Question

A student looked up the \(\Delta G_{\mathrm{i}}^{\circ}, \Delta H_{\mathrm{f}}^{\circ}\), and \(\Delta S^{\circ}\) values for \(\mathrm{CO}_{2}\) in Appendix 2. Plugging these values into Equation \(18.10,\) the student found that \(\Delta G_{\mathrm{f}}^{\circ} \neq \Delta H_{\mathrm{i}}^{\circ}-\) \(T \Delta S^{\circ}\) at \(298 \mathrm{~K}\). What is wrong with this approach?

Short answer

The approach may have used incorrect terms, units, or made calculation errors.

Step-by-step solution

Understand the Gibbs Free Energy Equation

The correct equation is \( \Delta G = \Delta H - T \Delta S \). This is used to calculate the Gibbs free energy change (\( \Delta G \)) from the enthalpy change (\( \Delta H \)) and the entropy change (\( \Delta S \)) at a temperature \( T \). Ensure that all values are consistent with the equation.

Check the Meaning of Terms

Ensure that \( \Delta G_{\text{f}}^{\circ}, \Delta H_{\text{f}}^{\circ} \), and \( \Delta H_{\text{i}}^{\circ} \) are interpreted correctly. \( \Delta G_{\text{f}}^{\circ} \) and \( \Delta H_{\text{f}}^{\circ} \) are for the formation, while \( \Delta H_{\text{i}}^{\circ} \) must be checked if it should also be \( \Delta H_{\text{f}}^{\circ} \).

Verify Temperature Consistency

Ensure that the temperature \( T \) used in the equation is accurate. For calculations at standard conditions, \( T = 298 \text{ K} \) is typically used. Double-check if all the values in the standard state were taken at this temperature.

Analyze Units

Check if all values were given and calculated in consistent units (usually joules or kilojoules). Inconsistent units can lead to errors in calculating \( \Delta G \).

Examine Calculation Mistakes

Re-calculate \( \Delta G \) using the equation \( \Delta G = \Delta H - T \Delta S \) with the given values, making sure no calculation mistakes are made or incorrectly plugged values are used.

Key concepts

enthalpy change

Understanding enthalpy change is crucial when you study the Gibbs Free Energy equation. Enthalpy, symbolized as \( \Delta H \), represents the heat content change of a system during a reaction. This change involves the absorption or release of heat and is relevant in chemical processes. If a reaction releases heat, \( \Delta H \) will be negative, indicating an exothermic reaction. Conversely, if it absorbs heat, \( \Delta H \) is positive and the reaction is endothermic. In the context of calculating Gibbs Free Energy, you need accurate \( \Delta H \) values to ensure the calculation is sound. These values, whether they are for individual reactants or overall reactions, are usually gathered from standard tables. Remember, ensuring that \( \Delta H \) values align with other related measurements is critical. Misused or misunderstood enthalpy changes can result in incorrect Gibbs Free Energy calculations.

entropy change

Entropy change, denoted by \( \Delta S \), is a measure of disorder or randomness in a system. In chemical terms, it's the extent of energy distribution in a system and plays a crucial role in predicting the spontaneity of a reaction.A positive \( \Delta S \) suggests increased disorder, such as when a solid dissolves into a liquid, while a negative \( \Delta S \) indicates decreased disorder, often observed when gas molecules condense into a liquid. For accurate calculations using the Gibbs Free Energy equation, ensure that the \( \Delta S \) values reflect standard conditions, just like \( \Delta H \). Standard values typically indicate these changes without needing additional context from experimental conditions. Being careful with the units and signs of \( \Delta S \) helps prevent errors, leading to trustworthy results when applying the $$ \Delta G = \Delta H - T \Delta S $$ equation.

temperature consistency

Temperature plays a significant role in the Gibbs Free Energy calculation. The temperature \( T \), measured in Kelvin, directly influences \( \Delta G \) by scaling the entropy change term \( T \Delta S \). When calculations are carried out, it's vital to use consistent temperature values.Since standard conditions typically employ a temperature of 298 K (25°C), you must ensure the values you are using match this reference point or correctly adjust them if not. Ignoring temperature consistency can lead to significant discrepancies in your results.When your calculations don't seem to work out, it's helpful to double-check that all your enthalpy and entropy inputs correspond to the same temperature. Ensuring this consistency provides a reliable basis for your subsequent calculations and interpretations.

calculation errors

Calculation errors are common pitfalls when dealing with thermodynamic equations like Gibbs Free Energy. These errors can emerge from various sources, often leading to confusing or incorrect results. The key to minimizing them is diligence and thoroughness.

• Double-check each term in the equation. Verify that each value, such as \( \Delta H \) and \( \Delta S \), is appropriately entered and aligns with the corresponding temperature \( T \).
• Pay attention to units, especially when \( \Delta H \) and \( \Delta S \) are expressed in different units (e.g., kilojoules vs. joules). Convert all measurements to a common unit before proceeding.
• Review mathematical operations. Simple arithmetic or algebra mistakes can result in significant issues. Carefully recompute each step if necessary.

A meticulous approach to managing these factors will prevent errors, leading to precise calculations and a better understanding of the involved chemical reactions.