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This isn’t a webpage or lesson plan, at this point.

Right now it is just my online notes on entropy

Main ideas

two types of entropy

Rod Vance, on Physics.Stackexchange.com, writes:

There are two definitions of entropy, which physicists believe to be the same (modulo the dimensional Boltzman scaling constant) and a postulate of their sameness has so far yielded agreement between what is theoretically foretold and what is experimentally observed. There are theoretical grounds, namely most of the subject of statistical mechanics, for our believing them to be the same, but ultimately their sameness is an experimental observation

  1. (Boltzmann / Shannon): Given a thermodynamic system with a known macrostate, the entropy is the size of the document, in bits, you would need to write down to specify the system’s full quantum state. Otherwise put, it is proportional to the logarithm of the number of full quantum states that could prevail and be consistent with the observed macrostate. Yet another version: it is the (negative) conditional Shannon entropy (information content) of the maximum likelihood probability distribution of the system’s microstate conditioned on the knowledge of the prevailing macrostate;
  2. (Clausius / Carnot): Let a quantity δQδQ of heat be input to a system at temperature TT. Then the system’s entropy change is δQTδQT. This definition requires background, not the least what we mean by temperature; the well-definedness of entropy (i.e. that it is a function of state alone so that changes are independent of path between endpoint states) follows from the definition of temperature, which is made meaningful by the following steps in reasoning: (see my answer here for details). (1) Carnot’s theorem shows that all reversible heat engines working between the same two hot and cold reservoirs must work at the same efficiency, for an assertion otherwise leads to a contradiction of the postulate that heat cannot flow spontaneously from the cold to the hot reservoir. (2) Given this universality of reversible engines, we have a way to compare reservoirs: we take a “standard reservoir” and call its temperature unity, by definition. If we have a hotter reservoir, such that a reversible heat engine operating between the two yields TT units if work for every 1 unit of heat it dumps to the standard reservoir, then we call its temperature TT. If we have a colder reservoir and do the same (using the standard as the hot reservoir) and find that the engine yields TT units of work for every 1 dumped, we call its temperature T1T−1. It follows from these definitions alone that the quantity δQTδQT is an exact differential because badQT∫abdQT between positions aa and bb in phase space must be independent of path (otherwise one can violate the second law). So we have this new function of state “entropy” definied to increase by the exact differential dS=δQ/TdS=δQ/T when the a system reversibly absorbs heat δQδQ.

As stated at the outset, it is an experimental observation that these two definitions are the same; we do need a dimensional scaling constant to apply to t





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External links

Entropy Sites — A Guide. Frank L. Lambert, Professor Emeritus

Entropy (order and disorder) (Wikipedia)

What Is Entropy? By Johannes Koelman

What is entropy? Thermodynamics of chemical equilibrium

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

High School Chemistry

PS3.A and 3.B Definition and conservation of energy and energy transfer

HS-PS3-4b. Provide evidence from informational text or available data to illustrate that the transfer of energy during a chemical reaction in a closed system involves changes in energy dispersal (enthalpy change) and heat content (entropy change) while assuming the overall energy in the system is conserved.

Mass Science Curriculum 2006

6. States of Matter, Kinetic Molecular Theory, and Thermochemistry

Central Concepts: Gas particles move independently of each other and are far apart. The behavior of gas particles can be modeled by the kinetic molecular theory. In liquids and solids, unlike gases, particles are close to each other. The driving forces of chemical reactions are energy and entropy. The reorganization of atoms in chemical reactions results in the release or absorption of heat energy.

6. States of Matter, Kinetic Molecular Theory, and Thermochemistry

6.5 Recognize that there is a natural tendency for systems to move in a direction of disorder or randomness (entropy).

SAT Subject Test in Chemistry

Thermochemistry: Including conservation of energy, calorimetry and specific heats, enthalpy (heat) changes associated with phase changes and chemical reactions, heating and cooling curves, entropy.

AP Chemistry

5.E: Chemical or physical processes are driven by a decrease in enthalpy or an increase in entropy, or both.


5.A.1: Temperature is a measure of the average kinetic energy of atoms and molecules.

5.E: One of the most powerful applications of thermodynamic principles is the ability to determine whether a process corresponding to a physical or chemical change will lie toward the reactant or product side when the process reaches a steady equilibrium state. The standard change in Gibbs free energy, ΔG° = ΔH° – TΔS°, is used to make this determination. If ΔG° < 0, then products are favored at equilibrium, and the forward process is considered to be “thermodynamically favored.” Conversely, if ΔG° > 0, then reactants are favored at equilibrium, and the reverse process is considered to be “thermodynamically favored.” Both the enthalpy change (ΔH°) and the entropy change (ΔS°) are closely related to the structure and nature of the components of the system; for this reason, it is often possible to make qualitative determinations concerning the sign (and magnitude) of ΔG° without explicit calculation…. Importantly, in biochemical systems, some reactions that oppose the thermodynamically favored direction are driven by coupled reactions. Thus, a cell can use energy to create order (a direction that is not thermodynamically favored) via coupling with thermodynamically favored reactions….

5.E.1: Entropy is a measure of the dispersal of matter and energy.

5.E.1: a. Entropy may be understood in qualitative terms rather than formal statistical terms. Although this is not the most rigorous approach to entropy, the use of qualitative reasoning emphasizes that the goal is for students to be able to make predictions about the direction of entropy change, ΔS°, for many typical chemical and physical processes.

b. Entropy increases when matter is dispersed. The phase change from solid to liquid, or from liquid to gas, results in a dispersal of matter in the sense that the individual particles become more free to move, and generally occupy a larger volume. Another way in which entropy increases in this context is when the number of individual particles increases when a chemical reaction precedes whose stoichiometry results in a larger number of product species than reacting species. Also, for a gas, the entropy increases when there is an increase in volume (at constant temperature), and the gas molecules are able to move within a larger space.

c. Entropy increases when energy is dispersed. From KMT, we know that the distribution of kinetic energy among the particles of a gas broadens as the temperature increases. This is an increase in the dispersal of energy, as the total kinetic energy of the system becomes spread more broadly among all of the gas molecules. Thus, as temperature increases, the entropy increases.

5.E.2: a. For the purposes of thermodynamic analysis in this course, the enthalpy and the internal energy will not be distinguished.

b. The phrase “thermodynamically favored” means that products are favored at equilibrium (K > 1).

c. Historically, the term “spontaneous” has been used to describe processes for which ΔG° < 0. The phrase “thermodynamically favored” is used here to avoid misunderstanding and confusion that can occur because of the common connotation of the term “spontaneous,” which students may believe means “immediately” or “without cause.”

d. For many processes, students will be able to determine, either quantitatively or qualitatively, the signs of both ΔH° and ΔS° for a physical or chemical process. In those cases where ΔH° < 0 and ΔS° > 0, there is no need to calculate ΔG° in order to determine that the process is thermodynamically favored.

e. As noted below in 5.E.5, the fact that a process is thermodynamically favored does not mean that it will proceed at a measurable rate.

f. Any process in which both ΔH° > 0 and ΔS° < 0 are not thermodynamically favored, (ΔG° > 0) and the process must favor reactants at equilibrium (K < 1). Because the signs of ΔS° and ΔH° reverse when a chemical or physical process is reversed, this must be the case.