D22.1 Entropy and Microstates

Jia Zhou; John Moore; Etienne Garand

D22.1 Entropy and Microstates

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)	Δ_rH° = 25.69 kJ/mol
NaCl(s) → Na⁺(aq) + Cl⁻(aq)	Δ_rH° = 3.87 kJ/mol
NaOH(s) → Na⁺(aq) + OH⁻(aq)	Δ_rH° = −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.

The other student is incorrect. All of the reactions are spontaneous as written (that is, each substance dissolves in water) but two of the reactions are endothermic and one is exothermic. In an exothermic reaction energy is transferred to the surroundings; in an endothermic reaction energy is transferred to the system from the surroundings. Thus in an endothermic process, the system is at higher energy after the process occurs. In an exothermic process the system goes to a lower energy state, but whether a reaction is exothermic is not a sufficient condition for the reaction to be spontaneous. At least one other factor must be involved.

Deciding whether a process is spontaneous requires knowledge of the system’s enthalpy change, but that is not sufficient. It also requires knowledge of change in another property: entropy. (Similarly, deciding whether a process is product-favored requires knowledge of the system’s standard enthalpy change and standard entropy change.) The entropy change, ΔS, at constant temperature is defined as:

${\Delta}S = \dfrac{q_{\text{rev}}}{T}$

Here, q_rev is the heat transfer of energy for a reversible process, a theoretical process that takes place such that its direction can be changed (it can be “reversed”) by an infinitesimally small change in some condition. An example of a reversible process is melting water at 0 °C and 1 bar, where liquid water and ice are at equilibrium. Raising the temperature a tiny bit causes the ice to melt; lowering the temperature a tiny bit reverses the process, causing liquid water to freeze.

On a molecular scale, the entropy of a system can be related to the number of possible microstates (W). A microstate is a specific configuration of the energies of a system. The relationship is:

S = k_B·ln(W)

where k_B is the Boltzmann constant with a value of 1.38×10⁻²³ J/K (or 8.314 J/mol·K).

Similar to enthalpy, the change in entropy for a process is the difference between its final (S_f) and initial (S_i) values:

ΔS = S_f – S_i = k_B·ln(W_f) – k_B·ln(W_i) = $k_\text{B}\cdot\text{ln}\left(\dfrac{W_{\text{f}}}{W_{\text{i}}}\right)$

For processes involving an increase in the number of microstates, W_f > W_i, the entropy of the system increases, ΔS > 0. Conversely, processes that reduce the number of microstates, W_f < W_i, yield a decrease in system entropy, ΔS < 0.

Consider a simplistic case of a system comprised of four distinguishable particles distributed among two energetically-equivalent states, as illustrated in the figure below. The number of microstates possible for such a system is 2⁴ = 16. Microstates with equivalent particle arrangements (not considering individual particle identities) are grouped together and are called distributions. The probability that a system exists as a given distribution is proportional to the number of microstates within the distribution. Because entropy increases logarithmically with the number of microstates, the most probable distribution is the one of greatest entropy.

**Figure: Microstates**. The sixteen microstates associated with placing four distinguishable particles in two states are shown. The microstates are collected into five distributions, (a)-(e), based on the numbers of particles in each of the two states.

For the system in the figure above, the most probable distribution is (c), with the probability of finding the system in this configuration being $\frac{6}{16}$ or 37.5%. The least probable configuration of the system is distributions (a) and (e), where each distribution has a probability of $\frac{1}{16}$ .

Activity: Entropy Change and Microstates

Consider another system as shown in the next figure. This system consists of two objects, AB and CD, and two units of energy (represented as “*”). Distribution (a) shows the three microstates possible for the initial state of the system, where both units of energy are contained within the hot AB object. If one energy unit is transferred, the result is distribution (b) consisting of four microstates. If both energy units are transferred, the result is distribution (c) consisting of three microstates.

Three rows labeled a, b, and c are shown and each contains rectangles with two sides where the left side is labeled, “A,” and “B,” and the right is labeled, “C,” and “D.” Row a has three rectangles where the first has a dot above and below the letter A, the second has a dot above the A and B, and the third which has a dot above and below the letter B. Row b has four rectangles; the first has a dot above A and C, the second has a dot above A and D, the third has a dot above B and C and the fourth has a dot above B and D. Row c has three rectangles; the first has a dot above and below the letter C, the second has a dot above C and D and the third has a dot above and below the letter D. — **Figure: Microstate model for heat transfer of energy.** This shows a microstate model describing the heat transfer of energy from a hot object to a cold object. (a) Before the heat transfer occurs, the AB object contains both units of energy and the distribution has three microstates. (b) If the heat transfer results in an even dispersal of energy (one energy unit transferred), a distribution of four microstates results. (c) If both energy units are transferred, the resulting distribution has three microstates.

Hence, we may describe this system by a total of ten microstates. The probability that there is no heat transfer of energy when the two objects are brought into contact (the system remains in distribution (a)) is 30%. It is much more likely for heat transfer to occur and yield either distribution (b) or (c), the combined probability being 70%. The most likely result, with 40% probability, is heat transfer to yield the uniform dispersal of energy represented by distribution (b).

This simple example supports the common observation that placing hot and cold objects in contact results in heat transfer that ultimately equalizes the objects’ temperatures, at which point thermal equilibrium has been achieved. Such a process is characterized by an increase in system entropy.

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Chem 109 Fall 2024 Copyright © by Jia Zhou; John Moore; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.