Methanol (CH₃OH) is made industrially in two steps from CO and H₂. It is so cheap to make that it is being considered for use as a precursor to hydrocarbon fuels, such as methane (CH₄):
Step 1. CO(g) + 2 H₂(g) S CH₃OH(l) ΔS° = - 332 J/K
Step 2. CH₃OH(l) → CH₄(g) + 1/2 O₂(g) ΔS° = 162 J/K
(e) In what temperature range is step 1 spontaneous?

Verified step by step guidance

1. Identify the condition for spontaneity: A reaction is spontaneous when the change in Gibbs free energy (\( \Delta G \)) is negative. The relationship is given by \( \Delta G = \Delta H - T \Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.
2. For step 1, we are given \( \Delta S^\circ = -332 \text{ J/K} \). We need to find the temperature range where the reaction is spontaneous, i.e., where \( \Delta G < 0 \).
3. Rearrange the equation for \( \Delta G \) to find the temperature: \( T < \frac{\Delta H}{\Delta S} \). Since \( \Delta S \) is negative, \( \Delta H \) must also be negative for the reaction to be spontaneous at some temperature.
4. Assume \( \Delta H \) is known or can be estimated from standard enthalpies of formation. Substitute \( \Delta H \) and \( \Delta S \) into the inequality \( T < \frac{\Delta H}{\Delta S} \) to solve for the temperature range.
5. Interpret the result: The temperature range obtained will indicate the maximum temperature below which the reaction is spontaneous.

Key Concepts

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic potential that helps predict the spontaneity of a process at constant temperature and pressure. A reaction is spontaneous if the change in Gibbs Free Energy (ΔG) is negative. The relationship between ΔG, enthalpy (ΔH), and entropy (ΔS) is given by the equation ΔG = ΔH - TΔS, where T is the temperature in Kelvin. Understanding this concept is crucial for determining the conditions under which a reaction, such as the formation of methanol, will occur spontaneously.

Entropy (ΔS)

Entropy (ΔS) is a measure of the disorder or randomness in a system. In the context of chemical reactions, a negative ΔS indicates that the reaction leads to a decrease in disorder, while a positive ΔS suggests an increase in disorder. The entropy changes for the steps in the methanol production process are essential for evaluating the spontaneity of the reactions. A reaction with a significant negative ΔS may require lower temperatures to remain spontaneous, as indicated by the Gibbs Free Energy equation.

Temperature and Spontaneity

Temperature plays a critical role in determining the spontaneity of a reaction, particularly when considering the entropy change. According to the Gibbs Free Energy equation, as temperature increases, the TΔS term becomes more significant. For a reaction to be spontaneous, the negative contribution from ΔH must outweigh the positive contribution from TΔS. Therefore, analyzing the temperature range where the entropy changes are favorable is essential for understanding when step 1 of methanol production will be spontaneous.