Question

Arrange the following substances ( 1 mole each) in order of increasing entropy at \(25^{\circ} \mathrm{C}:\) (a) \(\mathrm{Ne}(g)\), (b) \(\mathrm{SO}_{2}(g),\) (c) \(\mathrm{Na}(s)\) (d) \(\mathrm{NaCl}(s)\) (e) \(\mathrm{H}_{2}(g)\). Give the reasons for your arrangement.

Short answer

The substances arranged in order of increasing entropy at \(25^{\circ} \mathrm{C}\) are \(\mathrm{Na}(s)\), \(\mathrm{NaCl}(s)\), \(\mathrm{Ne}(g)\), \(\mathrm{H}_{2}(g)\), and \(\mathrm{SO}_{2}(g)\). This is based on their phases of matter and level of molecular complexity.

Step-by-step solution

Order based on phase

Consider the phase of matter. Generally, gases have more entropy than solids because molecules in a gas can move freely and randomly fill their container, while molecules in a solid are fixed in a lattice. Therefore, placing \(\mathrm{Na}(s)\) and \(\mathrm{NaCl}(s)\) at the lower end of the scale, \(\mathrm{Ne}(g)\), \(\mathrm{SO}_{2}(g)\) and \(\mathrm{H}_{2}(g)\) go towards the higher end.

Order within the same phase

Among the gases, distinguish them based on molecular complexity. More complex molecules have more entropy due to the higher number of possible configurations or arrangements of their atoms. \(\mathrm{SO}_{2}(g)\) is a more complex molecule than \(\mathrm{H}_{2}(g)\) and \(\mathrm{Ne}(g)\) are, so it will have more entropy. Among the two remaining gases, \(\mathrm{H}_{2}(g)\) has two atoms while neon is a monatomic gas, so \(\mathrm{H}_{2}(g)\) will have slightly more entropy.

Final Order

The final order of substances in increasing entropy would therefore be: \(\mathrm{Na}(s)\) < \(\mathrm{NaCl}(s)\) < \(\mathrm{Ne}(g)\) < \(\mathrm{H}_{2}(g)\) < \(\mathrm{SO}_{2}(g)\)

Key concepts

Molecular Complexity

Understanding the concept of molecular complexity is crucial when exploring entropy, which is a measure of a system's disorder. Entropy can be thought of as the number of ways a system can be arranged. The more complex a molecule is, the greater its entropy, as there are more possible configurations for its atoms.

Consider the example of sulfur dioxide (\textbf{SO}\(_2\)), hydrogen (\textbf{H}\(_2\)), and neon (\textbf{Ne}). Each of these substances has a different level of molecular complexity. Sulfur dioxide, with its three atoms and multiple vibrational modes, has a higher molecular complexity compared to the simpler diatomic hydrogen or the monatomic neon.

As a result, \textbf{SO}\(_2\) has more ways to store energy within its structure, resulting in higher entropy. This is important when arranging substances in order of increasing entropy; the more complex the molecular structure, the higher the position on the scale.

States of Matter

When discussing entropy, it's essential to differentiate between the states of matter: solids, liquids, and gases. In the context of the exercise, we focus on solids and gases, where the behavior of particles differs greatly.

Molecules in a solid are closely packed and organized in a fixed lattice, which restricts their motion and limits the entropy. In contrast, gases are composed of molecules that move freely and randomly, which allows for a higher number of accessible microstates, resulting in greater entropy.

Therefore, when arranging substances by entropy, a solid such as sodium (\textbf{Na}) or sodium chloride (\textbf{NaCl}) would be placed at the lower end, while gases like neon (\textbf{Ne}), hydrogen (\textbf{H}\(_2\)), and sulfur dioxide (\textbf{SO}\(_2\)) rank higher, based on their ability to disperse and occupy a volume with less restriction.

Thermodynamics

The field of thermodynamics provides a framework for understanding energy transfer within systems, and entropy is a cornerstone concept within this domain. It thereby connects molecular complexity and the states of matter through the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.

This law ensures that systems tend to move towards a state of greater disorder or randomness, which is why, given the chance, substances transform from solids to gases. The exercise illustrates this fundamental thermodynamic principle by arranging substances in the order of increasing entropy.

In practice, not only the state of matter (solid, liquid, or gas) determines entropy but also temperature, pressure, and intrinsic molecular properties. These factors collectively dictate how energy is distributed within a system, shaping the system's entropy profile. Therefore, accurate predictions on entropy order require consideration of all these interrelated thermodynamic elements.