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. What assumptions must you make in arriving at this estimate? (b) Use a reference source, such as WebElements (www.webelements.com), to find the experimental melting and boiling points of \(\mathrm{I}_{2}\). (c) Which of the values in part (b) is closer to the value you obtained in part (a)? Can you explain why this is so?

Short answer

Based on the given solution, the estimated temperature at which the free-energy change for the transformation from I₂(s) to I₂(g) is zero can be calculated using the formula \(T = \frac{\Delta H_{1}}{\Delta S_{1}}\). The assumptions made include that the transformation occurs in a single step and follows the given formula for free-energy change, and that ΔH and ΔS are constant over the range of temperature considered. Upon comparing the calculated temperature with the experimental melting and boiling points of iodine, the temperature at which the free-energy change is zero is expected to be closer to the boiling point, as this is when the solid phase transitions to the gaseous phase, reaching an equilibrium between the phases.

Step-by-step solution

Understand the relationship between free energy change and temperature

First, it is important to note that the free-energy change for a transformation between solid and gaseous states of a substance is related to temperature. The relationship is given by the formula: \[ \Delta G = \Delta H - T \Delta S \] Where ΔG denotes the change in free energy, ΔH represents the change in enthalpy, T is the temperature, and ΔS stands for the change in entropy. We are looking for the temperature at which the free-energy change for the transformation from I₂(s) to I₂(g) is zero. To find this temperature, we'll set ΔG to 0 and solve for T: \[ 0 = \Delta H - T \Delta S \]

Obtain the values of ΔH and ΔS from Appendix C

We are given that the data for enthalpy change (ΔH) and entropy change (ΔS) can be found in Appendix C. Look for the values of ΔH and ΔS for the transformation from solid iodine to gaseous iodine, and denote them as ΔH₁ and ΔS₁.

Estimate the temperature at which the free-energy change is zero

Using the values of ΔH₁ and ΔS₁ obtained from Appendix C, we can now plug them into the formula established in Step 1 and solve for the temperature T: \[ 0 = \Delta H_{1} - T \Delta S_{1} \] \[ T = \frac{\Delta H_{1}}{\Delta S_{1}} \] Calculate the temperature T using the values of ΔH₁ and ΔS₁.

Identify the assumptions made

To arrive at this estimate, we assumed that the transformation from solid iodine to gaseous iodine occurs in a single step and follows the given formula for free-energy change. We have also assumed that ΔH and ΔS are constant over the range of temperature considered.

Find the experimental melting and boiling points of iodine

Use a reference source, such as WebElements (www.webelements.com), to find the experimental melting and boiling points of iodine. Note these values.

Compare the estimated temperature with experimental values

Identify which of the experimental values (melting point or boiling point) is closest to the temperature obtained in Step 3.

Explain the result

The temperature at which the free-energy change is zero corresponds to an equilibrium between the solid and gaseous phases of iodine. This equilibrium is reached when the solid phase is in the process of transitioning to the gaseous phase. Hence, we would expect the temperature found in Step 3 to be closer to the boiling point, since boiling involves a phase transition from a solid or liquid to a gas.

Key concepts

Enthalpy Change

Enthalpy change, often represented by \( \Delta H \), is a crucial concept in chemistry when we discuss energy changes in systems. Enthalpy is a measure of the total energy of a thermodynamic system, including internal energy and energy required for a system to make room for itself via pressure and volume environments. When a chemical reaction or phase change occurs, the difference between the enthalpy of the products and the enthalpy of the reactants gives us the enthalpy change.

• A positive \( \Delta H \) means the process is endothermic - the system absorbs energy from the surroundings, like in melting or boiling.
• A negative \( \Delta H \) indicates an exothermic process - releasing energy, such as freezing or condensing.

In the problem with iodine transitioning from solid \( \mathrm{I}_2(s) \) to gaseous \( \mathrm{I}_2(g) \), we find the enthalpy change by obtaining values of enthalpy in standard conditions from resources like Appendix C in a textbook. Then, these values are applied in the formula \( \Delta G = \Delta H - T \Delta S \) to determine the temperature where the phase change renders the free energy zero.

Entropy Change

Entropy change, symbolized as \( \Delta S \), refers to the measure of disorder or randomness in a system. When a system undergoes a transformation, \( \Delta S \) quantifies how the level of disorder alters.

• A process where entropy increases \( \Delta S > 0 \) implies higher disorganization, typical in transitions from solid to liquid or gas.
• If entropy decreases \( \Delta S < 0 \), there is less disorder, such as gas condensing into a liquid.

For iodine's phase transition from solid to gas, entropy change indicates increased disorder because gaseous iodine molecules are more randomly distributed than in its solid form. Using Appendix C data, we can derive the entropy change and install it into the free energy formula to uncover the temperature where \( \Delta G = 0 \), pointing to a critical phase change point. It is paramount to assume that \( \Delta S \) remains constant despite temperature variation to simplify calculations.

Phase Transition

Phase transition is an essential phenomenon marking the change of substance states, such as solid, liquid, and gas, due to temperature alterations. During a phase transition, certain properties of matter change, often abruptly, like density or specific heat capacity. Understanding these transitions can illuminate pivotal concepts in thermodynamics and physical chemistry.

• In boiling, like reading \( \mathrm{I}_2(s) \) to \( \mathrm{I}_2(g) \), the disruption of molecular bonds leads to a new energy state with increased entropy.
• Phase transition points, like melting and boiling points, are crucial. They specify conditions under which a substance's distinct phases coexist in balance.

To find when iodine's solid or gaseous phases equilibrate, you assess the precise point when free energy shift \( \Delta G \) equals zero: steadying the energy input and entropy change product. This equilibrium typically aligns closer to the boiling point, as the material transforms from a more ordered to a less ordered state.