Question

The value of \(\log _{10} \mathrm{~K}\) for a reaction \(\mathrm{A} \rightleftharpoons \mathrm{B}\) is (Given : \(\Delta_{\mathrm{r}} \mathrm{H}_{298 \mathrm{~K}}^{\circ}=-54.07 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{\mathrm{r}} \mathrm{S}_{298 \mathrm{~K}}^{\circ}=10 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) and \(\left.\mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} ; \quad 2.303 \times 8.314 \times 298=5705\right)\) (A) 5 (B) 10 (C) 95 (D) 100

Short answer

(A) 5

Step-by-step solution

Understand the Gibbs Free Energy Equation

Use the Gibbs free energy equation \(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ\) to calculate the standard Gibbs free energy change for the reaction. The temperature (T) is given as 298 K.

Calculate Standard Gibbs Free Energy Change

Plug in the given values to find \(\Delta G^\circ\): \(\Delta G^\circ = (-54.07 \times 10^3 \mathrm{~J} \mathrm{~mol}^{-1}) - (298 \mathrm{~K})(10 \mathrm{~JK}^{-1} \mathrm{~mol}^{-1})\).

Convert Gibbs Free Energy to Equilibrium Constant

The relationship between Gibbs free energy and the equilibrium constant is given by the equation \(\Delta G^\circ = -RT\ln K\). To find \(\log_{10} K\), use the relation \(\ln K = 2.303 \log_{10} K\) and the provided value for \(RT\).

Solve for \(\log_{10} K\)

Rearrange the equation to solve for \(\log_{10} K\) and substitute the calculated value of \(\Delta G^\circ\) and the given value of \(RT\): \(\log_{10} K = -\frac{\Delta G^\circ}{2.303 \times RT}\)

Calculate and Choose the Answer

After substituting all the values into the equation, calculate \(\log_{10} K\) to determine the closest answer among the choices.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as K, is a profound concept in chemistry that holds the key to understanding how chemical reactions reach a state of balance. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant over time.

For a generic reaction where A and B are reactants and products, respectively, written as \(A \rightleftharpoons B\), the equilibrium constant (K) is expressed in terms of the concentration of the products over the reactants, raised to their stoichiometric coefficients.

In the provided exercise, the relationship between Gibbs free energy change (\(\Delta G^\text{\circ}\)) and the equilibrium constant is essential. This relationship can be mathematically represented by the equation \(\Delta G^\text{\circ} = -RT\ln K\), where R is the gas constant and T is the temperature in Kelvin. When we solve for K, we can therefore predict the direction of the reaction and the extent to which reactants are converted into products under standard conditions.

Thermodynamics in Chemistry

Thermodynamics in chemistry is the study of energy changes accompanying chemical reactions and physical changes. One of the core principles is the concept of Gibbs free energy (G), a thermodynamic potential that measures the maximum amount of work that can be obtained from a closed system at constant pressure and temperature.

According to the second law of thermodynamics, the natural progression of any closed system moves towards a state of maximum entropy or disorder. The Gibbs free energy equation, \(\Delta G^\text{\circ} = \Delta H^\text{\circ} - T\Delta S^\text{\circ}\), elegantly combines enthalpy (\(\Delta H^\text{\circ}\), a measure of the total energy of a system), and entropy (\(\Delta S^\text{\circ}\), a measure of system disorder) to predict the direction of a chemical reaction and determine whether the reaction is spontaneous.

Standard Gibbs Free Energy Change

The standard Gibbs free energy change (\(\Delta G^\text{\circ}\)) of a reaction is a crucial concept that quantifies the spontaneity of a reaction under standard conditions (1 bar pressure and the temperature of interest, often 298 K). When \(\Delta G^\text{\circ}\) is negative, the process is exergonic, indicating that the reaction can proceed spontaneously. Conversely, a positive \(\Delta G^\text{\circ}\) suggests an endergonic reaction, which requires energy input to proceed.

In the context of our exercise, the calculated value of \(\Delta G^\text{\circ}\) can be plugged into the equation \(\Delta G^\text{\circ} = -RT\ln K\) to determine the equilibrium constant. The equilibrium constant reflects the ratio of product to reactant concentrations and thus indicates the extent to which a reaction has progressed. By converting the natural logarithm of K to a base-10 logarithm, we can derive the provided exercise's \(\log _{10} K\), enabling us to gauge the reaction's position of equilibrium and better understand the underlying thermodynamics.