Problem 36 - General Chemistry: Principles and Modern Applications

Question

Calculate the equilibrium constant and Gibbs energy for the reaction \(\mathrm{CO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})\) at \(483 \mathrm{K}\) by using the data tables from Appendix D. Are the values determined here different from or the same as those in exercise \(35 ?\) Explain.

Step-by-Step Solution

Answer

The Gibbs energy and equilibrium constant for the reaction are calculated from the provided data in Appendix D. We then compare the obtained values to those from exercise 35 to determine if they are the same or different.

Step 1: Identify the relevant data from Appendix D

Identify the standard Gibbs free energy of formation(\( \Delta G_{f}^{0} \)) for each compound involved in the reaction from the Appendix D. These values should be provided for the compounds CO, H2, and CH3OH at the temperature of 483K.

Step 2: Calculate the Gibbs free energy(ΔG) for the reaction

Calculate the standard Gibbs free energy of the reaction (\( \Delta G^{0} \)) using the equation \( \Delta G^{0} = \sum \Delta G_{f}^{0} (products) - \sum \Delta G_{f}^{0} (reactants) \). Here, multiply the \( \Delta G_{f}^{0} \) value of each compound by its stoichiometric coefficient in the balanced chemical equation.

Step 3: Calculate the equilibrium constant (K)

Use the obtained value of \( \Delta G^{0} \) to calculate the equilibrium constant (K) for the reaction at the given temperature. The Van't Hoff Equation relates these two quantities: \( \Delta G^{0} = -RT \ln K \), where R is the gas constant (8.314 J/mol*K) and T is the absolute temperature in Kelvin. Solving for K gives \( K = e^{-\Delta G^{0}/RT} \) .

Step 4: Compare the results with exercise 35

Compare the calculated values of \( \Delta G^{0} \) and K for this reaction with the values obtained in exercise 35. Express whether they match or differ and offer a probable explanation why.

Key Concepts

Gibbs Free Energy

Gibbs Free Energy (denoted as \( \Delta G \)) is a key concept in chemical equilibrium and thermodynamics. It tells us whether a reaction can occur spontaneously. A negative \( \Delta G \) indicates that a process can happen without any outside energy, thereby being considered spontaneous. To calculate the Gibbs free energy of a reaction, you first need the standard Gibbs free energy of formation (\( \Delta G_f^0 \)) values for each reactant and product, found in data tables like Appendix D.

Here's a simplified way to think about it:

• Subtract the total \( \Delta G_f^0 \) of the reactants from the total \( \Delta G_f^0 \) of the products.
• Consider stoichiometric coefficients from the balanced chemical equation, which show the molar amounts participating in the reaction.

This difference gives the standard Gibbs free energy change for the overall reaction (\( \Delta G^0 \)). If our \( \Delta G^0 \) value is negative, the reaction tends to favor products under standard conditions, i.e., it is spontaneous. Conversely, a positive \( \Delta G^0 \) suggests the reaction is non-spontaneous and predominantly favors reactants.

Equilibrium Constant

The equilibrium constant \( K \) is a crucial parameter in determining the position of equilibrium for a given chemical reaction. It measures how far a reaction proceeds before reaching equilibrium, where the rates of the forward and reverse reactions are equal. Simply put, it indicates the concentration ratio of products to reactants at equilibrium.

Derived from the reaction's Gibbs free energy \( \Delta G^0 \), \( K \) is dictated by the equation \( \Delta G^0 = -RT \ln K \), where \( R \) is the gas constant (8.314 J/mol*K) and \( T \) is the temperature in Kelvin. This equation highlights the profound connection between thermodynamics (in the form of \( \Delta G^0 \)) and equilibrium chemistry (through \( K \)).

• A large \( K \) value indicates that the reaction heavily favors the formation of products.
• A small \( K \) value suggests reactants are favored.
• If \( K \) is close to 1, the reaction is balanced at equilibrium, with neither reactants nor products being significantly more present.

Understanding \( K \) helps predict the direction of the reaction under specific conditions and guides chemists in manipulating those conditions to obtain desired outcomes.

Van't Hoff Equation

The Van't Hoff Equation provides a vital link between the temperature of a system and its equilibrium constant. It helps us predict how changes in temperature affect the position of equilibrium. The equation is expressed as:\[\Delta G^0 = -RT \ln K\]This formula illustrates how the standard Gibbs free energy change (\( \Delta G^0 \)) relates directly to the temperature (\( T \)) and the equilibrium constant (\( K \)). Essentially, Van't Hoff lets us see the temperature dependency of the equilibrium state.

Important points to consider with the Van't Hoff Equation:

• If temperature increases for an endothermic reaction (one that absorbs heat), \( K \) typically increases, favoring more product formation.
• For an exothermic reaction (one that releases heat), increasing temperature usually decreases \( K \), shifting the equilibrium to favor reactants.
• It provides crucial insights into reaction kinetics and mechanisms, aiding in industrial processes where temperature control is essential.

By understanding the Van't Hoff Equation, chemists can better manage and optimize reactions in both laboratory and industrial settings to achieve optimal product yields.