# Second Law of Thermodynamics

#### By Mark Ciotola

First published on May 16, 2019. Last updated on May 19, 2020.

## The Law

The Second Law of Thermodynamics states that the entropy on an isolated system can never decrease. More specifically,

$$\frac{dS}{dt} ≥ 0$$,

where $$S$$ is entropy and $$t$$ is time.

## Definition of Entropy

Our discussion of entropy shall pertain to physical entropyrather than information entropy. The two concepts are related but differ in ways that are beyond this discussion.

A rough rule of thumb is that entropy is what increases in real life systems when heat flows, pressures equalize or substances become more mixed. For example, when you let the air out of a tire or balloon, and you hear the hiss as the compressed air escapes, entropy is increasing.

With the advent of modern physics, the term entropy $$S$$ has a precise definition. It is Boltzmann’s constant $$k$$ multiplied by the log of multiplicity $$\Omega$$.

Boltzmann’s~constant: $$k = 1.381 x 10^{-23} m^2~kg~s^{-2}$$.

Entropy: $$S = k \ln{\Omega}$$.

### Multiplicity

Multiplicity is the number of ways a state can be produced in a given system. For example, in a system of three US pennies, a state comprising two heads can be comprised three ways:

Coin 1 Coin 2 Coin 3
H H T
H T H
T H H

Hence, the multiplicity of this system is three.

### A Few More Points

For entropy calculations, we are typically considering how the instances of energy can be arranged, but types of particles also have an effect.

It is important to note that the Second Law does not prohibit the entropy of a system from increasing.

## Ways Entropy Increases

There are commonly several situations where entropy increases. When heat flows result in no work, or less than the Carnot ideal, entropy increases. Many chemical reactions result in increases of entropy, such as when gasoline is combusted to propel an automobile. Entropy is increased when substances become more mixed even where no chemical reaction occurs, such as when helium and neon gasses become mixed together.

## Cosmological Perspective

The Second Law requires that the total entropy of the universe must increase over time.[1]Yet, the expansion of the universe result in decreasing, not increasing, mean entropy density of the universe. These two trends are not inconsistent. Total entropy of the universe is indeed increasing, but it is being spread out more quickly than it increases.[2]

Yet, locally, gravity pulls together matter and produces local regions of higher entropy such as stars and planets. So really, there are several contrasting trends. The total entropy of the universe increases. Yet as the universe expands, the mean entropy density decreases. Nevertheless, locally, gravity may result in local clumps of high entropy. Then, eventually the entropy of those clumps dissipates into the surrounding universe.

## Notes & References

[1]Such a trend extrapolated into the distant suggests that the universe will die a classic heat death, in which no work or life is possible.

[2]As long as this continues to be the case, reports of the universe’s impending heat death may be greatly exaggerated, or at least further off than once thought.