For a certain chemical reaction, the equilibrium constant [tex]K=6.6 \times 10^6[/tex] at [tex]25.0{ }^{\circ} C[/tex]. Calculate the standard Gibbs free energy of reaction. Round your answer to 2 significant digits.
Answer (2)
Convert the temperature from Celsius to Kelvin: T = 25.0 + 273.15 = 298.15 K .
Use the formula relating Gibbs free energy to the equilibrium constant: Δ G ∘ = − RT ln K .
Substitute the values: Δ G ∘ = − ( 8.314 ) ( 298.15 ) ln ( 6.6 × 1 0 6 ) ≈ − 38909.7 J / m o l .
Convert to kJ/mol and round to 2 significant digits: Δ G ∘ ≈ − 39 k J / m o l .
Explanation
Problem Analysis We are given the equilibrium constant K for a chemical reaction at a specific temperature T , and we are asked to calculate the standard Gibbs free energy of reaction, Δ G ∘ .
Formula We will use the relationship between the standard Gibbs free energy and the equilibrium constant: Δ G ∘ = − RT ln K where R is the ideal gas constant, T is the temperature in Kelvin, and K is the equilibrium constant.
Convert Celsius to Kelvin First, we need to convert the temperature from Celsius to Kelvin: T ( K ) = T ( ∘ C ) + 273.15 So, T = 25.0 + 273.15 = 298.15 K
Calculate Gibbs Free Energy Now, we can substitute the values into the formula: Δ G ∘ = − ( 8.314 m o l K J ) ( 298.15 K ) ln ( 6.6 × 1 0 6 ) Δ G ∘ = − ( 8.314 ) ( 298.15 ) ln ( 6.6 × 1 0 6 ) m o l J Δ G ∘ = − ( 8.314 ) ( 298.15 ) ( 15.699 ) m o l J Δ G ∘ = − 38909.7 m o l J To convert to kJ/mol, we divide by 1000: Δ G ∘ = − 38.9097 m o l k J
Round the Answer Rounding to 2 significant digits, we get: Δ G ∘ ≈ − 39 m o l k J
Final Answer Therefore, the standard Gibbs free energy of reaction is approximately -39 kJ/mol.
Examples
The Gibbs free energy is a crucial concept in thermodynamics that helps predict the spontaneity of a chemical reaction. For instance, in the Haber-Bosch process, which synthesizes ammonia from nitrogen and hydrogen, understanding the Gibbs free energy change is essential. By manipulating temperature and pressure, engineers can optimize the reaction conditions to ensure a negative Gibbs free energy, making the reaction spontaneous and maximizing ammonia production. This has significant implications for fertilizer production and global food supply.
The standard Gibbs free energy of reaction is calculated using the formula Δ G ∘ = − RT ln K . After substituting the values and performing the calculation, we find that Δ G ∘ is approximately − 39 kJ/mol .
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