H2SO4 + KMnO4 + Ca3P2 = H2O + K2SO4 + MnSO4 + Ca3(PO4)2

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + Ca_3P_2 calcium phosphide ⟶ H_2O water + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + Ca_3(PO_4)_2 tricalcium diphosphate

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + Ca_3P_2 ⟶ H_2O + K_2SO_4 + MnSO_4 + Ca_3(PO_4)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 Ca_3P_2 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 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, S, K, Mn, Ca and P: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_6 + 8 c_7 S: | c_1 = c_5 + c_6 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 Ca: | 3 c_3 = 3 c_7 P: | 2 c_3 = 2 c_7 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 = 24/5 c_2 = 16/5 c_3 = 1 c_4 = 24/5 c_5 = 8/5 c_6 = 16/5 c_7 = 1 Multiply by the least common denominator, 5, to eliminate fractional coefficients: c_1 = 24 c_2 = 16 c_3 = 5 c_4 = 24 c_5 = 8 c_6 = 16 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 24 H_2SO_4 + 16 KMnO_4 + 5 Ca_3P_2 ⟶ 24 H_2O + 8 K_2SO_4 + 16 MnSO_4 + 5 Ca_3(PO_4)_2

+ + ⟶ + + +

sulfuric acid + potassium permanganate + calcium phosphide ⟶ water + potassium sulfate + manganese(II) sulfate + tricalcium diphosphate

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + Ca_3P_2 ⟶ H_2O + K_2SO_4 + MnSO_4 + 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: 24 H_2SO_4 + 16 KMnO_4 + 5 Ca_3P_2 ⟶ 24 H_2O + 8 K_2SO_4 + 16 MnSO_4 + 5 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_2SO_4 | 24 | -24 KMnO_4 | 16 | -16 Ca_3P_2 | 5 | -5 H_2O | 24 | 24 K_2SO_4 | 8 | 8 MnSO_4 | 16 | 16 Ca_3(PO_4)_2 | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 24 | -24 | ([H2SO4])^(-24) KMnO_4 | 16 | -16 | ([KMnO4])^(-16) Ca_3P_2 | 5 | -5 | ([Ca3P2])^(-5) H_2O | 24 | 24 | ([H2O])^24 K_2SO_4 | 8 | 8 | ([K2SO4])^8 MnSO_4 | 16 | 16 | ([MnSO4])^16 Ca_3(PO_4)_2 | 5 | 5 | ([Ca3(PO4)2])^5 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 = ([H2SO4])^(-24) ([KMnO4])^(-16) ([Ca3P2])^(-5) ([H2O])^24 ([K2SO4])^8 ([MnSO4])^16 ([Ca3(PO4)2])^5 = (([H2O])^24 ([K2SO4])^8 ([MnSO4])^16 ([Ca3(PO4)2])^5)/(([H2SO4])^24 ([KMnO4])^16 ([Ca3P2])^5)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + Ca_3P_2 ⟶ H_2O + K_2SO_4 + MnSO_4 + 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: 24 H_2SO_4 + 16 KMnO_4 + 5 Ca_3P_2 ⟶ 24 H_2O + 8 K_2SO_4 + 16 MnSO_4 + 5 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_2SO_4 | 24 | -24 KMnO_4 | 16 | -16 Ca_3P_2 | 5 | -5 H_2O | 24 | 24 K_2SO_4 | 8 | 8 MnSO_4 | 16 | 16 Ca_3(PO_4)_2 | 5 | 5 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_2SO_4 | 24 | -24 | -1/24 (Δ[H2SO4])/(Δt) KMnO_4 | 16 | -16 | -1/16 (Δ[KMnO4])/(Δt) Ca_3P_2 | 5 | -5 | -1/5 (Δ[Ca3P2])/(Δt) H_2O | 24 | 24 | 1/24 (Δ[H2O])/(Δt) K_2SO_4 | 8 | 8 | 1/8 (Δ[K2SO4])/(Δt) MnSO_4 | 16 | 16 | 1/16 (Δ[MnSO4])/(Δt) Ca_3(PO_4)_2 | 5 | 5 | 1/5 (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 = -1/24 (Δ[H2SO4])/(Δt) = -1/16 (Δ[KMnO4])/(Δt) = -1/5 (Δ[Ca3P2])/(Δt) = 1/24 (Δ[H2O])/(Δt) = 1/8 (Δ[K2SO4])/(Δt) = 1/16 (Δ[MnSO4])/(Δt) = 1/5 (Δ[Ca3(PO4)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | calcium phosphide | water | potassium sulfate | manganese(II) sulfate | tricalcium diphosphate formula | H_2SO_4 | KMnO_4 | Ca_3P_2 | H_2O | K_2SO_4 | MnSO_4 | Ca_3(PO_4)_2 Hill formula | H_2O_4S | KMnO_4 | Ca_3P_2 | H_2O | K_2O_4S | MnSO_4 | Ca_3O_8P_2 name | sulfuric acid | potassium permanganate | calcium phosphide | water | potassium sulfate | manganese(II) sulfate | tricalcium diphosphate IUPAC name | sulfuric acid | potassium permanganate | calcium phosphanidylidenecalcium | water | dipotassium sulfate | manganese(+2) cation sulfate | tricalcium diphosphate