Question

For the majority of the compounds listed in Appendix C, the value of \(\Delta G_{f}^{\circ}\) is more positive (or less negative) than the value of \(\Delta H_{f}^{\circ}\). (a) Explain this observation, using \(\mathrm{NH}_{3}(g), \mathrm{CCl}_{4}(l)\), and \(\mathrm{KNO}_{3}(s)\) as examples. (b) \(\mathrm{An}\) exception to this observation is \(\mathrm{CO}(g)\). Explain the trend in the \(\Delta H_{f}^{\circ}\) and \(\Delta G_{f}^{\circ}\) values for this molecule.

Short answer

In general, the formation of compounds involves exothermic bond formation, making \(\Delta H_{f}^{\circ}\) negative. For most compounds, like ammonia, carbon tetrachloride, and potassium nitrate, \(\Delta G_{f}^{\circ}\) is more positive or less negative than \(\Delta H_{f}^{\circ}\) due to a negative entropy change, indicating a decrease in the disorder of the system. However, carbon monoxide is an exception to this trend, with \(\Delta G_{f}^{\circ}\) being less positive or more negative than \(\Delta H_{f}^{\circ}\) because its formation involves a positive entropy change, increasing the disorder of the system.

Step-by-step solution

General trend for compounds

In general, the formation of a compound involves bond formation, which is an exothermic process. This means \(\Delta H_{f}^{\circ}\) is negative. Now, in order to understand the relationship between \(\Delta G_{f}^{\circ}\) and \(\Delta H_{f}^{\circ}\), we should look at the equation: \[ \Delta G = \Delta H - T\Delta S \] Since \(\Delta H_{f}^{\circ}\) is negative, in order for \(\Delta G_{f}^{\circ}\) to be more positive or less negative, the entropy change must be positive. This means that the formation of the compound increases the disorder of the system. Now let's analyze the given examples to understand the reason behind this observation.

Example 1 - Ammonia (NH3)

In the formation reaction of ammonia (NH3), one nitrogen molecule (N2) reacts with three hydrogen molecules (H2) to form two ammonia molecules (NH3): \[ N_{2}(g) + 3H_{2}(g) \to 2NH_{3}(g) \] In this reaction, there are four moles of gas molecules on the reactant side and two moles of gas molecules on the product side. Therefore, the entropy change is negative, indicating a decrease in the disorder of the system. Since both the enthalpy change and entropy change are negative, the term \(T\Delta S\) in the equation for \(\Delta G\) will be positive, making \(\Delta G_{f}^{\circ}\) more positive (or less negative) than \(\Delta H_{f}^{\circ}\) for ammonia.

Example 2 - Carbon tetrachloride (CCl4)

In the formation reaction of carbon tetrachloride (CCl4), one carbon molecule (C) reacts with four chlorine molecules (Cl2) to form one carbon tetrachloride molecule (CCl4): \[ C(s) + 2Cl_{2}(g) \to CCl_{4}(l) \] In this reaction, there are two moles of gas molecules on the reactant side and no gas molecules on the product side, as CCl4 is a liquid. Therefore, the entropy change is again negative, indicating a decrease in the disorder of the system. As seen with ammonia, both enthalpy and entropy changes are negative, making \(\Delta G_{f}^{\circ}\) more positive (or less negative) than \(\Delta H_{f}^{\circ}\) for carbon tetrachloride.

Example 3 - Potassium nitrate (KNO3)

In the formation reaction of potassium nitrate (KNO3), potassium (K), nitrogen (N), and oxygen (O) atoms react in a solid state to form one potassium nitrate molecule (KNO3): \[ K(s) + \frac{1}{2}N_{2}(g) + \frac{3}{2}O_{2}(g) \to KNO_{3}(s) \] In this reaction, there are two moles of gas molecules on the reactant side and no gas molecules on the product side, as KNO3 is a solid. Therefore, the entropy change is again negative, indicating a decrease in the disorder of the system. As in the previous two examples, both enthalpy and entropy changes are negative, making \(\Delta G_{f}^{\circ}\) more positive (or less negative) than \(\Delta H_{f}^{\circ}\) for potassium nitrate.

Exception - Carbon monoxide (CO)

In the formation reaction of carbon monoxide (CO), one carbon molecule (C) reacts with half a mole of oxygen molecules (O2) to form one carbon monoxide molecule (CO): \[ C(s) + \frac{1}{2}O_{2}(g) \to CO(g) \] In this reaction, there is half a mole of gas molecules on the reactant side and one mole of gas molecules on the product side. Therefore, the entropy change is positive, indicating an increase in the disorder of the system. Since the enthalpy change is negative due to bond formation and the entropy change is positive, the term \(T\Delta S\) in the equation for \(\Delta G\) will be negative, making \(\Delta G_{f}^{\circ}\) less positive (or more negative) than \(\Delta H_{f}^{\circ}\) for carbon monoxide. This is the exception to the general trend observed for most compounds.

Key concepts

Entropy and Enthalpy

Understanding the interplay between entropy and enthalpy is crucial for grasping how chemical reactions occur. In simple terms, enthalpy, represented by \(\Delta H\), is a measure of the total energy of a system, often associated with heat absorbed or released during a chemical reaction. A negative \(\Delta H\) indicates that the reaction releases heat to the surroundings, thus being exothermic, while a positive one means it absorbs heat, which is endothermic.

Entropy, denoted as \(\Delta S\), quantifies the amount of disorder within a system. An increase in entropy reflects a state change towards more disorder or randomness. For most natural processes, entropy tends to increase, following the second law of thermodynamics. In chemical reactions, the change from more structured reactants to less structured products typical announces an increase in entropy.

However, it's not always straightforward. In the textbook examples, we find cases where forming a compound decreases entropy. So, despite sometimes contradicting our expectations, the relationship between enthalpy and entropy plays a decisive role in determining the spontaneity of a reaction through the Gibbs Free Energy equation.

Chemical Thermodynamics

Chemical thermodynamics involves studying the energy changes associated with chemical reactions. Gibbs Free Energy, abbreviated as \(\Delta G\), is a key concept here. It combines enthalpy and entropy to predict whether a reaction will occur spontaneously under constant temperature and pressure. The formula \[ \Delta G = \Delta H - T\Delta S \] integrates these variables, with \(T\) being the temperature in Kelvin.

The terms involved in this equation have unique implications. A negative \(\Delta G\) means the reaction is spontaneous; a positive value indicates it’s non-spontaneous. However, temperature can influence spontaneity. For instance, at higher temperatures, the \(T\Delta S\) term could be large enough to make \(\Delta G\) negative, even if \(\Delta H\) is positive. Understanding the interrelation of these variables equips students with predicting not just if but under what conditions a reaction may proceed.

Formation Reactions

A formation reaction is one where a substance is formed from its elements in their standard states. The standard enthalpy change of formation, \(\Delta H_{f}^{\circ}\), and the standard Gibbs Free Energy of formation, \(\Delta G_{f}^{\circ}\), are of particular interest because they provide insights into the stability and spontaneity of compounds when formed under standard conditions.

Interpreting Standard Formation Values

In analyzing \(\Delta H_{f}^{\circ}\) and \(\Delta G_{f}^{\circ}\), we observe that compounds with highly negative \(\Delta G_{f}^{\circ}\) values are typically more stable and likely to form spontaneously. Conversely, a more positive \(\Delta G_{f}^{\circ}\) suggests a less spontaneous process. Through the given examples, like ammonia or carbon tetrachloride, we can discern how changes in enthalpy and entropy during the formation dictate the \(\Delta G_{f}^{\circ}\) outcome, reinforcing the essence of understanding these values in the context of chemical reactions.

Standard States

The concept of standard states pertains to the reference state of a material used in thermodynamics to define properties such as enthalpy, entropy, and Gibbs Free Energy. For a pure substance, the standard state is its most stable form at 1 bar (approximately 1 atmosphere) of pressure and a specified temperature, often 25°C (298.15 K). The significance of standard states lies in their role in determining standard reaction enthalpies and Gibbs Free Energies.

Relevance in Calculations

We rely on the values of substances in their standard states to calculate enthalpy and Gibbs Free Energy changes during formation reactions. It provides a consistent basis for comparison. Any deviation from these conditions needs accounting for in calculations, which could lead to entirely different outcomes. The concept of standard states thus establishes a foundational baseline that guides chemists in predicting reaction behavior and stability of compounds.