Question

Consider the reaction $$ \mathrm{PbCO}_{3}(s) \rightleftharpoons \mathrm{PbO}(s)+\mathrm{CO}_{2}(g) $$ Using data in Appendix C, calculate the equilibrium pres- sure of \(\mathrm{CO}_{2}\) in the system at (a) \(400^{\circ} \mathrm{C}\) and \((\mathbf{b}) 180^{\circ} \mathrm{C} .\)

Short answer

The equilibrium pressures of CO2 in the system at (a) 400°C and (b) 180°C are calculated using the relationship between Gibbs free energy change and the equilibrium constant. For 400°C, the equilibrium pressure of CO2 is 6.42 atm, and for 180°C, it is 0.183 atm.

Step-by-step solution

Write the expression for the equilibrium constant K

For the reaction, \[PbCO_3(s) \rightleftharpoons PbO(s) + CO_2(g)\] The equilibrium constant K is defined as: \[K = \frac{P_{CO_2}}{1}\] Since the partial pressures of solids is unity, only the partial pressure of CO2 gas affects the value of K.

Determine the Gibbs free energy changes (∆G°) associated with the reaction

We will obtain the standard Gibbs free energy change values (∆G°) for the reaction from Appendix C. ∆G°(PbCO3) = -699.3 kJ/mol ∆G°(PbO) = -217.3 kJ/mol ∆G°(CO2) = -394.4 kJ/mol

Calculate the Gibbs free energy change for the reaction (∆G°rxn)

Using the given ∆G° values, let's calculate the ∆G° for the reaction: \[\Delta G_{rxn}^{\circ} = \sum \Delta G_{products}^{\circ} - \sum \Delta G_{reactants}^{\circ}\] \[\Delta G_{rxn}^{\circ} = ( \Delta G_{PbO}^{\circ} + \Delta G_{CO_2}^{\circ}) - \Delta G_{PbCO_3}^{\circ}\] \[\Delta G_{rxn}^{\circ} = (-217.3 - 394.4) - (-699.3)\] ΔG°rxn = -87.6 kJ/mol

Calculate the equilibrium constant K using Gibbs free energy change

The relationship between the standard Gibbs free energy change and the equilibrium constant K is given by: \[\Delta G_{rxn}^{\circ} = -RT \ln K\] Where R is the gas constant (8.314 J/(mol K)) and T is the temperature in Kelvin. Now, let's calculate the equilibrium constant K for both temperatures: (a) At 400°C, T = 673 K. \[-87.6 \times 10^3 = -8.314 \times 673 \times \ln K_{a}\] \[K_{a} = e^{\frac{-87.6 \times 10^3}{-8.314 \times 673}}\] \[K_{a} = 6.42\] (b) At 180°C, T = 453 K. \[-87.6 \times 10^3 = -8.314 \times 453 \times \ln K_{b}\] \[K_{b} = e^{\frac{-87.6 \times 10^3}{-8.314 \times 453}}\] \[K_{b} = 0.183\]

Calculate the equilibrium pressure of CO2

We previously obtained that the K = P_CO2, so at both temperatures: (a) At 400°C, P_CO2 = K_a = 6.42 atm (b) At 180°C, P_CO2 = K_b = 0.183 atm Hence, the equilibrium pressures of CO2 at 400°C and 180°C are 6.42 atm and 0.183 atm, respectively.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that occurs when a chemical reaction and its reverse reaction proceed at the same rate, leading to a balance in the concentration of reactants and products. It's important to recognize that this dynamic state does not imply that the reactants and products are in equal concentrations, but rather that their concentrations have stabilized in a ratio that does not change over time.

In the context of our example, the reaction involving the decomposition of lead carbonate into lead oxide and carbon dioxide gas reaches chemical equilibrium when the rate of decomposition equals the rate at which lead oxide and carbon dioxide react to form lead carbonate again. Although, in this case, we are focused on a particular part of the equilibrium involving the pressure of carbon dioxide gas, the principle is the same. Even in a state of equilibrium, reactions continue to occur in both directions; however, there is no net change in the concentrations of reactants and products.

Understanding chemical equilibrium is crucial in various applications such as predicting the extent of a reaction, determining the optimal conditions for a reaction, and synthesizing products in the chemical industry.

Gibbs Free Energy

Gibbs free energy, often denoted as \(\Delta G\), is a thermodynamic quantity that is used to predict the direction of chemical reactions and to determine whether a process is spontaneous at constant temperature and pressure. The 'free' portion of the term refers to the energy within a system that is available to do work.

The Gibbs free energy for a reaction at equilibrium is zero; however, the change in Gibbs free energy (\(\Delta G_{rxn}^\circ\)) for a process is a useful predictor of spontaneity. If \(\Delta G_{rxn}^\circ\) is negative, the reaction will proceed spontaneously in the forward direction under standard conditions. On the other hand, a positive value indicates that the reaction is not spontaneous and will proceed in the reverse direction.

In our exercise, we used the changes in Gibbs free energy for lead carbonate, lead oxide, and carbon dioxide to calculate the overall change in Gibbs free energy for the reaction. This value is then utilized to determine the equilibrium constant \(K\), which in turn allows for the calculation of equilibrium pressure of \(\mathrm{CO}_2\).

Equilibrium Constant

The equilibrium constant, represented by the symbol \(K\), quantifies the relationship between the concentrations of reactants and products at chemical equilibrium for a reversible reaction. For reactions in which gases are involved, the equilibrium constant can be expressed in terms of partial pressures, as seen in our example, which results in a \(K_p\) value.

For the reaction at hand, since lead carbonate and lead oxide are in solid states, their concentrations remain constant and are therefore not included in the expression for \(K\). As a result, the equilibrium constant is simply the partial pressure of \(\mathrm{CO}_2\) gas in atmospheres. The calculation of equilibrium constant involves the use of the standard Gibbs free energy change, as shown in the exercise, where we found that \(K\) can be calculated from \(\Delta G_{rxn}^\circ\) using the relationship \(\Delta G_{rxn}^\circ = -RT \ln K\).

The exercise demonstrates how the equilibrium constant changes with temperature: when we calculated \(K_a\) and \(K_b\) for 400°C and 180°C, respectively. The value of \(K\) directly reflects the extent to which a reaction favors the production of products over reactants, leading to higher \(K\) values signifying a greater degree of conversion to products at equilibrium. This information is essential when working with chemical reactions and determining the conditions necessary for desired product yield.