Problem 72 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Which of the following would be expected to have the largest entropy per mole? (a) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{~s})\) (b) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{I})\) (c) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{~g})\) (d) \(\mathrm{SO}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

\(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{~g})\) is expected to have the largest entropy per mole.

Step 1: Understand the Concept of Entropy

Entropy is a measure of the disorder or randomness in a system. The more ways the particles that make up a system can be arranged without changing the overall state of the system, the higher the entropy. Generally, the entropy increases from solids to liquids to gases, and it is higher for substances with larger, more complex molecules.

Step 2: Compare Physical States

Among the given options, the substances are in three different physical states: solid, liquid, and gas. We know gases have higher entropy than liquids and solids because the particles in a gas can move more freely and have a greater number of accessible microstates.

Step 3: Compare Molecular Complexity

For substances in the same physical state, we compare their molecular complexity. A more complex molecule will generally have higher entropy because it has more vibrational modes and can store more thermal energy.

Step 4: Identify the Substance with the Highest Entropy

Considering the physical states and molecular complexity, (a) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{~s})\) - solid, complex molecule(b) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{I})\) - liquid, complex molecule(c) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{~g})\) - gas, complex molecule(d) \(\mathrm{SO}_{2}(\mathrm{~g})\) - gas, less complex moleculeOption (c) is a gas and has a complex molecule, so it is expected to have the highest entropy per mole.

Key Concepts

Entropy and Molecular Complexity

When discussing the concept of entropy and molecular complexity, it's essential to realize that molecules are like puzzles with varying levels of difficulty. The more complex the molecule, the more ways it can be put together, or in scientific terms, the more microstates it can have. This versatility in arrangement at a molecular scale is a key indicator of higher entropy.

Imagine a LEGO set; a set with more pieces can be built in more ways than a set with fewer pieces. In chemical terms, larger molecules with complex atomic arrangements have many more vibrational, rotational, and translational movements available to them. Each of these movements represents a possible state the molecule can exist in, directly increasing the entropy of the system.

In chemistry, a bigger molecule doesn't always mean it's more complex. The type and arrangement of atoms, and the possible interactions they can have with one another, also factor into the complexity. The multitude of interactions and possible states contribute to a substance's overall entropy. Therefore, in comparing substances, a larger and structurally more intricate molecule will generally have higher entropy than a simpler molecule.

Entropy and Physical States

Diving into entropy and physical states takes us to the heart of how substances behave in different forms. Think of a substance like water: as ice, it's solid and orderly; as liquid water, it's more disordered, and as steam, it's quite chaotic. This order-to-chaos transition is a reflection of a substance's entropy varying with its physical state.

Solids are the strict librarians of the physical states; their particles are rigidly arranged and move the least, resulting in the lowest number of microstates and, subsequently, the lowest entropy. As we heat things up to the liquid state, the particles break free from their strict positions and meander around more freely, leading to an increase in entropy.

Gases are the wild party-goers, with particles zipping around in all directions, bumping into one another and the container walls. This high degree of freedom allows for the maximum number of microstates, making gases have the highest entropy among the physical states. When we discuss changes in entropy, the transition from solid to liquid to gas usually corresponds to an increase in entropy.

Microstates and Entropy

At the microscopic level, the concept of microstates and entropy is a profound one. Each microstate represents a unique way in which the components of a system, like atoms or molecules, can be arranged while still maintaining the overall macroscopic properties of the system. For instance, if you have a box of colored balls, every different arrangement of those balls would be a microstate.

Entropy is a measure of these microstates – the more microstates there are, the higher the entropy. It's almost like the number of ways you can mix up a deck of cards–more arrangements mean more disorder. In chemistry, as temperature increases, substances generally have more available energy, and their particles can access more microstates, leading to higher entropy.

In summary, entropy can be thought of as a quantitative expression of the number of microstates that a system can explore. With more microstates comes a greater potential for disorder and higher entropy, which is integral to understanding spontaneous processes and the second law of thermodynamics.