Question

For the reaction: \(2 \mathrm{NOCl}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})\) \(+\mathrm{Cl}_{2}(\mathrm{~g}), \Delta \mathrm{H}^{\circ}=18 \mathrm{kcal}\) and \(\Delta S^{\circ}=30 \mathrm{cal} / \mathrm{K}\) at \(300 \mathrm{~K}\). The equilibrium constant, \(K_{\mathrm{p}}^{\circ}\) of the reaction at \(300 \mathrm{~K}\) is (a) \(\mathrm{e}^{15}\) (b) \(\mathrm{e}^{-15}\) (c) \(\mathrm{e}^{-18}\) (d) \(\mathrm{e}^{-12}\)

Short answer

\(e^{-15}\)

Step-by-step solution

Understand the Gibbs Free Energy Equation

The equilibrium constant, \(K_p\), for a reaction can be calculated from the standard Gibbs free energy change, \(\Delta G^\circ\), using the equation \(\Delta G^\circ = -RT\ln(K_p)\), where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\ln\) is the natural logarithm.

Calculate the Standard Gibbs Free Energy Change

The standard Gibbs free energy change \(\Delta G^\circ\) can be determined using the relationship \(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ\), where \(\Delta H^\circ\) is the standard enthalpy change and \(\Delta S^\circ\) is the standard entropy change for the reaction.

Convert Units for Entropy

The entropy is given in \(\mathrm{cal} / \mathrm{K}\), but the gas constant \(R\) will be used in \(\mathrm{kcal} / (\mathrm{K}\cdot\mathrm{mol})\). Therefore, convert \(\Delta S^\circ\) from calories to kilocalories by dividing by 1000.

Insert Values into the Gibbs Free Energy Equation

Insert the given values \(\Delta H^\circ = 18 \mathrm{kcal}\) and the converted \(\Delta S^\circ\) value into the equation \(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ\), and calculate \(\Delta G^\circ\).

Use the Gibbs Free Energy to Calculate \(K_p\)

Use the calculated \(\Delta G^\circ\) value and the temperature \(T = 300 \mathrm{K}\) in the equation \(\Delta G^\circ = -RT\ln(K_p)\) to solve for \(K_p\).

Determine the Correct Answer

The value of \(K_p\) will be in the form of an exponential. Compare the calculated \(K_p\) value to the answer choices provided to determine which one matches the calculated value.

Key concepts

Equilibrium Constant

In the context of chemical reactions, the equilibrium constant (represented as K_p for gas-phase reactions or K_c for concentration-based reactions) is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium. Each concentration is raised to the power of its stoichiometric coefficient from the balanced equation. The higher the value of K, the more the reaction favors the formation of products under standard conditions.

Understanding the equilibrium constant is crucial for predicting the extent of a reaction and for calculating related thermodynamic quantities, such as the standard Gibbs free energy change. By using the equation
\[\Delta G^\circ = -RT\ln(K_p)\]
where R is the universal gas constant and T is the temperature in Kelvin, students can relate the equilibrium constant to the energy change of the system.

Enthalpy Change

Enthalpy change (ΔH) is defined as the total heat content of a system and is a measure of the energy absorbed or released during a chemical reaction at constant pressure. It is a crucial component in the determination of the standard Gibbs free energy change. The sign of ΔH indicates whether the reaction is exothermic (negative ΔH, releases heat) or endothermic (positive ΔH, absorbs heat).

In our exercise, the reaction has a positive ΔH of 18 kcal, which suggests that the reaction absorbs heat from the surroundings. This is a critical factor in predicting whether a reaction will be spontaneous under certain conditions. The enthalpy change is used together with entropy change to determine the Gibbs free energy change.

Entropy Change

Entropy change (ΔS) quantifies the degree of disorder or randomness in a system. During a reaction, entropy can either increase or decrease, and this change is essential for assessing the spontaneity of a reaction according to the second law of thermodynamics.

For the reaction given in the exercise, the ΔS is positive, indicating an increase in disorder as the reaction proceeds. Positive entropy changes are common in reactions where gas molecules are produced since gases have higher entropy than solids or liquids. When calculating the Gibbs free energy, the unit of entropy must align with the unit of enthalpy; hence, it's important to convert ΔS from cal/K to kcal/K when R is expressed in kcal/(K·mol) as in our exercise.

Standard Gibbs Free Energy Change

The standard Gibbs free energy change (ΔG) combines the concepts of enthalpy and entropy to determine whether a process is thermodynamically favorable under standard conditions. A negative ΔG indicates a spontaneous process, while a positive value suggests non-spontaneity.

The equation ΔG = ΔH - TΔS reflects the energy balance between the heat exchange (enthalpy) and the dispersal of energy (entropy) at a given temperature T. In the step-by-step solution, after calculating ΔG from the given enthalpy and entropy values, one can determine the equilibrium constant K_p using the relationship ΔG = -RT ln(K_p). This provides a quantitative link between the thermodynamics of a reaction and its equilibrium properties.