Problem 72 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Using data in Appendix \(C\), estimate the temperature at which the free- energy change for the transformation from \(\mathrm{I}_{2}(s)\) to \(\mathrm{I}_{2}(g)\) is zero. (b) Use a reference source, such as Web Elements (www.webelements.com), to find the experimental melting and boiling points of \(I_{2}\). (c) Which of the values in part (b) is closer to the value you obtained in part (a)?

Step-by-Step Solution

Answer

To find the temperature at which the free-energy change for the transformation from \(I_2(s)\) to \(I_2(g)\) is zero, we use the equation \(T = \frac{\Delta H}{\Delta S}\), where \(\Delta H\) is the enthalpy change and \(\Delta S\) is the entropy change. Using the data from Appendix C, calculate the temperature T. Next, consult a reference source like Web Elements to find the experimental melting and boiling points of iodine. Finally, compare the estimated temperature with the experimental values to determine which one is closer.

Step 1: Recall the Gibbs free energy change equation

Remember that the gibbs free energy change is given by: \[ \Delta G = \Delta H - T \Delta S \] Here, \(\Delta H\) is the enthalpy change and \(\Delta S\) is the entropy change.

Step 2: Set the Gibbs free energy change to zero

As the problem asks for the temperature where the free energy change is zero, we will set \(\Delta G\) to 0 and solve for T: \[ 0 = \Delta H - T \Delta S \]

Step 3: Solve for T

Rearrange the equation and solve for T: \[ T = \frac{\Delta H}{\Delta S} \]

Step 4: Use the data from Appendix C

Find the enthalpy change (\(\Delta H\)) and entropy change (\(\Delta S\)) for the transformation from solid iodine to gaseous iodine in Appendix C.

Step 5: Calculate the temperature

Plug the values of \(\Delta H\) and \(\Delta S\) into the equation: \[ T = \frac{\Delta H}{\Delta S} \] Calculate the temperature T. #b. Find the experimental melting and boiling points#

Step 1: Consult a reference source

Go to a reliable reference source, such as Web Elements (www.webelements.com), to find the experimental melting and boiling points of iodine (I2). #c. Compare the estimated temperature with the experimental values#

Step 1: Identify the closer value

Compare the estimated temperature obtained in part a with the experimental melting and boiling points found in part b to decide which one is closer.

Key Concepts

Enthalpy Change

Enthalpy change, denoted as \( \Delta H \), is a measure of the total heat content in a system. When a substance undergoes a phase change, such as moving from solid iodine \(I_2(s)\) to gaseous iodine \(I_2(g)\), it absorbs or releases heat. This change reflects the energy needed to break the intermolecular forces holding the solid together.
Understanding \( \Delta H \) is crucial because it tells us how much energy is involved in the transformation.

• In endothermic processes: \( \Delta H > 0 \). Heat is absorbed.
• In exothermic processes: \( \Delta H < 0 \). Heat is released.

When studying phase transitions, such as sublimation (solid to gas), we calculate the enthalpy change using data from resources like tables in appendices. This change helps us predict temperature changes and energy requirements.

Entropy Change

Entropy change, represented by \( \Delta S \), measures the disorder or randomness of a system. Unlike enthalpy, which deals with energy, entropy focuses on the distribution of energy states and molecular positions.
In the context of iodine transitioning from solid to gas, \( \Delta S \) is positive as the molecules move further apart, increasing disorder.
Here are some key points:

• Higher entropy means higher disorder and vice versa.
• A positive \( \Delta S \) generally favors the transformation to higher temperatures.

By calculating \( \Delta S \), we can understand how likely a transition is to occur spontaneously. This connects directly to the Gibbs Free Energy equation, where \( \Delta S \) along with \( \Delta H \) helps in determining the feasibility of a reaction or phase change.

Temperature Estimation

Estimating the temperature where a phase transition occurs involves setting the Gibbs Free Energy \( \Delta G \) to zero. The relationship is given by:\[ \Delta G = \Delta H - T \Delta S \]To find the temperature \( T \) at which the change is roughly spontaneous, we rearrange to:\[ T = \frac{\Delta H}{\Delta S} \]This formula calculates the temperature at which the free energy change is zero, predicting the transition accurately between solid iodine to gaseous iodine.
Key considerations when estimating temperature include:

• Accurate values of \( \Delta H \) and \( \Delta S \) should be used.
• Proper data resources (such as appendices or reliable references) enhance precision.

By solving for \( T \), we acquire the estimated temperature, making scientific predictions about material phase transitions feasible.

Phase Transition

A phase transition occurs when a substance changes from one state of matter to another, such as from solid to liquid, liquid to gas, or solid to gas. The process for iodine transitioning from solid \(I_2(s)\) to gas \(I_2(g)\) is known as sublimation.
This transition involves changes in both enthalpy and entropy. During this process, the molecules gain enough energy to overcome the lattice structure, leading to higher entropy and requiring significant energy (enthalpy) input.
Some essential details include:

• Phase transitions require specific conditions of temperature and pressure.
• The transitions are driven by thermal inputs and the intrinsic properties of the molecules involved.

Understanding phase transitions helps us make accurate predictions about the behavior of substances under varying conditions, important for both scientific research and industrial applications.