Problem 103 - Chemistry Matter and Change

Question

Calculate the temperature at which \(\Delta G_{\text { system }}=-34.7 \mathrm{kJ}\) if \(\Delta H_{\text { system }}=-28.8 \mathrm{kJ}\) and \(\Delta S_{\text { system }}=22.2 \mathrm{J} / \mathrm{K}\) .

Step-by-Step Solution

Answer

The temperature is approximately 265.77 K.

Step 1: Gibbs Free Energy Equation

Start by using the Gibbs free energy equation: \(\Delta G = \Delta H - T \Delta S\). We are given \(\Delta G = -34.7\, \text{kJ}\), \(\Delta H = -28.8\, \text{kJ}\), and \(\Delta S = 22.2\, \text{J/K}\), which needs conversion to kJ for consistency.

Step 2: Convert Entropy Units

Convert the entropy change from \(\text{J/K}\) to \(\text{kJ/K}\) by dividing by 1000: \(\Delta S = \frac{22.2}{1000} = 0.0222\,\text{kJ/K}\).

Step 3: Rearrange the Equation

The Gibbs free energy equation \(\Delta G = \Delta H - T \Delta S\) can be rearranged to solve for temperature: \(T = \frac{\Delta H - \Delta G}{\Delta S}\).

Step 4: Substitute the Values

Substitute the values into the rearranged equation: \[T = \frac{-28.8\, \text{kJ} - (-34.7\, \text{kJ})}{0.0222\, \text{kJ/K}} = \frac{5.9\, \text{kJ}}{0.0222\, \text{kJ/K}}\].

Step 5: Calculate Temperature

Calculate \(T\): \[T = \frac{5.9\, \text{kJ}}{0.0222\, \text{kJ/K}} \approx 265.77\, \text{K}\].

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that studies the relationships between heat, work, and energy. It revolves around the principles of energy conversion and how energy flows in and out of systems. Understanding these principles is crucial for calculating energy changes in reactions and processes.
In thermodynamics, the concept of a system and its surroundings is key. A system is the part of the universe we are interested in, while everything else is considered the surroundings. Thermodynamic calculations often involve determining how much energy is transferred between the system and its surroundings. This is where Gibbs Free Energy comes into play, as it helps predict whether a process will occur spontaneously under constant pressure and temperature.

• The First Law of Thermodynamics, also known as the Law of Energy Conservation, states that energy cannot be created or destroyed, only transformed.
• The Second Law of Thermodynamics introduces the concept of entropy, which is a measure of disorder or randomness in a system.

By understanding these laws, we can predict and calculate the feasibility and extent of chemical reactions and physical changes.

Enthalpy

Enthalpy, represented by the symbol \( \Delta H \), is a measure of the total heat content of a system. It reflects the amount of energy absorbed or released during a reaction at constant pressure. When a reaction occurs, enthalpy change tells us whether the process is exothermic (releases heat) or endothermic (absorbs heat).
An exothermic reaction has a negative \( \Delta H \), indicating that energy is released to the surroundings. Conversely, an endothermic reaction has a positive \( \Delta H \), meaning that energy is absorbed from the surroundings.

• An example of an exothermic process is combustion, where heat is released as a fuel burns.
• An example of an endothermic process is photosynthesis, where plants absorb sunlight to convert carbon dioxide and water into glucose and oxygen.

In our exercise, the given \( \Delta H \) is -28.8 kJ, identifying it as an exothermic reaction. This information is crucial as it plays into the Gibbs Free Energy calculation by providing the enthalpy term.

Entropy

Entropy, symbolized as \( \Delta S \), quantifies the degree of disorder or randomness in a system. It is a central concept in the second law of thermodynamics and tends to increase over time in natural processes, reflecting the tilt towards greater disorder.
Entropy changes can be positive or negative, depending on whether a process creates or reduces disorder. For instance:

• Melting ice into water increases disorder, therefore increasing entropy.
• Condensing steam into water decreases disorder, thus decreasing entropy.

In the provided exercise, \( \Delta S \) is given as 22.2 J/K. To align the units with other energy quantities, we converted it to kilojoules per Kelvin, yielding 0.0222 kJ/K. This conversion allows for consistent calculations involving the Gibbs Free Energy equation. Understanding the impact of entropy helps determine the spontaneity of reactions within thermodynamic systems.

Temperature Calculation

Temperature plays a vital role in determining how a reaction proceeds and whether it will occur spontaneously. The Gibbs Free Energy equation, \( \Delta G = \Delta H - T \Delta S \), allows us to calculate the temperature at which a particular reaction becomes thermodynamically favorable.
For our exercise, we were tasked with finding the temperature at which the Gibbs Free Energy change, \( \Delta G \), is -34.7 kJ. Knowing both the enthalpy change and the entropy change, we rearranged the Gibbs equation to solve for temperature:

• First, ensure the consistency of units across all terms before calculation.
• Next, plug in the values: \( T = \frac{\Delta H - \Delta G}{\Delta S} \).

This equation provides an important link between energy changes and temperature. By calculating \( T = \frac{5.9 \, \text{kJ}}{0.0222 \, \text{kJ/K}} \approx 265.77 \, \text{K} \), we find that at approximately 266 Kelvin, the conditions favor the system’s Gibbs Free Energy being -34.7 kJ, indicating spontaneous reaction under these specific conditions.