A chemist fills a reaction vessel with 0.536 g aluminum hydroxide $[Al(OH)_3]$ solid, 0.337 M aluminum $[Al^{3+}]$ aqueous solution, and 0.196 M hydroxic $[OH^-]$ aqueous solution at a temperature of $25.0^{\circ} C$. Under these conditions, calculate the reaction free energy $\Delta G$ for the following chemical reaction: $Al(OH)_3(s) \rightleftharpoons Al^{3+}(aq)+3 OH^{-}(aq)$ Use the thermodynamic information in the ALEKS Data tab. Round your answer to the nearest kilojoule.
Answer (2)
Calculate the standard free energy change: Δ G ∘ = 179 kJ/mol .
Calculate the reaction quotient: Q = [ A l 3 + ] [ O H − ] 3 = ( 0.337 ) ( 0.196 ) 3 = 0.002537 .
Use the equation Δ G = Δ G ∘ + RTl n Q to find Δ G = 179000 + ( 8.314 ) ( 298.15 ) l n ( 0.002537 ) = 164204.3 J/mol .
Convert to kilojoules and round: Δ G ≈ 164 kJ/mol .
164
Explanation
Problem Setup We are asked to calculate the reaction free energy Δ G for the reaction A l ( O H ) 3 ( s ) ⇌ A l 3 + ( a q ) + 3 O H − ( a q ) under given conditions. We will use the equation Δ G = Δ G ∘ + RTl n Q , where Δ G ∘ is the standard free energy change, R is the ideal gas constant, T is the temperature in Kelvin, and Q is the reaction quotient.
Find Standard Free Energy Change First, we need to find the standard free energy change, Δ G ∘ . According to the ALEKS data tab, Δ G ∘ = 179 kJ/mol = 179000 J/mol .
Calculate Reaction Quotient Q Next, we calculate the reaction quotient Q. For the given reaction, Q = [ A l 3 + ] [ O H − ] 3 . We are given the concentrations of A l 3 + and O H − as 0.337 M and 0.196 M, respectively. Therefore, Q = ( 0.337 ) ( 0.196 ) 3 = 0.337 × 0.00752936 = 0.002537
Calculate Delta G Now, we can use the equation Δ G = Δ G ∘ + RTl n Q to calculate the reaction free energy Δ G . The ideal gas constant R is 8.314 J/(mol*K), and the temperature T is 25. 0 ∘ C = 298.15 K . Thus, Δ G = 179000 + ( 8.314 ) ( 298.15 ) l n ( 0.002537 ) = 179000 + ( 8.314 ) ( 298.15 ) ( − 5.974 ) = 179000 − 14795.7 = 164204.3 J/mol
Convert to kJ and Round Finally, we convert the result to kilojoules and round to the nearest kilojoule: Δ G = 1000 164204.3 = 164.2043 kJ/mol ≈ 164 kJ/mol
Final Answer Therefore, the reaction free energy Δ G under these conditions is approximately 164 kJ.
Examples
Understanding Gibbs free energy is crucial in many real-world applications. For instance, in designing new chemical reactions for drug synthesis, chemists need to know whether a reaction will occur spontaneously under certain conditions. By calculating the Gibbs free energy, they can predict the feasibility of the reaction and optimize conditions such as temperature and concentration to maximize product yield. This ensures efficient and cost-effective production of pharmaceuticals.
To calculate Δ G for the given chemical reaction, we first found Δ G ∘ = 179 kJ/mol and calculated the reaction quotient Q ≈ 0.002537 . Using the equation Δ G = Δ G ∘ + RT ln Q , we determined that Δ G ≈ 164 kJ/mol after performing the necessary calculations and conversions.
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