H2O + KMnO4 + NaNO = KOH + MnO2 + NaNO3

H_2O water + KMnO_4 potassium permanganate + NaNO ⟶ KOH potassium hydroxide + MnO_2 manganese dioxide + NaNO_3 sodium nitrate

Balance the chemical equation algebraically: H_2O + KMnO_4 + NaNO ⟶ KOH + MnO_2 + NaNO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 NaNO ⟶ c_4 KOH + c_5 MnO_2 + c_6 NaNO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn, Na and N: H: | 2 c_1 = c_4 O: | c_1 + 4 c_2 + c_3 = c_4 + 2 c_5 + 3 c_6 K: | c_2 = c_4 Mn: | c_2 = c_5 Na: | c_3 = c_6 N: | c_3 = 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 2 c_3 = 3/2 c_4 = 2 c_5 = 2 c_6 = 3/2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 2 c_2 = 4 c_3 = 3 c_4 = 4 c_5 = 4 c_6 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + 4 KMnO_4 + 3 NaNO ⟶ 4 KOH + 4 MnO_2 + 3 NaNO_3

+ + NaNO ⟶ + +

water + potassium permanganate + NaNO ⟶ potassium hydroxide + manganese dioxide + sodium nitrate

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + NaNO ⟶ KOH + MnO_2 + NaNO_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: 2 H_2O + 4 KMnO_4 + 3 NaNO ⟶ 4 KOH + 4 MnO_2 + 3 NaNO_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 H_2O | 2 | -2 KMnO_4 | 4 | -4 NaNO | 3 | -3 KOH | 4 | 4 MnO_2 | 4 | 4 NaNO_3 | 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 | 2 | -2 | ([H2O])^(-2) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) NaNO | 3 | -3 | ([NaNO])^(-3) KOH | 4 | 4 | ([KOH])^4 MnO_2 | 4 | 4 | ([MnO2])^4 NaNO_3 | 3 | 3 | ([NaNO3])^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])^(-2) ([KMnO4])^(-4) ([NaNO])^(-3) ([KOH])^4 ([MnO2])^4 ([NaNO3])^3 = (([KOH])^4 ([MnO2])^4 ([NaNO3])^3)/(([H2O])^2 ([KMnO4])^4 ([NaNO])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + NaNO ⟶ KOH + MnO_2 + NaNO_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: 2 H_2O + 4 KMnO_4 + 3 NaNO ⟶ 4 KOH + 4 MnO_2 + 3 NaNO_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 H_2O | 2 | -2 KMnO_4 | 4 | -4 NaNO | 3 | -3 KOH | 4 | 4 MnO_2 | 4 | 4 NaNO_3 | 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 | 2 | -2 | -1/2 (Δ[H2O])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) NaNO | 3 | -3 | -1/3 (Δ[NaNO])/(Δt) KOH | 4 | 4 | 1/4 (Δ[KOH])/(Δt) MnO_2 | 4 | 4 | 1/4 (Δ[MnO2])/(Δt) NaNO_3 | 3 | 3 | 1/3 (Δ[NaNO3])/(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 = -1/2 (Δ[H2O])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/3 (Δ[NaNO])/(Δt) = 1/4 (Δ[KOH])/(Δt) = 1/4 (Δ[MnO2])/(Δt) = 1/3 (Δ[NaNO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | NaNO | potassium hydroxide | manganese dioxide | sodium nitrate formula | H_2O | KMnO_4 | NaNO | KOH | MnO_2 | NaNO_3 Hill formula | H_2O | KMnO_4 | NNaO | HKO | MnO_2 | NNaO_3 name | water | potassium permanganate | | potassium hydroxide | manganese dioxide | sodium nitrate IUPAC name | water | potassium permanganate | | potassium hydroxide | dioxomanganese | sodium nitrate

| water | potassium permanganate | NaNO | potassium hydroxide | manganese dioxide | sodium nitrate molar mass | 18.015 g/mol | 158.03 g/mol | 52.996 g/mol | 56.105 g/mol | 86.936 g/mol | 84.994 g/mol phase | liquid (at STP) | solid (at STP) | | solid (at STP) | solid (at STP) | solid (at STP) melting point | 0 °C | 240 °C | | 406 °C | 535 °C | 306 °C boiling point | 99.9839 °C | | | 1327 °C | | density | 1 g/cm^3 | 1 g/cm^3 | | 2.044 g/cm^3 | 5.03 g/cm^3 | 2.26 g/cm^3 solubility in water | | | | soluble | insoluble | soluble surface tension | 0.0728 N/m | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.001 Pa s (at 550 °C) | | 0.003 Pa s (at 250 °C) odor | odorless | odorless | | | |