KMnO4 + H3PO4 + NaNO2 = H2O + NaNO3 + K3PO4 + Mn3(PO4)2

KMnO_4 potassium permanganate + H_3PO_4 phosphoric acid + NaNO_2 sodium nitrite ⟶ H_2O water + NaNO_3 sodium nitrate + K3PO4 + Mn3(PO4)2

Balance the chemical equation algebraically: KMnO_4 + H_3PO_4 + NaNO_2 ⟶ H_2O + NaNO_3 + K3PO4 + Mn3(PO4)2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KMnO_4 + c_2 H_3PO_4 + c_3 NaNO_2 ⟶ c_4 H_2O + c_5 NaNO_3 + c_6 K3PO4 + c_7 Mn3(PO4)2 Set the number of atoms in the reactants equal to the number of atoms in the products for K, Mn, O, H, P, N and Na: K: | c_1 = 3 c_6 Mn: | c_1 = 3 c_7 O: | 4 c_1 + 4 c_2 + 2 c_3 = c_4 + 3 c_5 + 4 c_6 + 8 c_7 H: | 3 c_2 = 2 c_4 P: | c_2 = c_6 + 2 c_7 N: | c_3 = c_5 Na: | 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_6 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 3 c_3 = 15/2 c_4 = 9/2 c_5 = 15/2 c_6 = 1 c_7 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 6 c_2 = 6 c_3 = 15 c_4 = 9 c_5 = 15 c_6 = 2 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 KMnO_4 + 6 H_3PO_4 + 15 NaNO_2 ⟶ 9 H_2O + 15 NaNO_3 + 2 K3PO4 + 2 Mn3(PO4)2

+ + ⟶ + + K3PO4 + Mn3(PO4)2

potassium permanganate + phosphoric acid + sodium nitrite ⟶ water + sodium nitrate + K3PO4 + Mn3(PO4)2

Construct the equilibrium constant, K, expression for: KMnO_4 + H_3PO_4 + NaNO_2 ⟶ H_2O + NaNO_3 + K3PO4 + Mn3(PO4)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: 6 KMnO_4 + 6 H_3PO_4 + 15 NaNO_2 ⟶ 9 H_2O + 15 NaNO_3 + 2 K3PO4 + 2 Mn3(PO4)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 KMnO_4 | 6 | -6 H_3PO_4 | 6 | -6 NaNO_2 | 15 | -15 H_2O | 9 | 9 NaNO_3 | 15 | 15 K3PO4 | 2 | 2 Mn3(PO4)2 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KMnO_4 | 6 | -6 | ([KMnO4])^(-6) H_3PO_4 | 6 | -6 | ([H3PO4])^(-6) NaNO_2 | 15 | -15 | ([NaNO2])^(-15) H_2O | 9 | 9 | ([H2O])^9 NaNO_3 | 15 | 15 | ([NaNO3])^15 K3PO4 | 2 | 2 | ([K3PO4])^2 Mn3(PO4)2 | 2 | 2 | ([Mn3(PO4)2])^2 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 = ([KMnO4])^(-6) ([H3PO4])^(-6) ([NaNO2])^(-15) ([H2O])^9 ([NaNO3])^15 ([K3PO4])^2 ([Mn3(PO4)2])^2 = (([H2O])^9 ([NaNO3])^15 ([K3PO4])^2 ([Mn3(PO4)2])^2)/(([KMnO4])^6 ([H3PO4])^6 ([NaNO2])^15)

Construct the rate of reaction expression for: KMnO_4 + H_3PO_4 + NaNO_2 ⟶ H_2O + NaNO_3 + K3PO4 + Mn3(PO4)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: 6 KMnO_4 + 6 H_3PO_4 + 15 NaNO_2 ⟶ 9 H_2O + 15 NaNO_3 + 2 K3PO4 + 2 Mn3(PO4)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 KMnO_4 | 6 | -6 H_3PO_4 | 6 | -6 NaNO_2 | 15 | -15 H_2O | 9 | 9 NaNO_3 | 15 | 15 K3PO4 | 2 | 2 Mn3(PO4)2 | 2 | 2 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 KMnO_4 | 6 | -6 | -1/6 (Δ[KMnO4])/(Δt) H_3PO_4 | 6 | -6 | -1/6 (Δ[H3PO4])/(Δt) NaNO_2 | 15 | -15 | -1/15 (Δ[NaNO2])/(Δt) H_2O | 9 | 9 | 1/9 (Δ[H2O])/(Δt) NaNO_3 | 15 | 15 | 1/15 (Δ[NaNO3])/(Δt) K3PO4 | 2 | 2 | 1/2 (Δ[K3PO4])/(Δt) Mn3(PO4)2 | 2 | 2 | 1/2 (Δ[Mn3(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 = -1/6 (Δ[KMnO4])/(Δt) = -1/6 (Δ[H3PO4])/(Δt) = -1/15 (Δ[NaNO2])/(Δt) = 1/9 (Δ[H2O])/(Δt) = 1/15 (Δ[NaNO3])/(Δt) = 1/2 (Δ[K3PO4])/(Δt) = 1/2 (Δ[Mn3(PO4)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium permanganate | phosphoric acid | sodium nitrite | water | sodium nitrate | K3PO4 | Mn3(PO4)2 formula | KMnO_4 | H_3PO_4 | NaNO_2 | H_2O | NaNO_3 | K3PO4 | Mn3(PO4)2 Hill formula | KMnO_4 | H_3O_4P | NNaO_2 | H_2O | NNaO_3 | K3O4P | Mn3O8P2 name | potassium permanganate | phosphoric acid | sodium nitrite | water | sodium nitrate | |

| potassium permanganate | phosphoric acid | sodium nitrite | water | sodium nitrate | K3PO4 | Mn3(PO4)2 molar mass | 158.03 g/mol | 97.994 g/mol | 68.995 g/mol | 18.015 g/mol | 84.994 g/mol | 212.26 g/mol | 354.75 g/mol phase | solid (at STP) | liquid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | | melting point | 240 °C | 42.4 °C | 271 °C | 0 °C | 306 °C | | boiling point | | 158 °C | | 99.9839 °C | | | density | 1 g/cm^3 | 1.685 g/cm^3 | 2.168 g/cm^3 | 1 g/cm^3 | 2.26 g/cm^3 | | solubility in water | | very soluble | | | soluble | | surface tension | | | | 0.0728 N/m | | | dynamic viscosity | | | | 8.9×10^-4 Pa s (at 25 °C) | 0.003 Pa s (at 250 °C) | | odor | odorless | odorless | | odorless | | |