H2 + C = CH4

H_2 (hydrogen) + C (activated charcoal) ⟶ CH_4 (methane)

Balance the chemical equation algebraically: H_2 + C ⟶ CH_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2 + c_2 C ⟶ c_3 CH_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H and C: H: | 2 c_1 = 4 c_3 C: | c_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 = 2 c_2 = 1 c_3 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2 + C ⟶ CH_4

+ ⟶

hydrogen + activated charcoal ⟶ methane

Construct the equilibrium constant, K, expression for: H_2 + C ⟶ CH_4 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_2 + C ⟶ CH_4 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_2 | 2 | -2 C | 1 | -1 CH_4 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2 | 2 | -2 | ([H2])^(-2) C | 1 | -1 | ([C])^(-1) CH_4 | 1 | 1 | [CH4] 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 = ([H2])^(-2) ([C])^(-1) [CH4] = ([CH4])/(([H2])^2 [C])

Construct the rate of reaction expression for: H_2 + C ⟶ CH_4 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_2 + C ⟶ CH_4 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_2 | 2 | -2 C | 1 | -1 CH_4 | 1 | 1 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_2 | 2 | -2 | -1/2 (Δ[H2])/(Δt) C | 1 | -1 | -(Δ[C])/(Δt) CH_4 | 1 | 1 | (Δ[CH4])/(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[H2])/(Δt) = -(Δ[C])/(Δt) = (Δ[CH4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| hydrogen | activated charcoal | methane formula | H_2 | C | CH_4 name | hydrogen | activated charcoal | methane IUPAC name | molecular hydrogen | carbon | methane

| hydrogen | activated charcoal | methane molar mass | 2.016 g/mol | 12.011 g/mol | 16.04 g/mol phase | gas (at STP) | solid (at STP) | gas (at STP) melting point | -259.2 °C | 3550 °C | -182.47 °C boiling point | -252.8 °C | 4027 °C | -161.48 °C density | 8.99×10^-5 g/cm^3 (at 0 °C) | 2.26 g/cm^3 | 6.67151×10^-4 g/cm^3 (at 20 °C) solubility in water | | insoluble | soluble surface tension | | | 0.0137 N/m dynamic viscosity | 8.9×10^-6 Pa s (at 25 °C) | | 1.114×10^-5 Pa s (at 25 °C) odor | odorless | | odorless