaction expression for: C + TiO_2 + B_4C ⟶ CO + TiB_2 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 TiO_2 + B_4C ⟶ 4 CO + 2 TiB_2 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 TiO_2 | 2 | -2 B_4C | 1 | -1 CO | 4 | 4 TiB_2 | 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 C | 3 | -3 | -1/3 (Δ[C])/(Δt) TiO_2 | 2 | -2 | -1/2 (Δ[TiO2])/(Δt) B_4C | 1 | -1 | -(Δ[B4C])/(Δt) CO | 4 | 4 | 1/4 (Δ[CO])/(Δt) TiB_2 | 2 | 2 | 1/2 (Δ[TiB2])/(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[TiO2])/(Δt) = -(Δ[B4C])/(Δt) = 1/4 (Δ[CO])/(Δt) = 1/2 (Δ[TiB2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| activated charcoal | titanium dioxide | boron carbide | carbon monoxide | titanium boride formula | C | TiO_2 | B_4C | CO | TiB_2 Hill formula | C | O_2Ti | CB_4 | CO | B_2Ti_1 name | activated charcoal | titanium dioxide | boron carbide | carbon monoxide | titanium boride IUPAC name | carbon | | | carbon monoxide |

| activated charcoal | titanium dioxide | boron carbide | carbon monoxide | titanium boride molar mass | 12.011 g/mol | 79.865 g/mol | 55.25 g/mol | 28.01 g/mol | 69.489 g/mol phase | solid (at STP) | solid (at STP) | solid (at STP) | gas (at STP) | melting point | 3550 °C | 1843 °C | 2350 °C | -205 °C | boiling point | 4027 °C | 2900 °C | 3500 °C | -191.5 °C | density | 2.26 g/cm^3 | 4.26 g/cm^3 | 2.51 g/cm^3 | 0.001145 g/cm^3 (at 25 °C) | 4.52 g/cm^3 solubility in water | insoluble | insoluble | insoluble | | dynamic viscosity | | | | 1.772×10^-5 Pa s (at 25 °C) | odor | | | odorless | odorless |