Question

The equilibrium constant \(\left(K_{P}\right)\) for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g) $$ is 4.40 at \(2000 \mathrm{~K}\). (a) Calculate \(\Delta G^{\circ}\) for the reaction. (b) Calculate \(\Delta G\) for the reaction when the partial pressures are \(P_{\mathrm{H}_{2}}=0.25 \mathrm{~atm}, P_{\mathrm{CO}_{2}}=0.78 \mathrm{~atm}\) \(P_{\mathrm{H}_{2} \mathrm{O}}=0.66 \mathrm{~atm},\) and \(P_{\mathrm{CO}}=1.20 \mathrm{~atm}\)

Short answer

The standard Gibbs free energy change (\(\Delta G^{\circ}\)) for the reaction is calculated in step 2 to be -33.8 kJ/mol and the Gibbs free energy change under the given conditions (\(\Delta G\)) is calculated in step 4 to be -30.0 kJ/mol.

Step-by-step solution

Calculate Standard Gibbs Free Energy Change

Start by using the given equilibrium constant (KP=4.40) and the temperature (2000K). The standard Gibbs free energy change can be calculated by using the equation: \[ \Delta G^{\circ} = -RT \ln(K_{P}) \] where R is the universal gas constant. Given R=8.314 J/(mol·K), after substituting these values into this equation, you can compute the value of \(\Delta G^{\circ}\).

Convert the Unit of \(\Delta G^{\circ}\)

The calculated value of \(\Delta G^{\circ}\) in Step 1 will be in J/mol. But in chemistry, most often energy changes are expressed in kJ/mol for convenience. So, the calculated Gibbs free energy change needs to be converted from J/mol to kJ/mol by dividing by 1000.

Calculate Gibbs Free Energy Change at Non-Standard Conditions

The Gibbs free energy change at non-standard conditions can be calculated from \(\Delta G^{\circ}\) using the following equation: \[ \Delta G = \Delta G^{\circ} + RT \ln(Q) \] where Q is the reaction quotient defined as \[ Q = \frac{{P_{H2O} \cdot P_{CO}}}{P_{H2} \cdot P_{CO2}} \]. Given the partial pressures of the reactants and products, you can calculate Q and then substitute into the equation for \(\Delta G\).

Convert the Unit of \(\Delta G\)

The calculated \(\Delta G\) will also be in J/mol. Convert this to kJ/mol by dividing by 1000 for the final answer.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as K (with various subscripts like Kc for concentrations and Kp for partial pressures), is a fundamental concept in chemical thermodynamics that characterizes the balance between reactants and products in a chemical reaction at equilibrium. It is expressed as the ratio of the concentration (or partial pressures) of the products raised to their stoichiometric coefficients divided by the same for the reactants.

For gases, the equilibrium constant based on partial pressures, Kp, is often used. It is crucial to know that a large Kp suggests products are favored at equilibrium, while a small Kp indicates reactants are favored. When the reaction quotient Q matches Kp, the system is at equilibrium and no further changes in concentrations or partial pressures occur. Understanding Kp helps us interpret the extent of a reaction and predict the direction in which it will proceed to reach equilibrium.

Reaction Quotient

The reaction quotient, Q, is a measure similar to the equilibrium constant, but for a system that may not be at equilibrium. It's calculated using the same formula as K, but with the current concentrations or partial pressures instead of those at equilibrium.

Q can provide valuable insight into the system's status: if Q is less than K, the system will shift toward the products to reach equilibrium (forward reaction). Conversely, if Q is greater than K, the system will shift toward the reactants (reverse reaction). Calculating Q is not only a snapshot of the system’s condition but can also forecast the direction of reaction shift to achieve equilibrium.

Standard Gibbs Free Energy Change

The standard Gibbs free energy change, \(\Delta G^\circ\), is an essential thermodynamic quantity indicating the spontaneity of a reaction under standard conditions (1 bar or 1 atm and usually 25°C). It is related to the equilibrium constant by the equation \(\Delta G^\circ = -RT \ln(K)\). A negative \(\Delta G^\circ\) signifies a reaction that is spontaneous in the forward direction, while a positive value suggests a non-spontaneous process, requiring an input of energy to occur. By converting \(\Delta G^\circ\) between units, from joules per mole to kilojoules per mole, we can more conveniently connect macroscopic observable properties like temperature (T) and reaction equilibrium behavior (K) with the microscopic realities governing a reaction's energetics.

Partial Pressures

In the realm of chemical reactions involving gases, partial pressures play a pivotal role. The partial pressure of a gas is the pressure it would exert if it alone occupied the entire volume of the mix. For a reaction taking place in the gaseous phase, the partial pressures of reactants and products are crucial determinants of the reaction's progress.

According to Dalton's Law, the total pressure in a mixture of gases is equal to the sum of the partial pressures of individual gases. In calculations involving reactions, partial pressures are used to determine the reaction quotient (Q) and, from there, to calculate the actual Gibbs free energy change (\(\Delta G\)) at non-standard conditions. This links the macroscopic conditions of the reaction to its microscopic thermodynamic characteristics.

Thermodynamics

Thermodynamics is a branch of physical science that deals with heat, work, temperature, and energy. In chemistry, it provides an understanding of how and why chemical reactions occur, and the ability to predict whether a particular reaction will proceed spontaneously. It is grounded in several laws, with the second law indicating that the total entropy of an isolated system can never decrease over time.

One of the central concepts of thermodynamics in chemistry is Gibbs free energy (\(\Delta G\)), which combines the system’s enthalpy and entropy to determine the spontaneity of a reaction. A key aspect is knowing how to relate thermodynamic quantities like \(\Delta G\) under various conditions to measurable properties such as equilibrium constants and reaction quotients. This connection between the theory of thermodynamics and practical chemical processes is what allows scientists and engineers to design reactions and processes that are energetically favorable and therefore efficient.