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H2O + KMnO4 + Cr2(SO4)3 = H2SO4 + K2Cr2O7 + MnO2

Input interpretation

H_2O water + KMnO_4 potassium permanganate + Cr_2(SO_4)_3 chromium sulfate ⟶ H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + MnO_2 manganese dioxide
H_2O water + KMnO_4 potassium permanganate + Cr_2(SO_4)_3 chromium sulfate ⟶ H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + MnO_2 manganese dioxide

Balanced equation

Balance the chemical equation algebraically: H_2O + KMnO_4 + Cr_2(SO_4)_3 ⟶ H_2SO_4 + K_2Cr_2O_7 + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 Cr_2(SO_4)_3 ⟶ c_4 H_2SO_4 + c_5 K_2Cr_2O_7 + c_6 MnO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn, Cr and S: H: | 2 c_1 = 2 c_4 O: | c_1 + 4 c_2 + 12 c_3 = 4 c_4 + 7 c_5 + 2 c_6 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 Cr: | 2 c_3 = 2 c_5 S: | 3 c_3 = c_4 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 1 c_4 = 3 c_5 = 1 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: |   | 3 H_2O + 2 KMnO_4 + Cr_2(SO_4)_3 ⟶ 3 H_2SO_4 + K_2Cr_2O_7 + 2 MnO_2
Balance the chemical equation algebraically: H_2O + KMnO_4 + Cr_2(SO_4)_3 ⟶ H_2SO_4 + K_2Cr_2O_7 + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 Cr_2(SO_4)_3 ⟶ c_4 H_2SO_4 + c_5 K_2Cr_2O_7 + c_6 MnO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn, Cr and S: H: | 2 c_1 = 2 c_4 O: | c_1 + 4 c_2 + 12 c_3 = 4 c_4 + 7 c_5 + 2 c_6 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 Cr: | 2 c_3 = 2 c_5 S: | 3 c_3 = c_4 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 1 c_4 = 3 c_5 = 1 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2O + 2 KMnO_4 + Cr_2(SO_4)_3 ⟶ 3 H_2SO_4 + K_2Cr_2O_7 + 2 MnO_2

Structures

 + + ⟶ + +
+ + ⟶ + +

Names

water + potassium permanganate + chromium sulfate ⟶ sulfuric acid + potassium dichromate + manganese dioxide
water + potassium permanganate + chromium sulfate ⟶ sulfuric acid + potassium dichromate + manganese dioxide

Equilibrium constant

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + Cr_2(SO_4)_3 ⟶ H_2SO_4 + K_2Cr_2O_7 + MnO_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 H_2O + 2 KMnO_4 + Cr_2(SO_4)_3 ⟶ 3 H_2SO_4 + K_2Cr_2O_7 + 2 MnO_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 H_2O | 3 | -3 KMnO_4 | 2 | -2 Cr_2(SO_4)_3 | 1 | -1 H_2SO_4 | 3 | 3 K_2Cr_2O_7 | 1 | 1 MnO_2 | 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 | 3 | -3 | ([H2O])^(-3) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) Cr_2(SO_4)_3 | 1 | -1 | ([Cr2(SO4)3])^(-1) H_2SO_4 | 3 | 3 | ([H2SO4])^3 K_2Cr_2O_7 | 1 | 1 | [K2Cr2O7] MnO_2 | 2 | 2 | ([MnO2])^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])^(-3) ([KMnO4])^(-2) ([Cr2(SO4)3])^(-1) ([H2SO4])^3 [K2Cr2O7] ([MnO2])^2 = (([H2SO4])^3 [K2Cr2O7] ([MnO2])^2)/(([H2O])^3 ([KMnO4])^2 [Cr2(SO4)3])
Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + Cr_2(SO_4)_3 ⟶ H_2SO_4 + K_2Cr_2O_7 + MnO_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 H_2O + 2 KMnO_4 + Cr_2(SO_4)_3 ⟶ 3 H_2SO_4 + K_2Cr_2O_7 + 2 MnO_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 H_2O | 3 | -3 KMnO_4 | 2 | -2 Cr_2(SO_4)_3 | 1 | -1 H_2SO_4 | 3 | 3 K_2Cr_2O_7 | 1 | 1 MnO_2 | 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 | 3 | -3 | ([H2O])^(-3) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) Cr_2(SO_4)_3 | 1 | -1 | ([Cr2(SO4)3])^(-1) H_2SO_4 | 3 | 3 | ([H2SO4])^3 K_2Cr_2O_7 | 1 | 1 | [K2Cr2O7] MnO_2 | 2 | 2 | ([MnO2])^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])^(-3) ([KMnO4])^(-2) ([Cr2(SO4)3])^(-1) ([H2SO4])^3 [K2Cr2O7] ([MnO2])^2 = (([H2SO4])^3 [K2Cr2O7] ([MnO2])^2)/(([H2O])^3 ([KMnO4])^2 [Cr2(SO4)3])

Rate of reaction

Construct the rate of reaction expression for: H_2O + KMnO_4 + Cr_2(SO_4)_3 ⟶ H_2SO_4 + K_2Cr_2O_7 + MnO_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 H_2O + 2 KMnO_4 + Cr_2(SO_4)_3 ⟶ 3 H_2SO_4 + K_2Cr_2O_7 + 2 MnO_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 H_2O | 3 | -3 KMnO_4 | 2 | -2 Cr_2(SO_4)_3 | 1 | -1 H_2SO_4 | 3 | 3 K_2Cr_2O_7 | 1 | 1 MnO_2 | 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 | 3 | -3 | -1/3 (Δ[H2O])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) Cr_2(SO_4)_3 | 1 | -1 | -(Δ[Cr2(SO4)3])/(Δt) H_2SO_4 | 3 | 3 | 1/3 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 1 | 1 | (Δ[K2Cr2O7])/(Δt) MnO_2 | 2 | 2 | 1/2 (Δ[MnO2])/(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[H2O])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -(Δ[Cr2(SO4)3])/(Δt) = 1/3 (Δ[H2SO4])/(Δt) = (Δ[K2Cr2O7])/(Δt) = 1/2 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)
Construct the rate of reaction expression for: H_2O + KMnO_4 + Cr_2(SO_4)_3 ⟶ H_2SO_4 + K_2Cr_2O_7 + MnO_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 H_2O + 2 KMnO_4 + Cr_2(SO_4)_3 ⟶ 3 H_2SO_4 + K_2Cr_2O_7 + 2 MnO_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 H_2O | 3 | -3 KMnO_4 | 2 | -2 Cr_2(SO_4)_3 | 1 | -1 H_2SO_4 | 3 | 3 K_2Cr_2O_7 | 1 | 1 MnO_2 | 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 | 3 | -3 | -1/3 (Δ[H2O])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) Cr_2(SO_4)_3 | 1 | -1 | -(Δ[Cr2(SO4)3])/(Δt) H_2SO_4 | 3 | 3 | 1/3 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 1 | 1 | (Δ[K2Cr2O7])/(Δt) MnO_2 | 2 | 2 | 1/2 (Δ[MnO2])/(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[H2O])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -(Δ[Cr2(SO4)3])/(Δt) = 1/3 (Δ[H2SO4])/(Δt) = (Δ[K2Cr2O7])/(Δt) = 1/2 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

Chemical names and formulas

 | water | potassium permanganate | chromium sulfate | sulfuric acid | potassium dichromate | manganese dioxide formula | H_2O | KMnO_4 | Cr_2(SO_4)_3 | H_2SO_4 | K_2Cr_2O_7 | MnO_2 Hill formula | H_2O | KMnO_4 | Cr_2O_12S_3 | H_2O_4S | Cr_2K_2O_7 | MnO_2 name | water | potassium permanganate | chromium sulfate | sulfuric acid | potassium dichromate | manganese dioxide IUPAC name | water | potassium permanganate | chromium(+3) cation trisulfate | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | dioxomanganese
| water | potassium permanganate | chromium sulfate | sulfuric acid | potassium dichromate | manganese dioxide formula | H_2O | KMnO_4 | Cr_2(SO_4)_3 | H_2SO_4 | K_2Cr_2O_7 | MnO_2 Hill formula | H_2O | KMnO_4 | Cr_2O_12S_3 | H_2O_4S | Cr_2K_2O_7 | MnO_2 name | water | potassium permanganate | chromium sulfate | sulfuric acid | potassium dichromate | manganese dioxide IUPAC name | water | potassium permanganate | chromium(+3) cation trisulfate | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | dioxomanganese

Substance properties

 | water | potassium permanganate | chromium sulfate | sulfuric acid | potassium dichromate | manganese dioxide molar mass | 18.015 g/mol | 158.03 g/mol | 392.2 g/mol | 98.07 g/mol | 294.18 g/mol | 86.936 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) melting point | 0 °C | 240 °C | | 10.371 °C | 398 °C | 535 °C boiling point | 99.9839 °C | | 330 °C | 279.6 °C | |  density | 1 g/cm^3 | 1 g/cm^3 | 1.84 g/cm^3 | 1.8305 g/cm^3 | 2.67 g/cm^3 | 5.03 g/cm^3 solubility in water | | | | very soluble | | insoluble surface tension | 0.0728 N/m | | | 0.0735 N/m | |  dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.021 Pa s (at 25 °C) | |  odor | odorless | odorless | odorless | odorless | odorless |
| water | potassium permanganate | chromium sulfate | sulfuric acid | potassium dichromate | manganese dioxide molar mass | 18.015 g/mol | 158.03 g/mol | 392.2 g/mol | 98.07 g/mol | 294.18 g/mol | 86.936 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) melting point | 0 °C | 240 °C | | 10.371 °C | 398 °C | 535 °C boiling point | 99.9839 °C | | 330 °C | 279.6 °C | | density | 1 g/cm^3 | 1 g/cm^3 | 1.84 g/cm^3 | 1.8305 g/cm^3 | 2.67 g/cm^3 | 5.03 g/cm^3 solubility in water | | | | very soluble | | insoluble surface tension | 0.0728 N/m | | | 0.0735 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.021 Pa s (at 25 °C) | | odor | odorless | odorless | odorless | odorless | odorless |

Units