Problem 100 - Chemistry Matter and Change

Question

Calculate the temperature at which \(\Delta G_{\text { system }}=0\) if \(\Delta H_{\text { system }}=4.88 \mathrm{kJ}\) and \(\Delta S_{\text { system }}=55.2 \mathrm{J} / \mathrm{K}\)

Step-by-Step Solution

Answer

The temperature at which \( \Delta G = 0 \) is approximately 88.41 K.

Step 1: Understanding the Formula

The formula to calculate free energy change is given by \[ \Delta G = \Delta H - T \Delta S \]Here, \( \Delta G \) is the change in free energy, \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.

Step 2: Setting Free Energy Change to Zero

Since the problem states that \( \Delta G = 0 \), we set up the equation:\[ 0 = \Delta H - T \Delta S \] We need to find the temperature \( T \) at which this holds true.

Step 3: Rearranging the Equation

Rearrange the formula to solve for \( T \):\[ \Delta H = T \Delta S \] Then divide both sides by \( \Delta S \) to isolate \( T \):\[ T = \frac{\Delta H}{\Delta S} \]

Step 4: Converting Units

Before calculation, ensure units are consistent. Convert \( \Delta H \) from \( \text{kJ} \) to \( \text{J} \) since \( \Delta S \) is given in \( \text{J/K} \). Multiply \( 4.88 \text{kJ} \) by \( 1000 \) to get \( 4880 \text{J} \).

Step 5: Substituting Values into the Equation

Plug the values into the rearranged equation:\[ T = \frac{4880}{55.2} \]Now, perform the division to find the temperature.

Step 6: Calculating the Temperature

Perform the division \( 4880 \div 55.2 \) to find \( T \). This calculation reveals:\[ T \approx 88.41 \text{ K} \]

Key Concepts

Enthalpy

Enthalpy is a fundamental concept in chemistry, especially significant in thermodynamics. It represents the total heat content of a system under constant pressure. The symbol for enthalpy is \( \Delta H \), which provides insight into how energy is absorbed or released during a chemical reaction or physical transformation. When a system undergoes a change, the change in enthalpy can tell us how much heat is taken in or given off.

• If \( \Delta H \) is positive, the process is endothermic, meaning it absorbs heat.
• If \( \Delta H \) is negative, it is exothermic, indicating heat is released.

Understanding enthalpy helps predict whether a reaction will be spontaneous. In practical terms, knowing \( \Delta H \) allows chemists to balance equations and evaluate reaction feasibility. When performing calculations, it’s crucial to keep the units consistent, especially when combining it with changes in entropy or when calculating temperature.

Entropy

Entropy, denoted as \( \Delta S \), is a measure of disorder or randomness in a system. It's an essential concept in understanding the second law of thermodynamics, which states that the total entropy of a system and its surroundings always increases for a spontaneous process. In simpler terms, natural processes tend to move toward a state of higher entropy or disorder.

Entropy helps explain phenomena like why ice melts or why a gas expands to fill a container. In calculations, \( \Delta S \) is a key component in determining Gibbs Free Energy, where it impacts the spontaneity of a reaction.

• A positive \( \Delta S \) implies increased disorder, often a characteristic of processes such as melting or evaporation.
• A negative \( \Delta S \) suggests a decrease in randomness, such as in freezing or condensation.

When combined with enthalpy in Gibbs Free Energy calculations, knowing \( \Delta S \) allows you to determine how temperature influences whether a process is favorable or not.

Temperature Calculation

The calculation of temperature using Gibbs Free Energy involves understanding the relationship between enthalpy and entropy. This relationship is framed by the equation \( \Delta G = \Delta H - T \Delta S \). When solving for temperature at which \( \Delta G = 0 \), the equation simplifies to \( T = \frac{\Delta H}{\Delta S} \). This is because when the free energy change is zero, the system is at equilibrium, meaning no net change occurs.

To find the equilibrium temperature:

• Ensure all units are consistent. Convert \( \Delta H \) from kilojoules to joules if needed, as in this exercise, to match the units of \( \Delta S \) given in \( \text{J/K} \).
• Plug the values into the formula \( T = \frac{\Delta H}{\Delta S} \).
• Perform the division to find the temperature in Kelvin.

Through this process, you can determine the temperature at which a reaction reaches equilibrium, providing critical insights into how reaction conditions might be optimized in a laboratory setting or industrial process.