Question

Mathematical Consider the reaction \(A \rightleftharpoons B+C,\) where \(\Delta G^{\circ}=0.00\) (a) What is the value of \(\Delta G\left(\operatorname{not} \Delta G^{\circ}\right)\) when the initial concentrations of \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) are \(1 \mathrm{M}, 10^{-3} \mathrm{M},\) and \(10^{-6} \mathrm{M} ?\) (b) Try the same calculations for the reaction \(\mathrm{D}+\mathrm{E} \rightleftharpoons \mathrm{F}\), for the same relative order of concentrations. (c) Try the same calculations for the reaction \(G \rightleftharpoons H,\) if the concentrations are \(1 \mathrm{M}\) and \(10^{-3} \mathrm{M}\) for \(\mathrm{G}\) and \(\mathrm{H}\), respectively

Short answer

For (a) Reaction A ⇌ B + C, ΔG = -51.457 kJ/mol. For (b) Reaction D + E ⇌ F, ΔG = 0.00 kJ/mol. For (c) Reaction G ⇌ H, ΔG = -17.129 kJ/mol.

Step-by-step solution

- Writing down the Gibbs Free Energy formula

Use the formula for Gibbs Free Energy change: oindent\texttt{\( \Delta G = \Delta G^{\circ} + RT \ln Q \)} where \(Q\) is the reaction quotient, \(R\) is the gas constant (8.314 J/(mol·K)), and \(T\) is the temperature in Kelvin. Assume standard temperature \(T = 298 K\).

- Calculate the reaction quotient Q for reaction A ⇌ B + C

For the reaction \(A \rightleftharpoons B+C\), the reaction quotient \(Q\) is given by: oindent\( Q = \frac{[B][C]}{[A]} \) Substitute the given concentrations: oindent\( Q = \frac{10^{-3} \times 10^{-6}}{1} = 10^{-9} \)

- Substitute into Gibbs Free Energy formula for reaction A ⇌ B + C

Using the equation \( \Delta G = \Delta G^{\circ} + RT \ln Q \) with \( \Delta G^{\circ} = 0.00\), \( R = 8.314 \) J/(mol·K), \( T = 298 \) K, and \( Q = 10^{-9}\) : oindent\( \Delta G = 0.00 + (8.314 \times 298) \ln(10^{-9}) \)Calculate \( \ln(10^{-9}) = -20.723 \).\thenoindent\( \Delta G = (8.314 \times 298) \times (-20.723) = -51457.497 \) J/moloindent\( \Delta G = -51.457 \) kJ/mol

- Calculate the reaction quotient Q for reaction D + E ⇌ F

For the reaction \( D + E \rightleftharpoons F \), the reaction quotient \( Q \) is given by: oindent\( Q = \frac{[F]}{[D][E]} \) Substituting the concentrations as per the same order (0.1, 0.001, and 0.0001 M respectively): oindent\( Q = \frac{10^{-6}}{10^{-3} \times 10^{-3}} = 1 \)

- Substitute into Gibbs Free Energy formula for reaction D + E ⇌ F

Using the equation \( \Delta G = \Delta G^{\circ} + RT \ln Q \) with \( \Delta G^{\circ} = 0.00 \) and \( Q = 1 \): oindent\( \Delta G = 0.00 + (8.314 \times 298) \ln(1) \) oindentSince \( \ln(1) = 0 \)oindent: \( \Delta G = 0.00 \)

- Calculate the reaction quotient Q for reaction G ⇌ H

For the reaction \( G \rightleftharpoons H \), the reaction quotient \( Q \) is given by: oindent\( Q = \frac{[H]}{[G]} \) Substituting the given concentrations: oindent\( Q = \frac{10^{-3}}{1} = 10^{-3}\)

- Substitute into Gibbs Free Energy formula for reaction G ⇌ H

Using the equation \( \Delta G = \Delta G^{\circ} + RT \ln Q \) with \( \Delta G^{\circ} = 0.00 \), \( R = 8.314 \) J/(mol·K), \( T = 298 \) K, and \( Q = 10^{-3} \): oindent\( \Delta G = 0.00 + (8.314 \times 298) \ln(10^{-3}) \) Calculate \( \ln(10^{-3}) = -6.908 \). oindent\( \Delta G = (8.314 \times 298) \times (-6.908) = -17129.255 \) J/moloindent\( \Delta G = -17.129 \) kJ/mol

Key concepts

thermodynamics

Thermodynamics is the branch of physical science that deals with the relationships and conversions between heat and other forms of energy. It encompasses several key principles:

• The First Law of Thermodynamics: Energy cannot be created or destroyed, only converted from one form to another.
• The Second Law of Thermodynamics: In any energy transfer, some energy becomes unavailable to do work, often lost as heat, and the total entropy of a system and its surroundings always increases.
• The Third Law of Thermodynamics: As the temperature of a system approaches absolute zero (0 Kelvin), the entropy of the system approaches a minimum value.

Through these principles, thermodynamics helps us to understand and predict the energy changes involved in chemical reactions.}}