C + SiO2 + Ca(PO4)2 = CO + P + CaSiO3

C activated charcoal + SiO_2 silicon dioxide + Ca(PO4)2 ⟶ CO carbon monoxide + P red phosphorus + CaSiO_3 calcium silicate

Balance the chemical equation algebraically: C + SiO_2 + Ca(PO4)2 ⟶ CO + P + CaSiO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 C + c_2 SiO_2 + c_3 Ca(PO4)2 ⟶ c_4 CO + c_5 P + c_6 CaSiO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for C, O, Si, Ca and P: C: | c_1 = c_4 O: | 2 c_2 + 8 c_3 = c_4 + 3 c_6 Si: | c_2 = c_6 Ca: | c_3 = c_6 P: | 2 c_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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 7 c_2 = 1 c_3 = 1 c_4 = 7 c_5 = 2 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 7 C + SiO_2 + Ca(PO4)2 ⟶ 7 CO + 2 P + CaSiO_3

+ + Ca(PO4)2 ⟶ + +

activated charcoal + silicon dioxide + Ca(PO4)2 ⟶ carbon monoxide + red phosphorus + calcium silicate

Construct the equilibrium constant, K, expression for: C + SiO_2 + Ca(PO4)2 ⟶ CO + P + CaSiO_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: 7 C + SiO_2 + Ca(PO4)2 ⟶ 7 CO + 2 P + CaSiO_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 | 7 | -7 SiO_2 | 1 | -1 Ca(PO4)2 | 1 | -1 CO | 7 | 7 P | 2 | 2 CaSiO_3 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression C | 7 | -7 | ([C])^(-7) SiO_2 | 1 | -1 | ([SiO2])^(-1) Ca(PO4)2 | 1 | -1 | ([Ca(PO4)2])^(-1) CO | 7 | 7 | ([CO])^7 P | 2 | 2 | ([P])^2 CaSiO_3 | 1 | 1 | [CaSiO3] 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])^(-7) ([SiO2])^(-1) ([Ca(PO4)2])^(-1) ([CO])^7 ([P])^2 [CaSiO3] = (([CO])^7 ([P])^2 [CaSiO3])/(([C])^7 [SiO2] [Ca(PO4)2])

Construct the rate of reaction expression for: C + SiO_2 + Ca(PO4)2 ⟶ CO + P + CaSiO_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: 7 C + SiO_2 + Ca(PO4)2 ⟶ 7 CO + 2 P + CaSiO_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 | 7 | -7 SiO_2 | 1 | -1 Ca(PO4)2 | 1 | -1 CO | 7 | 7 P | 2 | 2 CaSiO_3 | 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 C | 7 | -7 | -1/7 (Δ[C])/(Δt) SiO_2 | 1 | -1 | -(Δ[SiO2])/(Δt) Ca(PO4)2 | 1 | -1 | -(Δ[Ca(PO4)2])/(Δt) CO | 7 | 7 | 1/7 (Δ[CO])/(Δt) P | 2 | 2 | 1/2 (Δ[P])/(Δt) CaSiO_3 | 1 | 1 | (Δ[CaSiO3])/(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 = -1/7 (Δ[C])/(Δt) = -(Δ[SiO2])/(Δt) = -(Δ[Ca(PO4)2])/(Δt) = 1/7 (Δ[CO])/(Δt) = 1/2 (Δ[P])/(Δt) = (Δ[CaSiO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| activated charcoal | silicon dioxide | Ca(PO4)2 | carbon monoxide | red phosphorus | calcium silicate formula | C | SiO_2 | Ca(PO4)2 | CO | P | CaSiO_3 Hill formula | C | O_2Si | CaO8P2 | CO | P | CaO_3Si name | activated charcoal | silicon dioxide | | carbon monoxide | red phosphorus | calcium silicate IUPAC name | carbon | dioxosilane | | carbon monoxide | phosphorus | calcium dioxido-oxosilane

| activated charcoal | silicon dioxide | Ca(PO4)2 | carbon monoxide | red phosphorus | calcium silicate molar mass | 12.011 g/mol | 60.083 g/mol | 230.02 g/mol | 28.01 g/mol | 30.973761998 g/mol | 116.16 g/mol phase | solid (at STP) | solid (at STP) | | gas (at STP) | solid (at STP) | melting point | 3550 °C | 1713 °C | | -205 °C | 579.2 °C | boiling point | 4027 °C | 2950 °C | | -191.5 °C | | density | 2.26 g/cm^3 | 2.196 g/cm^3 | | 0.001145 g/cm^3 (at 25 °C) | 2.16 g/cm^3 | solubility in water | insoluble | insoluble | | | insoluble | dynamic viscosity | | | | 1.772×10^-5 Pa s (at 25 °C) | 7.6×10^-4 Pa s (at 20.2 °C) | odor | | odorless | | odorless | |