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The second law of thermodynamics, part 1

Thermodynamics

The second law of thermodynamics, part 1

Cover image: Natasha Vasiljeva

The second law of thermodynamics, part 1

Discussing three statements of the second law of thermodynamics and their equivalence, we prepare for a mathematical discussion of heat engines and entropy.

Having previously seen the zeroth law and the first law of thermodynamics, both of which set the stage up for more advanced discussions on the topic, we now proceed to the second law. Unlike its predecessors, the second law has numerous interpretations and brings a number of interesting ideas to the table, so we will move through them in three parts.

Cycles and more

The fundamental idea driving the second law is that of a cycle. The previous law told us that heat, work and energy are interconvertible. We now think of a series of processes that work to bring our system back to its initial state. For starters, we could think of it as heating our system and then cooling it. This is simple enough, and it is a cycle.

The heat flowing out of an object, though, has already been deemed to be due to some work done by it, at least in part. The pattern we recognise here is that a system in any such cycle will have a high temperature source and a low temperature sink. As some heat, Q_H, is exchanged from the source to the surroundings and some heat, Q_L, is exchanged from the sink to the surroundings, the remainder is manifested as the work, W, done by the system itself to “power” this heat exchange process.

In other words, we can think of this as a train going from the source to the sink (Q_H) but some energy is lost halfway as work (W) and the remaining (Q_L) continues onto the sink.  We are then left with

    \[Q_H-W = Q_L\;\Rightarrow\;Q_H - Q_L = W\]

in general and we can claim an efficiency (\eta, the ratio of output to input), of

    \[\eta = {W \over Q_H} = {Q_H - Q_L \over Q_H} = 1 - {Q_L \over Q_H}\]

which, as it turns out, is always less than 1.

It is further also possible to reverse this engine, taking heat from the sink to the source, to complete our cycle, and we can achieve utter efficiency by simply eliminating the “work” factor altogether. Now this type of a cycle is idealistic at best (since more energy is lost than can be accounted for these ways alone, to say nothing of eliminating the “work” factor) and such an engine is called a Carnot heat engine, or a Carnot reversible engine. We will encounter Carnot in part 2 when we talk more about cycles and delve deeper into heat engines.

Carnot and the efficiency of heat engines

The second law of thermodynamics has several different statements that all take different routes to the same ideology. It is important to discuss them all because of the various ideas they are founded on. For our purposes, we will focus on three famous statements. Having been introduced to Carnot, and because he first came up with the idea, we will begin with his words:

A reversible Carnot engine operating between two given temperatures is the most efficient engine possible; further, the efficiency of such a heat engine depends only on the temperatures between which it operates.

It may not be immediately apparent how the efficiency of any heat engine would depend on its operating temperatures alone, since we used work and heat to define \eta. The concept of temperature is something we have already discussed before. In brief, it is not unreasonable to assume that a hot body (say Q_1) has some temperature (say T_1) that is higher than the temperature (T_2) of a body with less heat (Q_2). By extension then,

    \[{Q_1 \over Q_2} = {T_1 \over T_2}\]

and, further, if the bodies in our source-sink system with Q_H and Q_L had temperatures of T_H and T_L respectively, we can simply re-write the efficiency as

    \[\eta = 1 - {T_L \over T_H}\]

and we quickly see how \eta is a function of the source and sink temperatures alone.

The origin of this definition between heat and temperature comes from the groundbreaking idea of entropy. This concept has its roots in the statements of the German scientist, Rudolf Clausius, which, on the surface may appear somewhat obvious.

Entropy and the Clausius statement

Imagine a system where a block of ice and a piece of burning coal are kept on top of each other. If we base our expectations on Carnot’s idealistic reversible heat engine, then common sense tells us that heat is transferred from the piece of coal to the block of ice until they achieve thermal equilibrium. However, absolutely nothing prevents the reverse from happening: why should the block of ice not continue to lose heat and become cooler while the piece of coal absorbs all of that heat and keeps getting hotter? Nothing, besides common sense, tells us that this is forbidden. It was Clausius who came up with a more scientific explanation, or at least the initial stages of it:

Heat can never pass from a colder body to a warmer body without some other change occurring at the same time.

As promised, this statement seems glaringly obvious at first, but is it? The first part is fine, almost common-sensical. But what “other change” is required to happen at the same time? To gain a better understanding of this, here is another form of this statement, but this time taking a statistical perspective:

Any system over a period of time spontaneously moves towards a thermodynamic macrostate corresponding to the largest microstates.

A macrostate is the the overall picture of how a system is arranged. Consider, for example, a collection of 25 coins. If we keep tossing them, they are hardly likely to all fall heads up or tails up. Instead we will end up with some fraction of heads and tails, say 13 heads and 12 tails. This is the macrostate of our system of 25 coins. The same macrostate may see any number of microstates, or specific arrangements: H-T-T-H-H-T-H-H-H-T … H-T-H, or T-H-H-T-T-T-T-H-H-H … T-T-H etc. We need not spend time or energy listing them all out; the idea is simply that of all these microstates that result in the same macrostate, the final result will always be one with a large number of microstates, with no discernible patterns in the mould. In other words, complete statistical disorder.

This idea of disorder will be a recurring one in physics and is called the entropy. Indeed this is one among other things, in Clausius’s statement, which is the “some other” change that he refers to. It is worth noting that the “some other” change may be as simple as a change in pressure or volume, but they will nonetheless all entail a change in entropy. While the change in entropy can well be negative, any spontaneous process, i.e. one that occurs without an external influence, sees an increase in entropy. The “some other” change, then, may also be the work done on the system to produce a negative entropy.

The Kelvin-Planck statement

Carnot’s statement was reiterated by Kelvin and once again by Planck to be more general. Carnot’s statement, if you will recall, was rather specific to heat engines.

No cyclic process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat to an equivalent amount of work.

This simply tells us that we can never have a process where Q_H = Q_L and that we will always have a work component. Note that this statement also, like Carnot’s, talks of a cyclic process, a process where a system undergoes a series of processes after which it returns to its original state.

Consequently, if Q_L is always involved and Q_H \neq W, we have Carnot’s other implication that \eta \neq 1. The Kelvin-Planck statement, therefore is a generalisation of Carnot’s reference to heat engines.

It is interesting that the second law of thermodynamics has so many statements. They should necessarily all be equivalent, even though they may not seem to be at first. This is particularly interesting because Clausius’s statement makes no mention of cyclic processes and, better yet, the statistical statement seems wholly disconnected from all our other discussions.

The last section of our discussion will be devoted to a qualitative understanding of how all these statements mean the same thing. This will then prove to be a good starting off point for parts 2 and 3 where we will treat these ideas, particularly entropy, on a mathematical footing.

Equivalence of statements

Let us try to picture all of these statements in reference to a machine, represented by \Box, which does various things to the heats and work involved moving from source to sink. Let us begin with some heat, Q_H output by a source. We will prove the equivalence of these statements not by showing how one maps onto another but, somewhat pessimistically, by proving that violating one statement means violating the other. Since we have already seen how the Kelvin-Planck statement is a careful generalisation of Carnot’s original, we shall focus on that and Clausius’s alone.

Consider the following action of our engine that violates the Kelvin-Planck statement:

    \[\textrm{source} \longrightarrow Q_H \longrightarrow \Box \longrightarrow  Q_H = W\]

Consider another machine, \Delta, which moves heat Q_L straight from sink to source, but not (yet) in violation of Clausius’ statement, since we do some work, W on the machine.

    \[\textrm{sink}\longrightarrow Q_L \longrightarrow \Delta + W \longrightarrow Q_L \longrightarrow \textrm{source}\]

So, combining them, a machine that violates the Kelvin-Planck statement and a machine that does not violate the Clausius statement, we now have

    \[\textrm{source} \longrightarrow Q_H \longrightarrow \Box \longrightarrow  Q_H + Q_L \textrm{ (sink)} \longrightarrow \Delta \longrightarrow Q_H + Q_L \longrightarrow \textrm{source}\]

or, since we are taking Q_H from the source and delivering it back, we end up with,

    \[\textrm{sink} \longrightarrow Q_L \longrightarrow \textrm{machine} \longrightarrow Q_L \longrightarrow \textrm{source}\]

in other words, there is a net heat exchange of Q_L with no external work or “some other” change, which is a clear violation of Clausius’s statement. Below is a more pictorial representation:

Equivalence of the various statements of the second law of thermodynamics.

The statistical interpretation is a little harder to equate directly to these physical statements, so we will retain it for discussion with a mathematical basis.

Part two of our discussion on the second law of thermodynamics will take us through what heat engines are and what Carnot cycles are, and part three will deal with what entropy is and how it explains various phenomena, from the statistical statement to the direction of time to the proposed grand theory of the heat death of our universe.

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V.H. Belvadi is an Assistant Professor of Physics. He teaches postgraduate courses in advanced classical mechanics, astrophysics and general relativity. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin. He currently serves as the Editor-in-Chief of Physics Capsule.

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