C + Na2Cr2O7 = CO2 + Na2CO3 + Cr2O3

C activated charcoal + Na_2Cr_2O_7 sodium bichromate ⟶ CO_2 carbon dioxide + Na_2CO_3 soda ash + Cr_2O_3 chromium(III) oxide

Balance the chemical equation algebraically: C + Na_2Cr_2O_7 ⟶ CO_2 + Na_2CO_3 + Cr_2O_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 C + c_2 Na_2Cr_2O_7 ⟶ c_3 CO_2 + c_4 Na_2CO_3 + c_5 Cr_2O_3 Set the number of atoms in the reactants equal to the number of atoms in the products for C, Cr, Na and O: C: | c_1 = c_3 + c_4 Cr: | 2 c_2 = 2 c_5 Na: | 2 c_2 = 2 c_4 O: | 7 c_2 = 2 c_3 + 3 c_4 + 3 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 = 2 c_5 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 C + 2 Na_2Cr_2O_7 ⟶ CO_2 + 2 Na_2CO_3 + 2 Cr_2O_3

+ ⟶ + +

activated charcoal + sodium bichromate ⟶ carbon dioxide + soda ash + chromium(III) oxide

Construct the equilibrium constant, K, expression for: C + Na_2Cr_2O_7 ⟶ CO_2 + Na_2CO_3 + Cr_2O_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: 3 C + 2 Na_2Cr_2O_7 ⟶ CO_2 + 2 Na_2CO_3 + 2 Cr_2O_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 C | 3 | -3 Na_2Cr_2O_7 | 2 | -2 CO_2 | 1 | 1 Na_2CO_3 | 2 | 2 Cr_2O_3 | 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) Na_2Cr_2O_7 | 2 | -2 | ([Na2Cr2O7])^(-2) CO_2 | 1 | 1 | [CO2] Na_2CO_3 | 2 | 2 | ([Na2CO3])^2 Cr_2O_3 | 2 | 2 | ([Cr2O3])^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) ([Na2Cr2O7])^(-2) [CO2] ([Na2CO3])^2 ([Cr2O3])^2 = ([CO2] ([Na2CO3])^2 ([Cr2O3])^2)/(([C])^3 ([Na2Cr2O7])^2)

Construct the rate of reaction expression for: C + Na_2Cr_2O_7 ⟶ CO_2 + Na_2CO_3 + Cr_2O_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: 3 C + 2 Na_2Cr_2O_7 ⟶ CO_2 + 2 Na_2CO_3 + 2 Cr_2O_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 C | 3 | -3 Na_2Cr_2O_7 | 2 | -2 CO_2 | 1 | 1 Na_2CO_3 | 2 | 2 Cr_2O_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 C | 3 | -3 | -1/3 (Δ[C])/(Δt) Na_2Cr_2O_7 | 2 | -2 | -1/2 (Δ[Na2Cr2O7])/(Δt) CO_2 | 1 | 1 | (Δ[CO2])/(Δt) Na_2CO_3 | 2 | 2 | 1/2 (Δ[Na2CO3])/(Δt) Cr_2O_3 | 2 | 2 | 1/2 (Δ[Cr2O3])/(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[Na2Cr2O7])/(Δt) = (Δ[CO2])/(Δt) = 1/2 (Δ[Na2CO3])/(Δt) = 1/2 (Δ[Cr2O3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| activated charcoal | sodium bichromate | carbon dioxide | soda ash | chromium(III) oxide formula | C | Na_2Cr_2O_7 | CO_2 | Na_2CO_3 | Cr_2O_3 Hill formula | C | Cr_2Na_2O_7 | CO_2 | CNa_2O_3 | Cr_2O_3 name | activated charcoal | sodium bichromate | carbon dioxide | soda ash | chromium(III) oxide IUPAC name | carbon | disodium oxido-(oxido-dioxo-chromio)oxy-dioxo-chromium | carbon dioxide | disodium carbonate |

| activated charcoal | sodium bichromate | carbon dioxide | soda ash | chromium(III) oxide molar mass | 12.011 g/mol | 261.96 g/mol | 44.009 g/mol | 105.99 g/mol | 151.99 g/mol phase | solid (at STP) | solid (at STP) | gas (at STP) | solid (at STP) | solid (at STP) melting point | 3550 °C | 356.7 °C | -56.56 °C (at triple point) | 851 °C | 2435 °C boiling point | 4027 °C | | -78.5 °C (at sublimation point) | 1600 °C | 4000 °C density | 2.26 g/cm^3 | 2.35 g/cm^3 | 0.00184212 g/cm^3 (at 20 °C) | | 4.8 g/cm^3 solubility in water | insoluble | | | soluble | insoluble dynamic viscosity | | | 1.491×10^-5 Pa s (at 25 °C) | 0.00355 Pa s (at 900 °C) | odor | | | odorless | |