HNO3 + MnO2 + NaHS = H2O + S + NaNO3 + Mn(NO3)2

HNO_3 nitric acid + MnO_2 manganese dioxide + NaHS sodium bisulfide ⟶ H_2O water + S mixed sulfur + NaNO_3 sodium nitrate + Mn(NO_3)_2 manganese(II) nitrate

Balance the chemical equation algebraically: HNO_3 + MnO_2 + NaHS ⟶ H_2O + S + NaNO_3 + Mn(NO_3)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 MnO_2 + c_3 NaHS ⟶ c_4 H_2O + c_5 S + c_6 NaNO_3 + c_7 Mn(NO_3)_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, Mn, Na and S: H: | c_1 + c_3 = 2 c_4 N: | c_1 = c_6 + 2 c_7 O: | 3 c_1 + 2 c_2 = c_4 + 3 c_6 + 6 c_7 Mn: | c_2 = c_7 Na: | c_3 = c_6 S: | 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 1 c_3 = 1 c_4 = 2 c_5 = 1 c_6 = 1 c_7 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 HNO_3 + MnO_2 + NaHS ⟶ 2 H_2O + S + NaNO_3 + Mn(NO_3)_2

+ + ⟶ + + +

nitric acid + manganese dioxide + sodium bisulfide ⟶ water + mixed sulfur + sodium nitrate + manganese(II) nitrate

Construct the equilibrium constant, K, expression for: HNO_3 + MnO_2 + NaHS ⟶ H_2O + S + NaNO_3 + Mn(NO_3)_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: 3 HNO_3 + MnO_2 + NaHS ⟶ 2 H_2O + S + NaNO_3 + Mn(NO_3)_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 HNO_3 | 3 | -3 MnO_2 | 1 | -1 NaHS | 1 | -1 H_2O | 2 | 2 S | 1 | 1 NaNO_3 | 1 | 1 Mn(NO_3)_2 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HNO_3 | 3 | -3 | ([HNO3])^(-3) MnO_2 | 1 | -1 | ([MnO2])^(-1) NaHS | 1 | -1 | ([NaHS])^(-1) H_2O | 2 | 2 | ([H2O])^2 S | 1 | 1 | [S] NaNO_3 | 1 | 1 | [NaNO3] Mn(NO_3)_2 | 1 | 1 | [Mn(NO3)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 = ([HNO3])^(-3) ([MnO2])^(-1) ([NaHS])^(-1) ([H2O])^2 [S] [NaNO3] [Mn(NO3)2] = (([H2O])^2 [S] [NaNO3] [Mn(NO3)2])/(([HNO3])^3 [MnO2] [NaHS])

Construct the rate of reaction expression for: HNO_3 + MnO_2 + NaHS ⟶ H_2O + S + NaNO_3 + Mn(NO_3)_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: 3 HNO_3 + MnO_2 + NaHS ⟶ 2 H_2O + S + NaNO_3 + Mn(NO_3)_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 HNO_3 | 3 | -3 MnO_2 | 1 | -1 NaHS | 1 | -1 H_2O | 2 | 2 S | 1 | 1 NaNO_3 | 1 | 1 Mn(NO_3)_2 | 1 | 1 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 HNO_3 | 3 | -3 | -1/3 (Δ[HNO3])/(Δt) MnO_2 | 1 | -1 | -(Δ[MnO2])/(Δt) NaHS | 1 | -1 | -(Δ[NaHS])/(Δt) H_2O | 2 | 2 | 1/2 (Δ[H2O])/(Δt) S | 1 | 1 | (Δ[S])/(Δt) NaNO_3 | 1 | 1 | (Δ[NaNO3])/(Δt) Mn(NO_3)_2 | 1 | 1 | (Δ[Mn(NO3)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 = -1/3 (Δ[HNO3])/(Δt) = -(Δ[MnO2])/(Δt) = -(Δ[NaHS])/(Δt) = 1/2 (Δ[H2O])/(Δt) = (Δ[S])/(Δt) = (Δ[NaNO3])/(Δt) = (Δ[Mn(NO3)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | manganese dioxide | sodium bisulfide | water | mixed sulfur | sodium nitrate | manganese(II) nitrate formula | HNO_3 | MnO_2 | NaHS | H_2O | S | NaNO_3 | Mn(NO_3)_2 Hill formula | HNO_3 | MnO_2 | HNaS | H_2O | S | NNaO_3 | MnN_2O_6 name | nitric acid | manganese dioxide | sodium bisulfide | water | mixed sulfur | sodium nitrate | manganese(II) nitrate IUPAC name | nitric acid | dioxomanganese | sodium sulfanide | water | sulfur | sodium nitrate | manganese(2+) dinitrate

| nitric acid | manganese dioxide | sodium bisulfide | water | mixed sulfur | sodium nitrate | manganese(II) nitrate molar mass | 63.012 g/mol | 86.936 g/mol | 56.06 g/mol | 18.015 g/mol | 32.06 g/mol | 84.994 g/mol | 178.95 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) | melting point | -41.6 °C | 535 °C | 43 °C | 0 °C | 112.8 °C | 306 °C | boiling point | 83 °C | | | 99.9839 °C | 444.7 °C | | density | 1.5129 g/cm^3 | 5.03 g/cm^3 | 1.79 g/cm^3 | 1 g/cm^3 | 2.07 g/cm^3 | 2.26 g/cm^3 | 1.536 g/cm^3 solubility in water | miscible | insoluble | | | | soluble | surface tension | | | | 0.0728 N/m | | | dynamic viscosity | 7.6×10^-4 Pa s (at 25 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | | 0.003 Pa s (at 250 °C) | odor | | | | odorless | | |