H2O + KMnO4 + Ca3P2 = KOH + MnO2 + Ca3(PO4)2

H_2O water + KMnO_4 potassium permanganate + Ca_3P_2 calcium phosphide ⟶ KOH potassium hydroxide + MnO_2 manganese dioxide + Ca_3(PO_4)_2 tricalcium diphosphate

Balance the chemical equation algebraically: H_2O + KMnO_4 + Ca_3P_2 ⟶ KOH + MnO_2 + Ca_3(PO_4)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 Ca_3P_2 ⟶ c_4 KOH + c_5 MnO_2 + c_6 Ca_3(PO_4)_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn, Ca and P: H: | 2 c_1 = c_4 O: | c_1 + 4 c_2 = c_4 + 2 c_5 + 8 c_6 K: | c_2 = c_4 Mn: | c_2 = c_5 Ca: | 3 c_3 = 3 c_6 P: | 2 c_3 = 2 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 8/3 c_2 = 16/3 c_3 = 1 c_4 = 16/3 c_5 = 16/3 c_6 = 1 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 8 c_2 = 16 c_3 = 3 c_4 = 16 c_5 = 16 c_6 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2O + 16 KMnO_4 + 3 Ca_3P_2 ⟶ 16 KOH + 16 MnO_2 + 3 Ca_3(PO_4)_2

+ + ⟶ + +

water + potassium permanganate + calcium phosphide ⟶ potassium hydroxide + manganese dioxide + tricalcium diphosphate

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + Ca_3P_2 ⟶ KOH + MnO_2 + Ca_3(PO_4)_2 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: 8 H_2O + 16 KMnO_4 + 3 Ca_3P_2 ⟶ 16 KOH + 16 MnO_2 + 3 Ca_3(PO_4)_2 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_2O | 8 | -8 KMnO_4 | 16 | -16 Ca_3P_2 | 3 | -3 KOH | 16 | 16 MnO_2 | 16 | 16 Ca_3(PO_4)_2 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 8 | -8 | ([H2O])^(-8) KMnO_4 | 16 | -16 | ([KMnO4])^(-16) Ca_3P_2 | 3 | -3 | ([Ca3P2])^(-3) KOH | 16 | 16 | ([KOH])^16 MnO_2 | 16 | 16 | ([MnO2])^16 Ca_3(PO_4)_2 | 3 | 3 | ([Ca3(PO4)2])^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 = ([H2O])^(-8) ([KMnO4])^(-16) ([Ca3P2])^(-3) ([KOH])^16 ([MnO2])^16 ([Ca3(PO4)2])^3 = (([KOH])^16 ([MnO2])^16 ([Ca3(PO4)2])^3)/(([H2O])^8 ([KMnO4])^16 ([Ca3P2])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + Ca_3P_2 ⟶ KOH + MnO_2 + Ca_3(PO_4)_2 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: 8 H_2O + 16 KMnO_4 + 3 Ca_3P_2 ⟶ 16 KOH + 16 MnO_2 + 3 Ca_3(PO_4)_2 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_2O | 8 | -8 KMnO_4 | 16 | -16 Ca_3P_2 | 3 | -3 KOH | 16 | 16 MnO_2 | 16 | 16 Ca_3(PO_4)_2 | 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 H_2O | 8 | -8 | -1/8 (Δ[H2O])/(Δt) KMnO_4 | 16 | -16 | -1/16 (Δ[KMnO4])/(Δt) Ca_3P_2 | 3 | -3 | -1/3 (Δ[Ca3P2])/(Δt) KOH | 16 | 16 | 1/16 (Δ[KOH])/(Δt) MnO_2 | 16 | 16 | 1/16 (Δ[MnO2])/(Δt) Ca_3(PO_4)_2 | 3 | 3 | 1/3 (Δ[Ca3(PO4)2])/(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 = -1/8 (Δ[H2O])/(Δt) = -1/16 (Δ[KMnO4])/(Δt) = -1/3 (Δ[Ca3P2])/(Δt) = 1/16 (Δ[KOH])/(Δt) = 1/16 (Δ[MnO2])/(Δt) = 1/3 (Δ[Ca3(PO4)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | calcium phosphide | potassium hydroxide | manganese dioxide | tricalcium diphosphate formula | H_2O | KMnO_4 | Ca_3P_2 | KOH | MnO_2 | Ca_3(PO_4)_2 Hill formula | H_2O | KMnO_4 | Ca_3P_2 | HKO | MnO_2 | Ca_3O_8P_2 name | water | potassium permanganate | calcium phosphide | potassium hydroxide | manganese dioxide | tricalcium diphosphate IUPAC name | water | potassium permanganate | calcium phosphanidylidenecalcium | potassium hydroxide | dioxomanganese | tricalcium diphosphate

| water | potassium permanganate | calcium phosphide | potassium hydroxide | manganese dioxide | tricalcium diphosphate molar mass | 18.015 g/mol | 158.03 g/mol | 182.18 g/mol | 56.105 g/mol | 86.936 g/mol | 310.17 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) | melting point | 0 °C | 240 °C | 0.16 °C | 406 °C | 535 °C | boiling point | 99.9839 °C | | | 1327 °C | | density | 1 g/cm^3 | 1 g/cm^3 | 2.51 g/cm^3 | 2.044 g/cm^3 | 5.03 g/cm^3 | 3.14 g/cm^3 solubility in water | | | decomposes | soluble | insoluble | surface tension | 0.0728 N/m | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.001 Pa s (at 550 °C) | | odor | odorless | odorless | | | |