H2O + Mn2O7 = Mn(OH)7

H_2O water + Mn_2O_7 manganese(VII) oxide ⟶ Mn(OH)7

Balance the chemical equation algebraically: H_2O + Mn_2O_7 ⟶ Mn(OH)7 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 Mn_2O_7 ⟶ c_3 Mn(OH)7 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O and Mn: H: | 2 c_1 = 7 c_3 O: | c_1 + 7 c_2 = 7 c_3 Mn: | 2 c_2 = c_3 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 = 7 c_2 = 1 c_3 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 7 H_2O + Mn_2O_7 ⟶ 2 Mn(OH)7

+ ⟶ Mn(OH)7

water + manganese(VII) oxide ⟶ Mn(OH)7

Construct the equilibrium constant, K, expression for: H_2O + Mn_2O_7 ⟶ Mn(OH)7 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: 7 H_2O + Mn_2O_7 ⟶ 2 Mn(OH)7 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 | 7 | -7 Mn_2O_7 | 1 | -1 Mn(OH)7 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 7 | -7 | ([H2O])^(-7) Mn_2O_7 | 1 | -1 | ([Mn2O7])^(-1) Mn(OH)7 | 2 | 2 | ([Mn(OH)7])^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 = ([H2O])^(-7) ([Mn2O7])^(-1) ([Mn(OH)7])^2 = ([Mn(OH)7])^2/(([H2O])^7 [Mn2O7])

Construct the rate of reaction expression for: H_2O + Mn_2O_7 ⟶ Mn(OH)7 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: 7 H_2O + Mn_2O_7 ⟶ 2 Mn(OH)7 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 | 7 | -7 Mn_2O_7 | 1 | -1 Mn(OH)7 | 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 H_2O | 7 | -7 | -1/7 (Δ[H2O])/(Δt) Mn_2O_7 | 1 | -1 | -(Δ[Mn2O7])/(Δt) Mn(OH)7 | 2 | 2 | 1/2 (Δ[Mn(OH)7])/(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 = -1/7 (Δ[H2O])/(Δt) = -(Δ[Mn2O7])/(Δt) = 1/2 (Δ[Mn(OH)7])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | manganese(VII) oxide | Mn(OH)7 formula | H_2O | Mn_2O_7 | Mn(OH)7 Hill formula | H_2O | Mn_2O_7 | H7MnO7 name | water | manganese(VII) oxide |

| water | manganese(VII) oxide | Mn(OH)7 molar mass | 18.015 g/mol | 221.87 g/mol | 173.99 g/mol phase | liquid (at STP) | | melting point | 0 °C | 5.9 °C | boiling point | 99.9839 °C | | density | 1 g/cm^3 | 2.4 g/cm^3 | solubility in water | | soluble | surface tension | 0.0728 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | odor | odorless | |