D22.1 Entropy
Activity: Entropy Change and Microstates
Atoms tend to spontaneously go to the lowest-energy state. For example, when an atom is excited by a high temperature, the excited state emits radiation and the atom returns to the ground state. Based on this observation, another student says that everything tends to go to the lowest energy state.
Consider these three thermochemical expressions for three substances that dissolve in water at 25 °C.
NH4NO3(s) ⟶ NH4+(aq) + NO3−(aq) | ΔrH° = 25.69 kJ/mol |
NaCl(s) ⟶ Na+(aq) + Cl−(aq) | ΔrH° = 3.87 kJ/mol |
NaOH(s) ⟶ Na+(aq) + OH−(aq) | ΔrH° = −44.50 kJ/mol |
Based on the data given here, write a brief explanation of whether you agree with the student.
Write in your notebook, then left-click here for an explanation.
Introduction to Entropy
To understand why chemical and physical processes occur, we must look beyond energy content alone. While changes in enthalpy (ΔH) tells us whether energy is absorbed or released, it does not fully explain why certain changes happen on their own even when energy seems to flow “against” intuition. A broader picture emerges when we consider how energy is distributed—that is, entropy.
Entropy is often misunderstood as “disorder,” a vague term that can obscure more than it reveals. In thermodynamics, entropy is better understood as a measure of energy dispersal—how energy is spread across the various motions and configurations available to particles in a system. For example, when ice melt to form liquid water, it isn’t becoming more chaotic, rather it is gaining access to more kinds of molecular motion (rotational, translational), and thus more ways to distribute energy.
A helpful analogy is a box of Lego bricks. Stack them into a tall tower and you’ve created a precise, low-entropy, structure: there are only a few ways to arrange them that result in a tall tower. On the other hand, scatter the same bricks loosely across the bottom of the box, and suddenly there are millions of ways to arrange them. This scattered arrangement has higher entropy because it corresponds to more possible configurations. Systems with higher entropy are more stable not because they’re “messier,” but because there are more ways to exist in those states—they’re simply more probable.
The same idea applies to molecular systems. In a solid, molecules are arranged in a highly ordered structure, like the stacked lego tower. In a gas, molecules are free to move in many ways, filling the available space, like the scattered bricks (in zero-g). The gas phase has a higher entropy because energy is more evenly spread across more configurations, and that makes it more thermodynamically stable. Systems in high-entropy states are already “settled in” among many possibilities and are less easily disrupted by small changes.
Molecular Motions and Energy Dispersal
To understand how energy spreads through a system, we need to look at the motions of its particles. Molecules can move in three fundamental ways: translation, rotation, and vibration.
- Translational motion involves whole molecules moving through space, which is the dominant form of motion in gases.
- Rotational motion involves molecules spinning around their axes.
- Vibrational motion is more localized: atoms within a molecule stretch and bend the bonds that connect them. Larger, more complex molecules have more vibrational modes than smaller or more rigid ones.
Not all motions are accessible in every state of matter. Solids mostly allow vibration; liquids additionally enable rotation and limited translation; gases offer the full suite of motions. As substances move from solid to liquid to gas, the number of accessible microstates, unique combinations of these motions, increases dramatically. That’s why entropy increases across phase changes, as we will see below.
Quantifying Entropy and Microstates
Entropy is represented by S. The standard molar entropy (S°) refers to the entropy of one mole of a substance at 1 bar of pressure and, if in solution, 1 M concentration. These values are measured experimentally and reflect how energy spreads through a system’s available motions. Consider water:
Substance | Physical State | S° (J/mol·K) |
H2O | Solid | 41 |
H2O | Liquid | 70 |
H2O | Gas | 189 |
From solid to liquid to gas, entropy increases due to expanding freedom of motion and more accessible configurations.
Boltzmann captured this idea with a famous equation:
S = kBlnW
Here, W represents the number of microstates—the distinct ways particles can be arranged and moving while still appearing macroscopically the same (e.g., same temperature, phase, etc.). More microstates mean more entropy. Boltzmann’s equation helps explain what entropy means, but it is not how chemists calculate entropy. For instance, 1 mole of ice at 0 °C has an S° of about 41 J/mol·K, which corresponds to approximately W ≈ 101,000,000,000,000,000,000,000,000 microstates, a number beyond comprehension. Instead of counting microstates, chemists use experimental data and tabulated S° values.
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