Question

Hydrogen is produced commercially by the reaction of methane with steam: $$\mathrm{CH}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}(g)+3 \mathrm{H}_{2}(g)$$ a. Calculate \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) for this reaction (use the data in Appendix 4). b. What temperatures will favor product formation at standard conditions? Assume \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature.

Short answer

a. Using the values from Appendix 4 and the equations: $$\Delta H^{\circ} = \sum \Delta f H^{\circ}(\text{products}) - \sum \Delta f H^{\circ}(\text{reactants}),$$ and $$\Delta S^{\circ} = \sum S^{\circ}(\text{products}) - \sum S^{\circ}(\text{reactants}),$$ calculate the standard enthalpy change (∆H°) and the standard entropy change (∆S°) for the reaction. b. To find the temperature that favors product formation at standard conditions, use the Gibbs free energy equation: $$T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}}$$ Calculate the temperature using the obtained values for ∆H° and ∆S°. Product formation is favored at temperatures above this point.

Step-by-step solution

Calculate the standard enthalpy change (∆H°) for the reaction

Refer to Appendix 4 for the standard enthalpies of formation (∆fH°) values for each of the species in the reaction: $$\mathrm{CH}_{4}(g) + \mathrm{H}_{2}\mathrm{O}(g) \rightleftharpoons \mathrm{CO}(g) + 3 \mathrm{H}_{2}(g)$$ Using the given values from Appendix 4, calculate the standard enthalpy change for the reaction (∆H°) using the following equation: $$\Delta H^{\circ} = \sum \Delta f H^{\circ}(\text{products}) - \sum \Delta f H^{\circ}(\text{reactants})$$

Calculate the standard entropy change (∆S°) for the reaction

Refer to Appendix 4 for the standard molar entropies (S°) values for each of the species in the reaction: $$\mathrm{CH}_{4}(g) + \mathrm{H}_{2}\mathrm{O}(g) \rightleftharpoons \mathrm{CO}(g) + 3 \mathrm{H}_{2}(g)$$ Using the given values from Appendix 4, calculate the standard entropy change for the reaction (∆S°) using the following equation: $$\Delta S^{\circ} = \sum S^{\circ}(\text{products}) - \sum S^{\circ}(\text{reactants})$$

Determine the temperatures that favor product formation

Using the Gibbs free energy equation, ∆G° = ∆H° - T∆S°, and the calculated values of ∆H° and ∆S° from steps 1 and 2, determine the temperature at which product formation is favored. At this temperature, ∆G° = 0. Rearrange the Gibbs free energy equation to solve for the temperature, T: $$T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}}$$ Calculate the temperature with the obtained values for ∆H° and ∆S°. This temperature is the point where the reaction becomes spontaneous, and higher temperatures will favor product formation, while lower temperatures will favor reactant formation. Based on your answer, you can conclude the temperatures that favor product formation at standard conditions.

Key concepts

Standard Enthalpy Change (∆H°)

In thermodynamics, the standard enthalpy change, denoted by \( \Delta H^{\circ} \), is a critical concept when studying chemical reactions. Enthalpy itself is a measure of heat change during a chemical process. The term "standard" implies that the measurements are made under standard conditions, which are 1 atm pressure and a temperature of 298.15 K (25°C).
To calculate \( \Delta H^{\circ} \) for a given reaction, you need to look up the standard enthalpies of formation of each reactant and product. This data is commonly found in scientific appendixes or tables.
Use this formula for calculation:

• \( \Delta H^{\circ} = \sum \Delta f H^{\circ}(\text{products}) - \sum \Delta f H^{\circ}(\text{reactants}) \)

By summing up the standard enthalpies of the products and subtracting the sum of the standard enthalpies of the reactants, you can determine whether the reaction absorbs or releases heat. A negative \( \Delta H^{\circ} \) indicates an exothermic reaction (releases heat), whereas a positive value indicates an endothermic reaction (absorbs heat). Understanding this concept is foundational in predicting the energy profile of a reaction.

Standard Entropy Change (∆S°)

Entropy is a measure of disorder or randomness. The standard entropy change, \( \Delta S^{\circ} \), is used to understand how the disorder changes during a reaction under standard conditions.
To calculate \( \Delta S^{\circ} \), you first gather the standard molar entropies of the reactants and products from a data source. The equation you use is:

• \( \Delta S^{\circ} = \sum S^{\circ}(\text{products}) - \sum S^{\circ}(\text{reactants}) \)

When \( \Delta S^{\circ} \) is positive, the reaction leads to an increase in disorder, which often makes the reaction more likely to proceed. If it is negative, the reaction creates a more ordered system, which can sometimes require additional energy to maintain. Entropy changes play a significant role in determining the favorability of reactions along with enthalpy changes.

Gibbs Free Energy (∆G°)

Gibbs Free Energy, represented by \( \Delta G^{\circ} \), is crucial in predicting whether a chemical reaction is spontaneous. It combines both the enthalpy and entropy changes to give a single value that tells you if a reaction can occur on its own or needs outside energy.
To calculate \( \Delta G^{\circ} \), use the equation:

• \( \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \)

Here, \( T \) represents temperature in Kelvin. If \( \Delta G^{\circ} \) is negative, the reaction tends to be spontaneous, meaning it can proceed without any added energy. If it's positive, the reaction is non-spontaneous as it can't happen without external energy.
This energy aspect is vital for chemists and engineers who want to design reactions that can efficiently produce desired products. Understanding how \( \Delta G^{\circ} \) works helps predict and control reaction paths in industry and laboratory settings.

Temperature Dependence of Reaction

Temperature can significantly influence whether a reaction is favorable or unfavorable. The key to understanding this lies in how temperature interacts with enthalpy and entropy in the Gibbs Free Energy equation. The temperature at which \( \Delta G^{\circ} = 0 \) is a pivotal point, marking the boundary between spontaneity and non-spontaneity.
To find this temperature, set \( \Delta G^{\circ} \) to zero in the Gibbs equation and solve for \( T \):

• \( T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} \)

This calculation shows the temperature at which the reaction changes its nature regarding spontaneity. Above this temperature, the reaction becomes spontaneous, favoring product formation. Below it, the reaction leans towards the formation of reactants.
Temperature dependence is a crucial factor in chemical engineering and environmental studies, as it dictates how reactions behave in various conditions. By understanding this dependence, better process controls and conditions can be devised for industrial applications.