C + SiO2 + Ca3(PO4)2 = CO + P + P4 + CaSiO3

C activated charcoal + SiO_2 silicon dioxide + Ca_3(PO_4)_2 tricalcium diphosphate ⟶ CO carbon monoxide + P red phosphorus + P_4 white phosphorus + CaSiO_3 calcium silicate

Balance the chemical equation algebraically: C + SiO_2 + Ca_3(PO_4)_2 ⟶ CO + P + P_4 + CaSiO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 C + c_2 SiO_2 + c_3 Ca_3(PO_4)_2 ⟶ c_4 CO + c_5 P + c_6 P_4 + c_7 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_7 Si: | c_2 = c_7 Ca: | 3 c_3 = c_7 P: | 2 c_3 = c_5 + 4 c_6 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_6 = 1 and solve the system of equations for the remaining coefficients: c_2 = (3 c_1)/5 c_3 = c_1/5 c_4 = c_1 c_5 = (2 c_1)/5 - 4 c_6 = 1 c_7 = (3 c_1)/5 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 15 and solve for the remaining coefficients: c_1 = 15 c_2 = 9 c_3 = 3 c_4 = 15 c_5 = 2 c_6 = 1 c_7 = 9 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 15 C + 9 SiO_2 + 3 Ca_3(PO_4)_2 ⟶ 15 CO + 2 P + P_4 + 9 CaSiO_3

activated charcoal + silicon dioxide + tricalcium diphosphate ⟶ carbon monoxide + red phosphorus + white phosphorus + calcium silicate

Construct the equilibrium constant, K, expression for: C + SiO_2 + Ca_3(PO_4)_2 ⟶ CO + P + P_4 + 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: 15 C + 9 SiO_2 + 3 Ca_3(PO_4)_2 ⟶ 15 CO + 2 P + P_4 + 9 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 | 15 | -15 SiO_2 | 9 | -9 Ca_3(PO_4)_2 | 3 | -3 CO | 15 | 15 P | 2 | 2 P_4 | 1 | 1 CaSiO_3 | 9 | 9 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression C | 15 | -15 | ([C])^(-15) SiO_2 | 9 | -9 | ([SiO2])^(-9) Ca_3(PO_4)_2 | 3 | -3 | ([Ca3(PO4)2])^(-3) CO | 15 | 15 | ([CO])^15 P | 2 | 2 | ([P])^2 P_4 | 1 | 1 | [P4] CaSiO_3 | 9 | 9 | ([CaSiO3])^9 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])^(-15) ([SiO2])^(-9) ([Ca3(PO4)2])^(-3) ([CO])^15 ([P])^2 [P4] ([CaSiO3])^9 = (([CO])^15 ([P])^2 [P4] ([CaSiO3])^9)/(([C])^15 ([SiO2])^9 ([Ca3(PO4)2])^3)

Construct the rate of reaction expression for: C + SiO_2 + Ca_3(PO_4)_2 ⟶ CO + P + P_4 + 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: 15 C + 9 SiO_2 + 3 Ca_3(PO_4)_2 ⟶ 15 CO + 2 P + P_4 + 9 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 | 15 | -15 SiO_2 | 9 | -9 Ca_3(PO_4)_2 | 3 | -3 CO | 15 | 15 P | 2 | 2 P_4 | 1 | 1 CaSiO_3 | 9 | 9 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 | 15 | -15 | -1/15 (Δ[C])/(Δt) SiO_2 | 9 | -9 | -1/9 (Δ[SiO2])/(Δt) Ca_3(PO_4)_2 | 3 | -3 | -1/3 (Δ[Ca3(PO4)2])/(Δt) CO | 15 | 15 | 1/15 (Δ[CO])/(Δt) P | 2 | 2 | 1/2 (Δ[P])/(Δt) P_4 | 1 | 1 | (Δ[P4])/(Δt) CaSiO_3 | 9 | 9 | 1/9 (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 = -1/15 (Δ[C])/(Δt) = -1/9 (Δ[SiO2])/(Δt) = -1/3 (Δ[Ca3(PO4)2])/(Δt) = 1/15 (Δ[CO])/(Δt) = 1/2 (Δ[P])/(Δt) = (Δ[P4])/(Δt) = 1/9 (Δ[CaSiO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| activated charcoal | silicon dioxide | tricalcium diphosphate | carbon monoxide | red phosphorus | white phosphorus | calcium silicate formula | C | SiO_2 | Ca_3(PO_4)_2 | CO | P | P_4 | CaSiO_3 Hill formula | C | O_2Si | Ca_3O_8P_2 | CO | P | P_4 | CaO_3Si name | activated charcoal | silicon dioxide | tricalcium diphosphate | carbon monoxide | red phosphorus | white phosphorus | calcium silicate IUPAC name | carbon | dioxosilane | tricalcium diphosphate | carbon monoxide | phosphorus | tetraphosphorus | calcium dioxido-oxosilane