Second Law of Thermodynamics: Definition, Equation & Examples

A sand castle on the beach slowly crumbles as the day moves on. But someone who witnesses the reverse – sand spontaneously jumping into the shape of a castle – would say they must be watching a recording, not reality. Similarly, a glass of iced tea in which the cubes melt over time matches our expectations, but not a glass of liquid in which ice cubes spontaneously form.

The reason that some natural processes seem to make sense happening forward in time but not backwards in time has to do with the second law of thermodynamics. This important law is the only physical description of the universe that depends on time having a particular direction, in which we can only move forwards.

In contrast, Newton's laws or the kinematics equations, both used to describe the motion of objects, work equally well whether a physicist decides to analyze a football's arc as it moves forward or in the reverse. This is why the second law of thermodynamics is sometimes also referred to as "the arrow of time."

Microstates and Macrostates

Statistical mechanics is the branch of physics that relates microscopic-scale behavior, such as the motion of air molecules in a closed room, to subsequent macroscopic observations, such as the room's overall temperature. In other words, connecting what a human could directly observe to the myriad invisible spontaneous processes that together make it happen.

A microstate is one possible arrangement and energy distribution of all of the molecules in a closed thermodynamic system. For example, a microstate could describe the location and kinetic energy of each sugar and water molecule inside a thermos of hot chocolate.

A macrostate, on the other hand, is the set of all possible microstates of a system: all the possible ways the sugar and water molecules inside the thermos could be arranged. The way a physicist describes a macrostate is by using variables such as temperature, pressure and volume.

This is necessary because the number of possible microstates in a given macrostate is far too large to deal with. A room at 30 degrees Celsius is a useful measurement, though knowing it is 30 degrees does not reveal the specific properties of each air molecule in the room.

Although macrostates are generally used when talking about thermodynamics, understanding microstates is relevant since they describe the underlying physical mechanisms that lead to those larger measurements.

What Is Entropy?

Entropy is often described in words as a measure of the amount of disorder in a system. This definition was first proposed by Ludwig Boltzmann in 1877.

In terms of thermodynamics, it can be defined more specifically as the amount of thermal energy in a closed system that is not available to do useful work.

The transformation of useful energy to thermal energy is an irreversible process. Because of this, it follows that the total amount of entropy in a closed system – including the universe as a whole – can only ​increase​.

This concept explains how entropy relates to the direction that time flows. If physicists were able to take several snapshots of a closed system with the data on how much entropy was in each one, they could put them in time order following "the arrow of time" – going from less to more entropy.

To get much more technical, mathematically, the entropy of a system is defined by the following formula, which Boltzmann also came up with:


where ​Y​ is the number of microstates in the system (the number of ways the system can be ordered), ​k​ is the Boltzmann constant (found by dividing the ideal gas constant by Avogadro's constant: 1.380649 × 10−23 J/K) and ​ln​ is the natural logarithm (a logarithm to the base ​e​).

The main takeaway from this formula is to show that, as the number of microstates, or ways of ordering a system, increases, so does its entropy.

The change in entropy of a system as it moves from one macrostate to another can be described in terms of the macrostate variables heat and time:

\Delta S = \int \dfrac {dQ}{T}

where ​T​ is temperature and ​Q​ is the heat transfer in a reversible process as the system moves between two states.

The Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of the universe or an isolated system never decreases. In thermodynamics, an isolated system is one in which neither heat nor matter can enter or exit the system's boundaries.

In other words, in any isolated system (including the universe), entropy change is always zero or positive. What this essentially means is that random thermodynamic processes tend to lead to more disorder than order.

An important emphasis falls on the ​tend to​ part of that description. Random processes ​could​ lead to more order than disorder without violating natural laws; it is just vastly less likely to happen.

For example, of all the microstates in which a randomly shuffled deck of cards could end up – 8.066 × 1067 – only one of those options is equal to the order they had in the original package. It ​could​ happen, but the odds are very, very small. On the whole, everything naturally tends towards disorder.

The Significance of the Second Law of Thermodynamics

Entropy can be thought of as a measure of disorder or the randomness of a system. The second law of thermodynamics states that it always stays the same or increases, but never decreases. This is a direct result of statistical mechanics, since the description depends not on the extremely rare instance where a deck of cards shuffles into perfect order, but on the overall tendency of a system to increase in disorder.

One simplified way of thinking about this concept is to consider that un-mixing two sets of objects takes more time and effort than mixing them up in the first place. Ask any parent of a toddler to verify; it's easier to make a big mess than to clean it up!

Plenty of other observations in the real world "make sense" to us happening in one way but not another because they follow the second law of thermodynamics:

  • Heat flows from objects at higher temperature to objects at lower temperature and not the other way around (ice cubes melt and hot coffee left out on the table gradually cools until it matches room temperature).
  • Abandoned buildings slowly crumble and don't rebuild themselves.
  • A ball rolling along the playground slows and eventually stops, as friction transforms its kinetic energy into unusable thermal energy.

The second law of thermodynamics is just another way to formally describe the concept of the arrow of time: Moving forward in time, the entropy change of the universe cannot be negative.

What About Non-Isolated Systems?

If order is only ever increasing, why does looking around the world seem to reveal plenty examples of ordered situations?

While entropy ​on the whole​ is always increasing, local ​decreases​ in entropy are possible within pockets of larger systems. For example, the human body is a very organized, ordered system – it even turns a messy soup into exquisite bones and other complex structures. However, to do that, the body takes in energy and creates waste as it interacts with its surroundings. So, even though the person doing all this might experience less entropy within their body at the end of an eating/building body parts/excreting wastes cycle, the ​total entropy of the system​ – the body plus everything around it – still ​increases​.

Similarly, a motivated kid might be able to clean their room, but they converted energy into heat during the process (think of their own sweat and the heat generated by friction between objects being moved around). They probably also threw out a lot of chaotic trash, possibly breaking pieces down in the process. Again, entropy increases overall in the zip code, even if that room ends up spic and span.

Heat Death of the Universe

On a large scale, the second law of thermodynamics predicts the eventual ​heat death​ of the universe. Not to be confused with a universe dying in fiery throes, the phrase more precisely refers to the idea that eventually all useful energy will be converted into thermal energy, or heat, since the irreversible process is happening nearly everywhere all the time. Moreover, all this heat will eventually reach a stable temperature, or thermal equilibrium, since nothing else will be happening to it.

A common misconception about the heat death of the universe is that it represents a time when there is no energy left in the universe. This is not the case! Rather, it describes a time when all the useful energy has been transformed to thermal energy that has all reached the same temperature, like a swimming pool filled with half hot and half cold water, then left outside all afternoon.

Other Laws of Thermodynamics

The second law may be the hottest (or at least the most emphasized) in introductory thermodynamics, but as the name implies, it's not the only one. The others are discussed in more detail in other articles on the site, but here's a brief outline of them:

The zeroth law of thermodynamics.​ So named because it underlies the other laws of thermodynamics, the zeroth law essentially describes what temperature is. It states that when two systems are each in thermal equilibrium with a third system, they must necessarily also be in thermal equilibrium with one another. In other words, all three systems must be the same temperature. James Clerk Maxwell described a main outcome of this law as "All heat is of the same kind."

The first law of thermodynamics.​ This law applies the conservation of energy to thermodynamics. It states that the change in internal energy for a system is equal to the difference between the heat added to the system and the work done by the system:

\Delta U=Q-W

Where ​U​ is energy, ​Q​ is heat and ​W​ is work, all typically measured in joules (though sometimes in Btus or calories).

The third law of thermodynamics.​ This law defines ​absolute zero​ in terms of entropy. It states that a perfect crystal has zero entropy when its temperature is absolute zero, or 0 Kelvins. The crystal must be perfectly arranged or else it would have some inherent disorder (entropy) in its structure. At this temperature, the molecules in the crystal have no motion (which would also be considered thermal energy, or entropy).

Note that when the universe reaches its final state of thermal equilibrium – its heat death – it will have reached a temperature ​higher​ than absolute zero.

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