H2O + Mn(NO3)2 + K2S2O8 = HNO3 + HMnO4 + KHSO4

H_2O water + Mn(NO_3)_2 manganese(II) nitrate + K_2S_2O_8 potassium persulfate ⟶ HNO_3 nitric acid + HMnO4 + KHSO_4 potassium bisulfate

Balance the chemical equation algebraically: H_2O + Mn(NO_3)_2 + K_2S_2O_8 ⟶ HNO_3 + HMnO4 + KHSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 Mn(NO_3)_2 + c_3 K_2S_2O_8 ⟶ c_4 HNO_3 + c_5 HMnO4 + c_6 KHSO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Mn, N, K and S: H: | 2 c_1 = c_4 + c_5 + c_6 O: | c_1 + 6 c_2 + 8 c_3 = 3 c_4 + 4 c_5 + 4 c_6 Mn: | c_2 = c_5 N: | 2 c_2 = c_4 K: | 2 c_3 = c_6 S: | 2 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 4 c_2 = 1 c_3 = 5/2 c_4 = 2 c_5 = 1 c_6 = 5 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 8 c_2 = 2 c_3 = 5 c_4 = 4 c_5 = 2 c_6 = 10 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2O + 2 Mn(NO_3)_2 + 5 K_2S_2O_8 ⟶ 4 HNO_3 + 2 HMnO4 + 10 KHSO_4

+ + ⟶ + HMnO4 +

water + manganese(II) nitrate + potassium persulfate ⟶ nitric acid + HMnO4 + potassium bisulfate

Construct the equilibrium constant, K, expression for: H_2O + Mn(NO_3)_2 + K_2S_2O_8 ⟶ HNO_3 + HMnO4 + KHSO_4 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 + 2 Mn(NO_3)_2 + 5 K_2S_2O_8 ⟶ 4 HNO_3 + 2 HMnO4 + 10 KHSO_4 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 Mn(NO_3)_2 | 2 | -2 K_2S_2O_8 | 5 | -5 HNO_3 | 4 | 4 HMnO4 | 2 | 2 KHSO_4 | 10 | 10 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) Mn(NO_3)_2 | 2 | -2 | ([Mn(NO3)2])^(-2) K_2S_2O_8 | 5 | -5 | ([K2S2O8])^(-5) HNO_3 | 4 | 4 | ([HNO3])^4 HMnO4 | 2 | 2 | ([HMnO4])^2 KHSO_4 | 10 | 10 | ([KHSO4])^10 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) ([Mn(NO3)2])^(-2) ([K2S2O8])^(-5) ([HNO3])^4 ([HMnO4])^2 ([KHSO4])^10 = (([HNO3])^4 ([HMnO4])^2 ([KHSO4])^10)/(([H2O])^8 ([Mn(NO3)2])^2 ([K2S2O8])^5)

Construct the rate of reaction expression for: H_2O + Mn(NO_3)_2 + K_2S_2O_8 ⟶ HNO_3 + HMnO4 + KHSO_4 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 + 2 Mn(NO_3)_2 + 5 K_2S_2O_8 ⟶ 4 HNO_3 + 2 HMnO4 + 10 KHSO_4 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 Mn(NO_3)_2 | 2 | -2 K_2S_2O_8 | 5 | -5 HNO_3 | 4 | 4 HMnO4 | 2 | 2 KHSO_4 | 10 | 10 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) Mn(NO_3)_2 | 2 | -2 | -1/2 (Δ[Mn(NO3)2])/(Δt) K_2S_2O_8 | 5 | -5 | -1/5 (Δ[K2S2O8])/(Δt) HNO_3 | 4 | 4 | 1/4 (Δ[HNO3])/(Δt) HMnO4 | 2 | 2 | 1/2 (Δ[HMnO4])/(Δt) KHSO_4 | 10 | 10 | 1/10 (Δ[KHSO4])/(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) = -1/2 (Δ[Mn(NO3)2])/(Δt) = -1/5 (Δ[K2S2O8])/(Δt) = 1/4 (Δ[HNO3])/(Δt) = 1/2 (Δ[HMnO4])/(Δt) = 1/10 (Δ[KHSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | manganese(II) nitrate | potassium persulfate | nitric acid | HMnO4 | potassium bisulfate formula | H_2O | Mn(NO_3)_2 | K_2S_2O_8 | HNO_3 | HMnO4 | KHSO_4 Hill formula | H_2O | MnN_2O_6 | K_2O_8S_2 | HNO_3 | HMnO4 | HKO_4S name | water | manganese(II) nitrate | potassium persulfate | nitric acid | | potassium bisulfate IUPAC name | water | manganese(2+) dinitrate | dipotassium sulfonatooxy sulfate | nitric acid | | potassium hydrogen sulfate

| water | manganese(II) nitrate | potassium persulfate | nitric acid | HMnO4 | potassium bisulfate molar mass | 18.015 g/mol | 178.95 g/mol | 270.31 g/mol | 63.012 g/mol | 119.94 g/mol | 136.16 g/mol phase | liquid (at STP) | | solid (at STP) | liquid (at STP) | | solid (at STP) melting point | 0 °C | | 100 °C | -41.6 °C | | 214 °C boiling point | 99.9839 °C | | | 83 °C | | density | 1 g/cm^3 | 1.536 g/cm^3 | 2.477 g/cm^3 | 1.5129 g/cm^3 | | 2.32 g/cm^3 solubility in water | | | soluble | miscible | | surface tension | 0.0728 N/m | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 7.6×10^-4 Pa s (at 25 °C) | | odor | odorless | | | | |