Problem 74 - Chemistry: The Central Science

Question

(a) Using data in Appendix C, estimate the temperature at which the free- energy change for the transformation from \(I_{2}(s)\) to \(I_{2}(g)\) is zero. What assumptions must you make in arriving at this estimate? (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)? Can you explain why this is so?

Step-by-Step Solution

Answer

The estimated temperature at which the free-energy change for the transformation from \(I_{2}(s)\) to \(I_{2}(g)\) is zero is approximately 308 K. We made assumptions that the enthalpy change (ΔH) and entropy change (ΔS) are constant, and that no side reactions occur. Experimental melting and boiling points of iodine are 386.85 K and 457.4 K, respectively. The melting point is closer to our estimated temperature because the changes in enthalpy and entropy are more significant during the transition between solid and liquid phases of \(I_{2}\).

Step 1: Understanding the problem

We need to find the temperature at which the free-energy change for the transformation from \(I_{2}(s)\) to \(I_{2}(g)\) is zero. We'll use the Gibbs-Helmholtz Equation to find the temperature, which is given by: \[ΔG = ΔH - TΔS\] Where ΔG is the change in Gibbs free energy, ΔH is the change in Enthalpy, T is the Temperature, and ΔS is the change in Entropy. The task is to find the temperature at which the free energy change is zero.

Step 2: Identify the assumptions

The assumptions we need to make for this estimate are: 1. The enthalpy change (ΔH) and entropy change (ΔS) are constant over the range of temperature in question. 2. Only the transformation between the mentioned phases (\(I_{2}(s)\) to \(I_{2}(g)\)) is considered; no side reactions take place.

Step 3: Calculate the temperature using given data

According to the given data in Appendix C, we have the values for ΔH and ΔS as follows: ΔH = 62.44 kJ/mol ΔS = 202.77 J/mol·K We first need to convert ΔH to J/mol: ΔH = 62440 J/mol Now, we can find the temperature (T) where ΔG is zero using the Gibbs-Helmholtz Equation: ΔG = ΔH - TΔS = 0 T = ΔH/ΔS T = 62440 J/mol / 202.77 J/mol·K

Step 4: Compute temperature T

Now, calculate the value of T: T ≈ 308 K So, the estimated temperature at which the free-energy change for the transformation from \(I_{2}(s)\) to \(I_{2}(g)\) is zero is approximately 308 K. (b)

Step 5: Use a reference to find experimental melting and boiling points of \(I_{2}\)

By referring to a trusted source like Web Elements (www.webelements.com), we find that the experimental values for melting and boiling points of iodine are as follows: Melting point = 386.85 K Boiling point = 457.4 K (c)

Step 6: Determine the closest value and justify your answer

Comparing the experimental values with our estimated temperature, we found that the melting point is closer to our estimated temperature at which the free energy change is zero. Melting point = 386.85 K (closer) Boiling point = 457.4 K The melting point is closer because the changes in enthalpy and entropy are more significant during the transition between solid and liquid phases of \(I_{2}\). Since the calculations are based on constant values of enthalpy and entropy, the results give a better approximation for phase transitions with larger changes in these quantities.

Key Concepts

Chemical Thermodynamics

Chemical thermodynamics involves the study of energy changes during chemical reactions. It revolves around the laws of thermodynamics, which describe how energy is transformed and conserved in systems. The central aspect of chemical thermodynamics is the Gibbs free-energy change, denoted by ΔG, which determines the spontaneity of a chemical process.

When ΔG is negative, the process can occur spontaneously. If ΔG is positive, the process is non-spontaneous and requires energy input to occur. At equilibrium, ΔG equals zero, indicating no net change in the system. The relationship of Gibbs free energy to enthalpy ( ΔH), entropy ( ΔS), and temperature (T) is crucial in predicting the behavior of chemical systems, and comes from the equation ΔG = ΔH - T ΔS.

In solving problems involving the calculation of temperature where the free-energy change is zero, such as the transition from solid iodine ( I_{2}(s)) to gaseous iodine ( I_{2}(g)), we apply this equation. Moreover, in chemical thermodynamics, it is essential to note the assumptions we make, such as constants for ΔH and ΔS over a temperature range, and the absence of side reactions, which simplifies complex real-world situations.

Phase Transition

Phase transitions are transformations from one state of matter to another, such as solid to liquid (melting) or liquid to gas (evaporation). In our example with iodine, understanding at what temperature the phase transition occurs allows for predicting when a solid will turn into a gas.

Significance of Phase Transition Temperature

Phase transition temperatures, such as melting and boiling points, mark where a substance changes its phase under standard atmospheric pressure. These temperatures are characteristic properties of substances and are used for their identification and description.

Estimation Through Gibbs Free Energy

By setting the Gibbs free-energy change to zero ( ΔG = 0), we can estimate the temperature at which the transition occurs. For iodine, comparing the estimated temperature using this method to the experimental melting and boiling points helps us understand which phase transition is better aligned with the calculated values.

Entropy and Enthalpy

Entropy ( ΔS) is a measure of disorder or randomness in a system. An increase in entropy generally means greater dispersal of energy and matter. Enthalpy ( ΔH), on the other hand, reflects the heat content of a system and is associated with the breaking and forming of chemical bonds.

Contributions to Phase Transitions

During phase transitions, such as the melting of solid iodine to liquid, there is an increase in both entropy and enthalpy. Entropy increases because the molecules move from an orderly solid state to the much freer liquid state, whereas the enthalpy increases because energy is absorbed to break the bonds keeping the solid intact.

By calculating these values in the context of Gibbs free energy, students can predict phase transition temperatures and understand the energy changes associated with such transitions. It's a way to quantify the nuanced interplay between the energetic and molecular complexity of chemical processes.