Al + Mn2O3 = Al2O3 + Mn

Al aluminum + Mn_2O_3 manganese(III) oxide ⟶ Al_2O_3 aluminum oxide + Mn manganese

Balance the chemical equation algebraically: Al + Mn_2O_3 ⟶ Al_2O_3 + Mn Add stoichiometric coefficients, c_i, to the reactants and products: c_1 Al + c_2 Mn_2O_3 ⟶ c_3 Al_2O_3 + c_4 Mn Set the number of atoms in the reactants equal to the number of atoms in the products for Al, Mn and O: Al: | c_1 = 2 c_3 Mn: | 2 c_2 = c_4 O: | 3 c_2 = 3 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 = 2 c_2 = 1 c_3 = 1 c_4 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 Al + Mn_2O_3 ⟶ Al_2O_3 + 2 Mn

+ ⟶ +

aluminum + manganese(III) oxide ⟶ aluminum oxide + manganese

| aluminum | manganese(III) oxide | aluminum oxide | manganese molecular enthalpy | 0 kJ/mol | -959 kJ/mol | -1676 kJ/mol | 0 kJ/mol total enthalpy | 0 kJ/mol | -959 kJ/mol | -1676 kJ/mol | 0 kJ/mol | H_initial = -959 kJ/mol | | H_final = -1676 kJ/mol | ΔH_rxn^0 | -1676 kJ/mol - -959 kJ/mol = -717 kJ/mol (exothermic) | | |

| aluminum | manganese(III) oxide | aluminum oxide | manganese molecular entropy | 28.3 J/(mol K) | 110 J/(mol K) | 51 J/(mol K) | 32 J/(mol K) total entropy | 56.6 J/(mol K) | 110 J/(mol K) | 51 J/(mol K) | 64 J/(mol K) | S_initial = 166.6 J/(mol K) | | S_final = 115 J/(mol K) | ΔS_rxn^0 | 115 J/(mol K) - 166.6 J/(mol K) = -51.6 J/(mol K) (exoentropic) | | |

Construct the equilibrium constant, K, expression for: Al + Mn_2O_3 ⟶ Al_2O_3 + Mn 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: 2 Al + Mn_2O_3 ⟶ Al_2O_3 + 2 Mn 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 Al | 2 | -2 Mn_2O_3 | 1 | -1 Al_2O_3 | 1 | 1 Mn | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression Al | 2 | -2 | ([Al])^(-2) Mn_2O_3 | 1 | -1 | ([Mn2O3])^(-1) Al_2O_3 | 1 | 1 | [Al2O3] Mn | 2 | 2 | ([Mn])^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 = ([Al])^(-2) ([Mn2O3])^(-1) [Al2O3] ([Mn])^2 = ([Al2O3] ([Mn])^2)/(([Al])^2 [Mn2O3])

Construct the rate of reaction expression for: Al + Mn_2O_3 ⟶ Al_2O_3 + Mn 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: 2 Al + Mn_2O_3 ⟶ Al_2O_3 + 2 Mn 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 Al | 2 | -2 Mn_2O_3 | 1 | -1 Al_2O_3 | 1 | 1 Mn | 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 Al | 2 | -2 | -1/2 (Δ[Al])/(Δt) Mn_2O_3 | 1 | -1 | -(Δ[Mn2O3])/(Δt) Al_2O_3 | 1 | 1 | (Δ[Al2O3])/(Δt) Mn | 2 | 2 | 1/2 (Δ[Mn])/(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 = -1/2 (Δ[Al])/(Δt) = -(Δ[Mn2O3])/(Δt) = (Δ[Al2O3])/(Δt) = 1/2 (Δ[Mn])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| aluminum | manganese(III) oxide | aluminum oxide | manganese formula | Al | Mn_2O_3 | Al_2O_3 | Mn name | aluminum | manganese(III) oxide | aluminum oxide | manganese IUPAC name | aluminum | oxo-(oxomanganiooxy)manganese | dialuminum;oxygen(2-) | manganese

| aluminum | manganese(III) oxide | aluminum oxide | manganese molar mass | 26.9815385 g/mol | 157.873 g/mol | 101.96 g/mol | 54.938044 g/mol phase | solid (at STP) | solid (at STP) | solid (at STP) | solid (at STP) melting point | 660.4 °C | 1347 °C | 2040 °C | 1244 °C boiling point | 2460 °C | | | 1962 °C density | 2.7 g/cm^3 | 4.5 g/cm^3 | | 7.3 g/cm^3 solubility in water | insoluble | | | insoluble surface tension | 0.817 N/m | | | dynamic viscosity | 1.5×10^-4 Pa s (at 760 °C) | | | odor | odorless | | odorless |