Problem 104 - Chemistry The Molecular Nature of Matter

Question

Methanol, a potential replacement for gasoline as an automotive fuel, can be made from \(\mathrm{H}_{2}\) and \(\mathrm{CO}\) by the reaction $$ \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}(g) $$ At \(500.0 \mathrm{~K},\) this reaction has \(K_{\mathrm{p}}=6.25 \times 10^{-3}\) . Calculate \(\Delta G_{500}^{\circ}\) for this reaction in units of kilojoules.

Step-by-Step Solution

Answer

\(\Delta G_{500}^\circ = 21.07 \text{ kJ/mol}\)

Step 1: Understand the relationship between Kp and ΔG°

Recall the equation that relates the standard change in Gibbs free energy \(\Delta G_{\text{standard}}^\circ\) at a given temperature T (in Kelvin) to the equilibrium constant Kp: \[ \Delta G_{\text{standard}}^\circ = -RT \ln(K_p) \] R is the universal gas constant and for this equation, it should be used in the unit of Joules per mole Kelvin (\(J/(mol\cdot K)\)).

Step 2: Convert the gas constant R to kilojoules

The value for the gas constant R in Joules per mole Kelvin is 8.314 J/(mol·K). To convert this value to kilojoules, we divide by 1000: \[ R = \frac{8.314 \text{ J}}{(\text{mol}\cdot\text{K})} \times \frac{1 \text{ kJ}}{1000 \text{ J}} = 0.008314 \text{ kJ}/(\text{mol}\cdot\text{K}) \]

Step 3: Calculate ΔG° using the given Kp and temperature

Substitute the given values for Kp and T into the equation for ΔG°, as well as the converted value for R: \[ \Delta G_{500}^\circ = -0.008314 \text{ kJ}/(\text{mol}\cdot\text{K}) \times 500.0 \text{ K} \times \ln(6.25 \times 10^{-3}) \] Calculate the value to find ΔG° in kilojoules.

Step 4: Perform the calculation

Performing the calculation with the natural logarithm gives the following: \[ \Delta G_{500}^\circ = -0.008314 \text{ kJ}/(\text{mol}\cdot\text{K}) \times 500.0 \text{ K} \times \ln(6.25 \times 10^{-3}) \] \[ \Delta G_{500}^\circ = -0.008314 \text{ kJ}/(\text{mol}\cdot\text{K}) \times 500.0 \text{ K} \times (-5.0707) \] \[ \Delta G_{500}^\circ = 0.008314 \text{ kJ}/(\text{mol}\cdot\text{K}) \times 500.0 \text{ K} \times 5.0707 \] \[ \Delta G_{500}^\circ = 21.07 \text{ kJ/mol} \] (Please note that the natural logarithm of 6.25 × 10^-3 should be calculated to provide the final value for ΔG°.)

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry, occurring when the rates of the forward and reverse reactions in a chemical process are equal, leading to a stable ratio of products and reactants. In this state, no net change in the concentration of the components occurs over time, but it's important to understand that reactions are still occurring on the molecular level. Equilibrium can be influenced by various factors, including changes in temperature, pressure, or concentration. For instance, the synthesis of methanol from hydrogen and carbon monoxide is an example of a reaction that can reach equilibrium under certain conditions.

Equilibrium Constant (Kp)

The equilibrium constant, expressed as Kp when dealing with gases, quantifies the ratio of the concentrations of the products to the reactants at equilibrium, raised to the power of their coefficients in the balanced chemical equation. The constant is specific to a particular reaction at a given temperature. In the methanol synthesis example, Kp is given as \( 6.25 \times 10^{-3} \) at 500 K, reflecting the position of equilibrium for this reaction under these conditions. A higher Kp value indicates a greater concentration of products at equilibrium, favoring the forward reaction.

Thermodynamics

Thermodynamics is a branch of chemistry that deals with the energy changes accompanying chemical and physical processes. It's built on several key concepts, including the internal energy of a system, enthalpy, entropy, and Gibbs free energy. In this context, the Gibbs free energy is of particular interest as it can predict the spontaneity of reactions under constant temperature and pressure; a negative change in Gibbs free energy indicates a spontaneous process. The synthesis of methanol in our example involves changes in Gibbs free energy that we can calculate using the equilibrium constant and temperature.

Reaction Quotient

The reaction quotient, denoted as Q, is a measure that tells us how far a system is from equilibrium. It is calculated using the same formula as the equilibrium constant, K, but for non-equilibrium conditions. When Q equals K, the system is at equilibrium. If Q is less than K, the reaction will proceed forward to reach equilibrium, whereas if Q is greater than K, the reaction will shift in the reverse direction to re-establish equilibrium. The concept of Q is central in determining the direction of a reaction's shift when there's a disturbance, as described by Le Châtelier's principle.

Gibbs Free Energy Equation

The Gibbs free energy equation, \( \Delta G = \Delta H - T\Delta S \), where G is Gibbs free energy, H is enthalpy, T is temperature, and S is entropy, defines the amount of work a system can do at constant pressure and temperature. Another useful form is the Gibbs free energy change for a reaction at standard conditions, helpful in our methanol example: \( \Delta G^\circ = -RT \ln(K_p) \. \), This equation relates the standard change in Gibbs free energy to the equilibrium constant, allowing us to calculate the Gibbs free energy change at a certain temperature using Kp. With the given Kp and temperature values for the methanol synthesis reaction, we use this relationship to calculate the change in Gibbs free energy, which is essential to understand whether the reaction is energetically favorable or not.