The Second Law of Thermodynamics made easy

The Second Law of Thermodynamics is one of the few scientific laws that has attained a status in wider culture, even featuring in rock tracks by Muse. Famously, C.P. Snow cited an understanding of the 2nd Law as something that every educated person should have.

The 2nd Law is often stated in technical language that makes its meaning hard to understand, but the basic principles are actually readily grasped. I was recently challenged to explain the 2nd Law at the level of a bright 13-year-old, and so here is my attempt.


The world is pretty complicated, so let’s make a simple “toy” version of it. This will still tell us the essential point about the 2nd law of thermodynamics. Take a box containing 100 coins. The coins are either heads-up or tails-up and there is enough room in the box to shake them so that they turn over.

So shake the box. About half the coins will be heads and half tails. There are lots and lots of possible states that have about half heads and about half tails. Shake the box again and you get another such state. And again; you get another one.

It doesn’t matter which particular coins are heads, and which particular coins are tails, which means that there are lots of possible ways of arranging each individual coin such that roughly half are heads and roughly half tails. There is nothing special about such an outcome, and so we call the overall state a disordered state.

Now consider the state where every coin is a heads. This is a special state because it requires every individual coin to be in a particular state — namely heads. There is only one possible way of arranging the coins such that all are heads. Thus we call this a highly ordered state.

Suppose we start in the all-heads state and we shake the box hard. What’s going to happen? Well, since there there is only one state with all heads (or, indeed, all tails) the chance of it landing back in that state are very low.

But, since there are lots and lots of states in which roughly half the coins are heads and half tails, it is overwhelmingly probable that you’ll end up with a nearly half-and-half state. In other words the system has moved from an ordered state to a disordered one. The more lop-sided an outcome is (heads outnumbering tails or vice versa) the fewer ways there are of producing that outcome, and so the less likely it is.

What happens if you’re in a disordered state to start? Well, the chances of moving to a highly ordered state are very low, just because there are few such states. Chances are you’ll land back in a disordered state, because there are lots of configurations that are disordered.

And that’s the second law of thermodynamics. It says that it is easy and likely for an ordered state to move naturally to a disordered state; but it’s highly unlikely for a disordered state to move spontaneously to an ordered one (that would take an intervention from outside, such as a kid coming along and turning over each coin as appropriate).

That’s the end of the explanation, and so far I’ve used no maths and no jargon. Depending on the kid, though, one could then introduce some jargon.

The “macrostate” is the overall ratio of heads to tails. The “microstate” is the particular configuration of every coin (imagine them numbered 1 to 100, and then the microstate is the list HTTHHTH … et cetera).

“Entropy” is then just a fancy word for how ordered the state is. Formally, it is a count of the number of microstates that would give the same macrostate (thus for the all-heads macrostate there is only one possible microstate, whereas for the 50:50 macrostate there are lots of microstates). A high-entropy macrostate has lots of possible microstates; a low-entropy macrostate has few possible microstates.

Saying “entropy always increases” is then just saying what I said above, namely that, since ordered macrostates have very few microstates, whereas disordered macrostates have lots, there is a natural tendency for systems to move from ordered states to disordered states, and it is very unlikely for a system to move spontaneously from a disordered state to an ordered state.

Note that, in the above reasoning, we have made the implicit assumption that each microstate is equally probable. This is usually a pretty good assumption.

The mathematics of counting microstates is accessible to a maths-able teenager. Start with four coins. There is one microstate that gives all-heads, four than give three heads and one tail, and six that give two heads and two tails (these are: HHTT, HTHT, HTTH, THTH, THHT & TTHH). The number of microstates for each of the macrostates (0, 1, 2, 3 and 4 heads) is thus 1, 4, 6, 4, 1 respectively. Thus two-of-each is six times more probable than all-heads.

One can generalise this into a “combinations formula” which says that for N coins the number of ways of having n heads is given by N!/[n!(N–n)!] where the ! indicates “factorial”.

Note that the 2nd law is probabilistic, not absolute — there is a tiny chance that a shake of the coins would indeed give all-heads, but that chance decreases as the number of coins increases. With four coins it would be one-in-24, and with 100 coins it would be one-in-2100.

In the real world we’re talking about molecules, which are tiny, and so there are about 1023 molecules (Avogadro’s number) in each teaspoon of stuff. Thus the chances of significant large-scale violations of the 2nd law effectively vanish to zero.

Nothing above says that a system cannot move from a disordered to an ordered state. There are lots of possible ways in which it can (the above possibility of a boy coming along and turning the coins over to attain a desired result is an example), but all of those possibilities require an input of energy into the system.


Thus they can occur in one part of a system, if that part receives energy from elsewhere in the system, but they cannot occur in the system overall if it is isolated (that is, not gaining energy). To quote Muse: “In an isolated system, entropy can only increase”.

(And by the way, if you’re wondering about the tiresome creationist claim that the evolution of complex and highly ordered life on Earth violates the 2nd law, no it doesn’t, since the Earth is not isolated, it gets a large energy input from the Sun.)


6 thoughts on “The Second Law of Thermodynamics made easy

  1. Dan Steeves

    Both the living cell with its staggering complexity and equally complex DNA genetic code, the periodic table that orders atomic structure and the awesome and extremely ordered universe composed of galaxies and groups of galaxies that absolutely overwhelm human understanding all operate by law. The laws of physics, the laws of genetics, the laws of chemistry and who knows what other laws operate in this vast, incredible universe giving convincing testimony of a Grand Designer and lawgiver with almighty creative power, infinite wisdom and other marvellous qualities that truly draw us to Him. Yet with all this overwhelming evidence people still believe in the darwinism fairytale which is what the great french biologist Pierre de Grasse said it to be.

    1. Dan Steeves

      I forgot to mention that the laws that govern the universe on the micro and macro level have to be legislated in order for the universe to function harmoniously and in an orderly way. Laws cannot create themselves. They must be created or legislated. Laws that protect human rights in the criminal and civil code are legislated by lawmakers. So much more so would be the case of laws that rule the universe. They would have to be legislated by a universal lawgiver. The Creator of the universe would be the one to legislate all laws, both physical and moral, for the peaceful and harmonious functioning of all creation.

    2. Coel Post author

      laws that govern the universe on the micro and macro level have to be legislated in order for the universe to function harmoniously and in an orderly way. Laws cannot create themselves.

      You are misunderstanding what scientific “laws” actually are. They are simply descriptions of how things are. They are not commandments, not entities that direct matter. Trying to understand why the universe is the way it is by invoking a God doesn’t get you anywhere, it just lands you with something even harder to explain.

    3. Coel Post author

      “I will explain something wonderful by … saying that it was all created by something even more wonderful”. That is not an explanation of anything. Nothing about the visible world is testimony of a sentient “law giver” behind it all, everything we see makes more sense without that hypothesis.

  2. Derek Freyberg

    Dan, the Second Law of Thermodynamics (as so ably explained by Coel) is nothing like a human law.
    The law that says “You may not park on this side of the street on Mondays” was created by people to serve their purpose – perhaps so the street cleaner can clean that side of the street on Mondays.
    The Second Law of Thermodynamics serves no person’s (or Creator’s, or Grand Designer’s) purpose – it’s simply a description of the way things are. There are more ways to have a bunch of coins half heads and half tails than there are all heads, or all tails; so if you shake the box, you’re more likely than not to end up nearer half-and-half; order tends to disorder.
    T-T-T-That’s all, folks.

  3. Peter (Oz) Jones

    @Dan Steeves
    mmm, this Grand Designer is probably even more staggeringly complex?
    So, I am just hazarding a guess here.
    There is a Yugely Bigger Designer who created this Grand Designer.
    Oh, wait a minute, I think there is a snag here . . . isn’t there?


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