H2O + H2SO4 + NaOH + KMnO4 = KOH + Na2SO4 + HMnO4

H_2O water + H_2SO_4 sulfuric acid + NaOH sodium hydroxide + KMnO_4 potassium permanganate ⟶ KOH potassium hydroxide + Na_2SO_4 sodium sulfate + HMnO4

Balance the chemical equation algebraically: H_2O + H_2SO_4 + NaOH + KMnO_4 ⟶ KOH + Na_2SO_4 + HMnO4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 H_2SO_4 + c_3 NaOH + c_4 KMnO_4 ⟶ c_5 KOH + c_6 Na_2SO_4 + c_7 HMnO4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Na, K and Mn: H: | 2 c_1 + 2 c_2 + c_3 = c_5 + c_7 O: | c_1 + 4 c_2 + c_3 + 4 c_4 = c_5 + 4 c_6 + 4 c_7 S: | c_2 = c_6 Na: | c_3 = 2 c_6 K: | c_4 = c_5 Mn: | c_4 = 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_3 = 2 c_2 c_4 = 2 c_2 + 1 c_5 = 2 c_2 + 1 c_6 = c_2 c_7 = 2 c_2 + 1 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_2 = 1 and solve for the remaining coefficients: c_1 = 1 c_2 = 1 c_3 = 2 c_4 = 3 c_5 = 3 c_6 = 1 c_7 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + H_2SO_4 + 2 NaOH + 3 KMnO_4 ⟶ 3 KOH + Na_2SO_4 + 3 HMnO4

+ + + ⟶ + + HMnO4

water + sulfuric acid + sodium hydroxide + potassium permanganate ⟶ potassium hydroxide + sodium sulfate + HMnO4

Construct the equilibrium constant, K, expression for: H_2O + H_2SO_4 + NaOH + KMnO_4 ⟶ KOH + Na_2SO_4 + HMnO4 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: H_2O + H_2SO_4 + 2 NaOH + 3 KMnO_4 ⟶ 3 KOH + Na_2SO_4 + 3 HMnO4 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 | 1 | -1 H_2SO_4 | 1 | -1 NaOH | 2 | -2 KMnO_4 | 3 | -3 KOH | 3 | 3 Na_2SO_4 | 1 | 1 HMnO4 | 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 | 1 | -1 | ([H2O])^(-1) H_2SO_4 | 1 | -1 | ([H2SO4])^(-1) NaOH | 2 | -2 | ([NaOH])^(-2) KMnO_4 | 3 | -3 | ([KMnO4])^(-3) KOH | 3 | 3 | ([KOH])^3 Na_2SO_4 | 1 | 1 | [Na2SO4] HMnO4 | 3 | 3 | ([HMnO4])^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])^(-1) ([H2SO4])^(-1) ([NaOH])^(-2) ([KMnO4])^(-3) ([KOH])^3 [Na2SO4] ([HMnO4])^3 = (([KOH])^3 [Na2SO4] ([HMnO4])^3)/([H2O] [H2SO4] ([NaOH])^2 ([KMnO4])^3)

Construct the rate of reaction expression for: H_2O + H_2SO_4 + NaOH + KMnO_4 ⟶ KOH + Na_2SO_4 + HMnO4 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: H_2O + H_2SO_4 + 2 NaOH + 3 KMnO_4 ⟶ 3 KOH + Na_2SO_4 + 3 HMnO4 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 | 1 | -1 H_2SO_4 | 1 | -1 NaOH | 2 | -2 KMnO_4 | 3 | -3 KOH | 3 | 3 Na_2SO_4 | 1 | 1 HMnO4 | 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 | 1 | -1 | -(Δ[H2O])/(Δt) H_2SO_4 | 1 | -1 | -(Δ[H2SO4])/(Δt) NaOH | 2 | -2 | -1/2 (Δ[NaOH])/(Δt) KMnO_4 | 3 | -3 | -1/3 (Δ[KMnO4])/(Δt) KOH | 3 | 3 | 1/3 (Δ[KOH])/(Δt) Na_2SO_4 | 1 | 1 | (Δ[Na2SO4])/(Δt) HMnO4 | 3 | 3 | 1/3 (Δ[HMnO4])/(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 = -(Δ[H2O])/(Δt) = -(Δ[H2SO4])/(Δt) = -1/2 (Δ[NaOH])/(Δt) = -1/3 (Δ[KMnO4])/(Δt) = 1/3 (Δ[KOH])/(Δt) = (Δ[Na2SO4])/(Δt) = 1/3 (Δ[HMnO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | sulfuric acid | sodium hydroxide | potassium permanganate | potassium hydroxide | sodium sulfate | HMnO4 formula | H_2O | H_2SO_4 | NaOH | KMnO_4 | KOH | Na_2SO_4 | HMnO4 Hill formula | H_2O | H_2O_4S | HNaO | KMnO_4 | HKO | Na_2O_4S | HMnO4 name | water | sulfuric acid | sodium hydroxide | potassium permanganate | potassium hydroxide | sodium sulfate | IUPAC name | water | sulfuric acid | sodium hydroxide | potassium permanganate | potassium hydroxide | disodium sulfate |

| water | sulfuric acid | sodium hydroxide | potassium permanganate | potassium hydroxide | sodium sulfate | HMnO4 molar mass | 18.015 g/mol | 98.07 g/mol | 39.997 g/mol | 158.03 g/mol | 56.105 g/mol | 142.04 g/mol | 119.94 g/mol phase | liquid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) | solid (at STP) | solid (at STP) | melting point | 0 °C | 10.371 °C | 323 °C | 240 °C | 406 °C | 884 °C | boiling point | 99.9839 °C | 279.6 °C | 1390 °C | | 1327 °C | 1429 °C | density | 1 g/cm^3 | 1.8305 g/cm^3 | 2.13 g/cm^3 | 1 g/cm^3 | 2.044 g/cm^3 | 2.68 g/cm^3 | solubility in water | | very soluble | soluble | | soluble | soluble | surface tension | 0.0728 N/m | 0.0735 N/m | 0.07435 N/m | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | 0.021 Pa s (at 25 °C) | 0.004 Pa s (at 350 °C) | | 0.001 Pa s (at 550 °C) | | odor | odorless | odorless | | odorless | | |