HNO3 + NaNO2 + NaMnO4 = NaNO3 + Mn(NO3)2 + HO

HNO_3 nitric acid + NaNO_2 sodium nitrite + NaMnO_4 sodium permanganate ⟶ NaNO_3 sodium nitrate + Mn(NO_3)_2 manganese(II) nitrate + (HO)^• hydroxyl radical

Balance the chemical equation algebraically: HNO_3 + NaNO_2 + NaMnO_4 ⟶ NaNO_3 + Mn(NO_3)_2 + (HO)^• Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 NaNO_2 + c_3 NaMnO_4 ⟶ c_4 NaNO_3 + c_5 Mn(NO_3)_2 + c_6 HO^• Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, Na and Mn: H: | c_1 = c_6 N: | c_1 + c_2 = c_4 + 2 c_5 O: | 3 c_1 + 2 c_2 + 4 c_3 = 3 c_4 + 6 c_5 + c_6 Na: | c_2 + c_3 = c_4 Mn: | 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 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 HNO_3 + NaNO_2 + NaMnO_4 ⟶ 2 NaNO_3 + Mn(NO_3)_2 + 3 HO^•

+ + ⟶ + + (HO)^•

nitric acid + sodium nitrite + sodium permanganate ⟶ sodium nitrate + manganese(II) nitrate + hydroxyl radical

Construct the equilibrium constant, K, expression for: HNO_3 + NaNO_2 + NaMnO_4 ⟶ NaNO_3 + Mn(NO_3)_2 + (HO)^• 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 + NaNO_2 + NaMnO_4 ⟶ 2 NaNO_3 + Mn(NO_3)_2 + 3 HO^• 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 NaNO_2 | 1 | -1 NaMnO_4 | 1 | -1 NaNO_3 | 2 | 2 Mn(NO_3)_2 | 1 | 1 HO^• | 3 | 3 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) NaNO_2 | 1 | -1 | ([NaNO2])^(-1) NaMnO_4 | 1 | -1 | ([NaMnO4])^(-1) NaNO_3 | 2 | 2 | ([NaNO3])^2 Mn(NO_3)_2 | 1 | 1 | [Mn(NO3)2] HO^• | 3 | 3 | ([HO•])^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 = ([HNO3])^(-3) ([NaNO2])^(-1) ([NaMnO4])^(-1) ([NaNO3])^2 [Mn(NO3)2] ([HO•])^3 = (([NaNO3])^2 [Mn(NO3)2] ([HO•])^3)/(([HNO3])^3 [NaNO2] [NaMnO4])

Construct the rate of reaction expression for: HNO_3 + NaNO_2 + NaMnO_4 ⟶ NaNO_3 + Mn(NO_3)_2 + (HO)^• 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 + NaNO_2 + NaMnO_4 ⟶ 2 NaNO_3 + Mn(NO_3)_2 + 3 HO^• 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 NaNO_2 | 1 | -1 NaMnO_4 | 1 | -1 NaNO_3 | 2 | 2 Mn(NO_3)_2 | 1 | 1 HO^• | 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 HNO_3 | 3 | -3 | -1/3 (Δ[HNO3])/(Δt) NaNO_2 | 1 | -1 | -(Δ[NaNO2])/(Δt) NaMnO_4 | 1 | -1 | -(Δ[NaMnO4])/(Δt) NaNO_3 | 2 | 2 | 1/2 (Δ[NaNO3])/(Δt) Mn(NO_3)_2 | 1 | 1 | (Δ[Mn(NO3)2])/(Δt) HO^• | 3 | 3 | 1/3 (Δ[HO•])/(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[NaNO2])/(Δt) = -(Δ[NaMnO4])/(Δt) = 1/2 (Δ[NaNO3])/(Δt) = (Δ[Mn(NO3)2])/(Δt) = 1/3 (Δ[HO•])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | sodium nitrite | sodium permanganate | sodium nitrate | manganese(II) nitrate | hydroxyl radical formula | HNO_3 | NaNO_2 | NaMnO_4 | NaNO_3 | Mn(NO_3)_2 | (HO)^• Hill formula | HNO_3 | NNaO_2 | MnNaO_4 | NNaO_3 | MnN_2O_6 | name | nitric acid | sodium nitrite | sodium permanganate | sodium nitrate | manganese(II) nitrate | hydroxyl radical IUPAC name | nitric acid | sodium nitrite | sodium permanganate | sodium nitrate | manganese(2+) dinitrate |

| nitric acid | sodium nitrite | sodium permanganate | sodium nitrate | manganese(II) nitrate | hydroxyl radical molar mass | 63.012 g/mol | 68.995 g/mol | 141.92 g/mol | 84.994 g/mol | 178.95 g/mol | 17.0073 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | | melting point | -41.6 °C | 271 °C | | 306 °C | | boiling point | 83 °C | | 100 °C | | | density | 1.5129 g/cm^3 | 2.168 g/cm^3 | 1.391 g/cm^3 | 2.26 g/cm^3 | 1.536 g/cm^3 | solubility in water | miscible | | | soluble | | dynamic viscosity | 7.6×10^-4 Pa s (at 25 °C) | | | 0.003 Pa s (at 250 °C) | |