C + SiO2 + Ca3(PO4)2 = CO + P4 + Ca2SiO4

C activated charcoal + SiO_2 silicon dioxide + Ca_3(PO_4)_2 tricalcium diphosphate ⟶ CO carbon monoxide + P_4 white phosphorus + Ca2SiO4

Balance the chemical equation algebraically: C + SiO_2 + Ca_3(PO_4)_2 ⟶ CO + P_4 + Ca2SiO4 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_4 + c_6 Ca2SiO4 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 + 4 c_6 Si: | c_2 = c_6 Ca: | 3 c_3 = 2 c_6 P: | 2 c_3 = 4 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_5 = 1 and solve the system of equations for the remaining coefficients: c_1 = 10 c_2 = 3 c_3 = 2 c_4 = 10 c_5 = 1 c_6 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 10 C + 3 SiO_2 + 2 Ca_3(PO_4)_2 ⟶ 10 CO + P_4 + 3 Ca2SiO4

+ + ⟶ + + Ca2SiO4

activated charcoal + silicon dioxide + tricalcium diphosphate ⟶ carbon monoxide + white phosphorus + Ca2SiO4

Construct the equilibrium constant, K, expression for: C + SiO_2 + Ca_3(PO_4)_2 ⟶ CO + P_4 + Ca2SiO4 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: 10 C + 3 SiO_2 + 2 Ca_3(PO_4)_2 ⟶ 10 CO + P_4 + 3 Ca2SiO4 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 | 10 | -10 SiO_2 | 3 | -3 Ca_3(PO_4)_2 | 2 | -2 CO | 10 | 10 P_4 | 1 | 1 Ca2SiO4 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression C | 10 | -10 | ([C])^(-10) SiO_2 | 3 | -3 | ([SiO2])^(-3) Ca_3(PO_4)_2 | 2 | -2 | ([Ca3(PO4)2])^(-2) CO | 10 | 10 | ([CO])^10 P_4 | 1 | 1 | [P4] Ca2SiO4 | 3 | 3 | ([Ca2SiO4])^3 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])^(-10) ([SiO2])^(-3) ([Ca3(PO4)2])^(-2) ([CO])^10 [P4] ([Ca2SiO4])^3 = (([CO])^10 [P4] ([Ca2SiO4])^3)/(([C])^10 ([SiO2])^3 ([Ca3(PO4)2])^2)

Construct the rate of reaction expression for: C + SiO_2 + Ca_3(PO_4)_2 ⟶ CO + P_4 + Ca2SiO4 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: 10 C + 3 SiO_2 + 2 Ca_3(PO_4)_2 ⟶ 10 CO + P_4 + 3 Ca2SiO4 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 | 10 | -10 SiO_2 | 3 | -3 Ca_3(PO_4)_2 | 2 | -2 CO | 10 | 10 P_4 | 1 | 1 Ca2SiO4 | 3 | 3 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 | 10 | -10 | -1/10 (Δ[C])/(Δt) SiO_2 | 3 | -3 | -1/3 (Δ[SiO2])/(Δt) Ca_3(PO_4)_2 | 2 | -2 | -1/2 (Δ[Ca3(PO4)2])/(Δt) CO | 10 | 10 | 1/10 (Δ[CO])/(Δt) P_4 | 1 | 1 | (Δ[P4])/(Δt) Ca2SiO4 | 3 | 3 | 1/3 (Δ[Ca2SiO4])/(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 = -1/10 (Δ[C])/(Δt) = -1/3 (Δ[SiO2])/(Δt) = -1/2 (Δ[Ca3(PO4)2])/(Δt) = 1/10 (Δ[CO])/(Δt) = (Δ[P4])/(Δt) = 1/3 (Δ[Ca2SiO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| activated charcoal | silicon dioxide | tricalcium diphosphate | carbon monoxide | white phosphorus | Ca2SiO4 formula | C | SiO_2 | Ca_3(PO_4)_2 | CO | P_4 | Ca2SiO4 Hill formula | C | O_2Si | Ca_3O_8P_2 | CO | P_4 | Ca2O4Si name | activated charcoal | silicon dioxide | tricalcium diphosphate | carbon monoxide | white phosphorus | IUPAC name | carbon | dioxosilane | tricalcium diphosphate | carbon monoxide | tetraphosphorus |

| activated charcoal | silicon dioxide | tricalcium diphosphate | carbon monoxide | white phosphorus | Ca2SiO4 molar mass | 12.011 g/mol | 60.083 g/mol | 310.17 g/mol | 28.01 g/mol | 123.89504799 g/mol | 172.24 g/mol phase | solid (at STP) | solid (at STP) | | gas (at STP) | solid (at STP) | melting point | 3550 °C | 1713 °C | | -205 °C | 44.15 °C | boiling point | 4027 °C | 2950 °C | | -191.5 °C | 280.5 °C | density | 2.26 g/cm^3 | 2.196 g/cm^3 | 3.14 g/cm^3 | 0.001145 g/cm^3 (at 25 °C) | 1.823 g/cm^3 | solubility in water | insoluble | insoluble | | | insoluble | dynamic viscosity | | | | 1.772×10^-5 Pa s (at 25 °C) | 0.00169 Pa s (at 50 °C) | odor | | odorless | | odorless | odorless |