Problem 75 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The entropy change in the fusion of one mole of a solid melting at \(300 \mathrm{~K}\) (latent heat of fusion, \(2930 \mathrm{~J} / \mathrm{mol}\) ) is (a) \(9.77 \mathrm{~J} / \mathrm{K}-\mathrm{mol}\) (b) \(10.73 \mathrm{~J} / \mathrm{K}-\mathrm{mol}\) (c) \(2930 \mathrm{~J} / \mathrm{K}-\mathrm{mol}\) (d) \(108.5 \mathrm{~J} / \mathrm{K}-\mathrm{mol}\)

Step-by-Step Solution

Answer

The entropy change in the fusion of one mole of the solid is (a) \(9.77 \mathrm{~J} / \mathrm{K}-\mathrm{mol}\).

Step 1: Understanding Entropy Change During Fusion

The entropy change (\(\triangle S\)) during the process of fusion is determined by the heat added or removed from the system (latent heat) and the temperature at which the change occurs, according to the formula \(\triangle S = \frac{Q}{T}\), where \(Q\) is the heat of fusion and \(T\) is the temperature in Kelvin.

Step 2: Calculate the Entropy Change

Here, the given latent heat of fusion \(Q\) is \(2930 \text{J/mol}\) and the melting occurs at \(T = 300 \text{K}\). To find the entropy change \(\triangle S\), plug in the values into the formula to get \(\triangle S = \frac{2930 \text{J/mol}}{300 \text{K}} = 9.77 \text{J/K-mol}\).

Key Concepts

Latent Heat of Fusion

Latent heat of fusion is a fundamental concept in physical chemistry and thermodynamics that describes the amount of energy in the form of heat that is needed to change a substance from the solid phase to the liquid phase without changing its temperature. It is an essential parameter when understanding phase transitions and dealing with the energy required to overcome the molecular forces holding the solid together.

For a given substance, the latent heat of fusion is constant at a particular temperature. It's measured in joules per mole (J/mol) and represents the entropic 'cost' of arranging the molecules into a less ordered state. This energy is not lost but converts the potential energy between particles into kinetic energy, enabling them to move more freely as the phase changes from solid to liquid. In the context of solving physical chemistry problems, knowing the latent heat of fusion is vital for calculating the entropy change during a melting process, as demonstrated in the given exercise.

Entropy in Thermodynamics

Entropy in thermodynamics is a measure of the disorder or randomness of particles in a system. It signifies the number of ways a system’s energy can be arranged, and how it influences the movement of heat and the performance of work. In essence, it's a key concept for understanding the second law of thermodynamics, which states that in an isolated system, entropy tends to increase over time, indicating a direction of spontaneous processes.

When a substance melts, entropy increases because the particles in the liquid state can arrange themselves in more ways than in a solid state, which corresponds to higher disorder. Calculating the change in entropy, symbolized as \(\Delta S\), is crucial for figuring out whether a process can occur spontaneously. The calculation involves using the formula \(\Delta S = \frac{Q}{T}\), where \(Q\) is the heat absorbed or released, in the case of fusion it's the latent heat, and \(T\) is the absolute temperature. This formula underpins many physical chemistry problems concerning energy changes and is illustrated in our exercise.

Physical Chemistry Problems

Physical chemistry problems, such as the one in our exercise, often involve understanding the principles of thermodynamics and applying them to calculate various parameters. The calculations are not just abstract figures; they represent real-world physical changes, such as melting ice or boiling water. Such problems test a student's ability to apply formulas, such as the entropy change formula, to different situations.

Mastering the solving of these problems requires a thorough understanding of concepts like latent heat, entropy, and other thermodynamic functions. One should approach these problems systematically, by first identifying the relevant concept, then understanding the physical process involved, and finally applying the correct formula with the given or derived values. As shown in the exercise, getting familiar with how to manipulate these values to solve for unknowns is a foundational skill in physical chemistry. An ability to do so can unlock a better understanding of the intricacies of energy and matter interactions at the molecular level.