What Is Entropy? Definition and Examples

Entropy is a key concept in physics and chemistry, with application in other disciplines, including cosmology, biology, and economics. In physics, it is part of thermodynamics. In chemistry, it is part of physical chemistry. Here is the entropy definition, a look at some important formulas, and examples of entropy.

Key Takeaways: Entropy

• Entropy is a measure of the number of possible microscopic arrangements (microstates) of a system’s macroscopic state (often described as randomness or disorder).
• Its symbol is the capital letter S. Typical units are joules per kelvin (J/K).
• Change in entropy can have a positive (more disordered) or negative (less disordered) value.
• In the natural world, entropy tends to increase. According to the second law of thermodynamics, the entropy of a system only decreases if the entropy of another system increases.

Entropy Definition

Entropy is a measure of the number of possible microscopic arrangements (microstates) that correspond to a system’s macroscopic state. It is often described informally as a measure of disorder, but a more precise definition is that it it quantifies the system’s multiplicity (the number of microstates) or probability of a given arrangement.

An ordered system has low entropy, while a system with many possible configurations has high entropy. In certain thermodynamic contexts, entropy is related to the portion of a system’s internal energy that is unavailable to do work, but this interpretation depends on conditions such as constant temperature and volume or pressure.

Entropy is an extensive property, meaning it depends on the amount of matter present. In equations, entropy is denoted S and its SI unit is the joule per kelvin (J⋅K⁻¹ or kg⋅m²⋅s⁻²⋅K⁻¹).

Microstates and Probability

In statistical mechanics, entropy links directly to probability. The Boltzmann equation,

S = k_Bln⁡W

relates entropy S to the number of microstates W available to a system, with k_B being the Boltzmann constant. Systems naturally evolve toward macrostates with more microstates because these are statistically more probable. This statistical perspective explains why entropy tends to increase over time in isolated systems.

Examples of Entropy

Here are several examples of entropy:

• Clean room vs. messy room – A tidy, organized room has low entropy. A messy room has high entropy. Restoring order requires input of energy, and without intervention, disorder tends to increase.
• Dissolving solids in liquids – Dissolving sugar in coffee increases entropy because ordered solid crystals become dispersed molecules in solution.
• Diffusion and osmosis – Molecules naturally spread from high to low concentration until evenly distributed. Spraying perfume in one corner of a room increases entropy as the scent molecules disperse throughout.
• Phase changes – Melting ice (solid → liquid) increases entropy as water molecules break out of the rigid lattice and move more freely. Boiling water (liquid → gas) increases entropy further as molecules spread far apart. Freezing water (liquid → solid) or condensing steam (gas → liquid) decreases entropy because molecular motion and available configurations are reduced.

Entropy Equation and Calculation

There are several entropy formulas:

Entropy of a Reversible Process

Calculating the entropy of a reversible process assumes that each configuration within the process is equally probable (which it may not actually be). Given equal probability of outcomes, entropy equals Boltzmann’s constant (k_B) multiplied by the natural logarithm of the number of microstates (W):

S = k_B ln W

Entropy of an Isothermal Process

For an isothermal process, the change in entropy (ΔS) equals the change in heat (ΔQ) divided by the absolute temperature (T):

ΔS = ΔQ / T

Applying calculus, entropy is the integral of dQ/T from the initial state to final state, where Q is heat and T is the absolute (Kelvin) temperature of a system.

Entropy and Internal Energy

In physical chemistry and thermodynamics, one useful entropy formula relates entropy to the internal energy (U) of a system:

dU = T dS – p dV

Here, the change in internal energy dU equals absolute temperature T multiplied by the change in entropy minus external pressure p and the change in volume V.

Gibbs Free Energy Connection

Entropy plays a key role in determining whether a process is spontaneous. In Gibbs free energy,

ΔG = ΔH − TΔS

A positive ΔS makes a process more likely to occur spontaneously at higher temperatures, while a negative ΔS can make it nonspontaneous unless offset by a favorable ΔH.

Entropy and the Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system cannot decrease. An isolated system cannot exchange matter or energy with its surroundings. For example, a scattered pile of papers never spontaneously reorders itself into a neat stack.

However, the entropy of one part of a system can decrease if the entropy of another part increases by an equal or greater amount. Freezing liquid water into ice decreases the entropy of the water, but the phase change releases heat to the surroundings, increasing their entropy. This keeps the total entropy of the system plus surroundings from decreasing, so there is no violation of the second law.

Entropy and Time

Physicists and cosmologists often call entropy “the arrow of time” because matter in isolated systems tends to move from order to disorder. When you look at the Universe as a whole, its entropy increases. Over time, ordered systems become more disordered and energy changes forms, ultimately getting lost as heat.

Entropy and Heat Death of the Universe

Some scientists predict the entropy of the universe eventually increases to the point useful work becomes impossible. When only thermal energy remains, the universe dies of heat death. However, other scientists dispute the heat death theory. An alternative theory views the universe as part of a larger system.

Entropy in Information Theory

Outside physics and chemistry, entropy describes uncertainty in information theory. Introduced by Claude Shannon, Shannon entropy quantifies the average information content or uncertainty in a message. The mathematics is analogous to thermodynamic entropy, with probability distributions replacing microstates. This cross-disciplinary link often helps students grasp that entropy is fundamentally about possibilities and likelihoods, not just “messiness.”

Common Misconceptions About Entropy

• “Entropy always increases” – This is true only for isolated systems. In open systems, entropy can locally decrease if compensated elsewhere.
• “Entropy equals disorder” – Disorder is a useful analogy, but entropy is better understood as a measure of the number of possible configurations.
• “High entropy means chaos” – High entropy systems can appear ordered at the macroscopic level but still have many microscopic arrangements.

Sources

• Atkins, Peter; Julio De Paula (2006). Physical Chemistry (8th ed.). Oxford University Press. ISBN 978-0-19-870072-2.
• Chang, Raymond (1998). Chemistry (6th ed.). New York: McGraw Hill. ISBN 978-0-07-115221-1.
• Clausius, Rudolf (1850). On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat. Poggendorff’s Annalen der Physick, LXXIX (Dover Reprint). ISBN 978-0-486-59065-3.
• Landsberg, P.T. (1984). “Can Entropy and “Order” Increase Together?”. Physics Letters. 102A (4): 171–173. doi:10.1016/0375-9601(84)90934-4
• Watson, J.R.; Carson, E.M. (May 2002). “Undergraduate students’ understandings of entropy and Gibbs free energy.” University Chemistry Education. 6 (1): 4. ISSN 1369-5614