Problem 99 - Chemistry Matter and Change

Question

Calculate \(\Delta G_{\text { system }}\) for each process, and state whether the process is spontaneous or nonspontaneous. a. \(\Delta H_{\text { system }}=145 \mathrm{kJ}, T=293 \mathrm{K}, \Delta S_{\mathrm{system}}=195 \mathrm{J} / \mathrm{K}\) b. \(\Delta H_{\text { system }}=-232 \mathrm{kJ}, T=273 \mathrm{K}, \Delta S_{\mathrm{system}}=138 \mathrm{J} / \mathrm{K}\) c. \(\Delta H_{\mathrm{system}}=-15.9 \mathrm{kJ}, T=373 \mathrm{K}, \Delta S_{\mathrm{system}}=-268 \mathrm{J} / \mathrm{K}\)

Step-by-Step Solution

Answer

Process (a) and (c) are nonspontaneous; process (b) is spontaneous.

Step 1: Write the Formula for Gibbs Free Energy Change

The Gibbs Free Energy Change (\(\Delta G\)) for a system can be determined using the formula: \(\Delta G = \Delta H - T \Delta S\), where \(\Delta G\) is in kJ, \(\Delta H\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the change in entropy.

Step 2: Convert Units for Entropy

Since \(\Delta S\) is given in J/K, it should be converted into kJ/K to be consistent with \(\Delta H\)'s units. Do this by dividing \(\Delta S\) by 1000. For example, \(195 \text{ J/K} = 0.195 \text{ kJ/K}\).

Step 3: Calculate \(\Delta G\) for Part (a)

Given \(\Delta H = 145 \text{ kJ}\), \(T = 293 \text{ K}\), and \(\Delta S = 0.195 \text{ kJ/K}\) (after conversion), apply the formula: \(\Delta G = 145 - 293 \times 0.195\). Calculate \(\Delta G = 145 - 57.135 = 87.865 \text{ kJ}\). This positive \(\Delta G\) indicates a nonspontaneous process.

Step 4: Calculate \(\Delta G\) for Part (b)

Given \(\Delta H = -232 \text{ kJ}\), \(T = 273 \text{ K}\), and \(\Delta S = 0.138 \text{ kJ/K}\), apply the formula: \(\Delta G = -232 - 273 \times 0.138\). Calculate \(\Delta G = -232 - 37.674 = -269.674 \text{ kJ}\). This negative \(\Delta G\) indicates a spontaneous process.

Step 5: Calculate \(\Delta G\) for Part (c)

Given \(\Delta H = -15.9 \text{ kJ}\), \(T = 373 \text{ K}\), and \(\Delta S = -0.268 \text{ kJ/K}\), apply the formula: \(\Delta G = -15.9 - 373 \times (-0.268)\). Calculate \(\Delta G = -15.9 + 99.964 = 84.064 \text{ kJ}\). This positive \(\Delta G\) indicates a nonspontaneous process.

Key Concepts

Enthalpy

Enthalpy is a measure of the heat content in a system at constant pressure. It is represented by the symbol \( \Delta H \) and is expressed in kilojoules (kJ). Understanding enthalpy is crucial to predicting the energy changes in a chemical reaction.

In essence, enthalpy tells us whether energy in the form of heat is absorbed or released during a chemical process.

• If \( \Delta H \) is positive, the process absorbs heat, referred to as an endothermic process.
• If \( \Delta H \) is negative, the process releases heat, known as an exothermic process.

Enthalpy is a vital component in calculating the Gibbs Free Energy change, which ultimately helps in determining if a process is spontaneous. A negative \( \Delta H \) generally favors spontaneity, but it's not the only factor to consider.

Entropy

Entropy, symbolized as \( \Delta S \), measures the disorder or randomness in a system. Expressed in joules per Kelvin (J/K), entropy reflects how dispersed the energy within a system is.

This concept can be a bit tricky, but an easy way to think about it is through examples. For instance, a messy room has higher entropy than a tidy one because the disorder is greater. Similarly, systems tend to move towards higher entropy or disorder over time.

• Positive \( \Delta S \) indicates an increase in disorder.
• Negative \( \Delta S \) reveals a decrease in disorder.

Entropy is essential when calculating Gibbs Free Energy because it is multiplied by the temperature to determine its influence on spontaneity. The relationship between entropy and Gibbs Free Energy is such that higher entropy at higher temperatures may make a process more spontaneous.

Spontaneous Processes

Spontaneous processes are those that occur naturally without any input of external energy. The spontaneity of a process is determined by the sign of the Gibbs Free Energy change, \( \Delta G \).

The formula \( \Delta G = \Delta H - T \Delta S \) helps in assessing this. Here,

• A negative \( \Delta G \) indicates a spontaneous process.
• A positive \( \Delta G \) suggests a nonspontaneous process.

Temperature plays a crucial role, as it can swing the outcome depending on the values of \( \Delta H \) and \( \Delta S \). For instance, a process might become spontaneous at higher temperatures due to the influence of the \( T \Delta S \) component.

Understanding these concepts is key to grasping why some reactions occur naturally and others require an energy input to proceed.