## Sunday, 23 February 2014

### Entropy a full, yet simple definition.

I am a third year physics student and have been given an assignment as part of my statistical physics call to discuss how I would explain entropy to a class of senior high school students in order to show my understanding it. This is what I have come up with and would appreciate any feed back and especially highlighting anything I got wrong.

Entropy

First I would define Entropy in the most succinct way “Energy depends to disperse”. So everything from a candle being lit that disperses heat energy out from the flame, to a rock being dropped into a pool of water and the energy going out in the form of the ripples and splashes.

Heat energy dispersing then leads on to the heat death of the universe. This is a suggested ultimate fate of the universe, in which all the heat from all the stars and other processes will have dispersed all there energy so that the universe is left in a thermal equilibrium. If this were to happen then there would be no energy left to produce work and so no more processes would be able to occur and the universe would more or less than come to a stop. The name “Heat Death” implies extraordinarily high temperatures however this is not the case is all the heat energy were to disperse it would result in a very low temperature due to the sheer size of the universe and the finite amount of heat energy being spread over it.

I would then say that there is more to entropy than that and that it involves several different definitions of different processes involved in thermodynamics, starting with the zeroth law of thermodynamic. The zeroth law is a very simple and somewhat obvious statement, it states “If two systems are in thermal equilibrium with a third system, then they must be in thermal equilibrium with each other”. In other words systems are in thermal equilibrium when they all have equal temperatures.

From this I would lead onto the first law of thermodynamics. The first law is simply a statement of the conservation of energy “(Change in U) = (Change in Q) + (Change in W)” where U is the internal energy, Q is the heat energy and W is the work done. This is why the “Heat Death” would occur as since there would be constant heat energy Q, there can subsequently be no change in the Work done W. It is important to note the Q and W are functions of state. This means that they do not depend on the history of how they got to that state. So, they can travel by any path and the path they took does not influence the final value.

Now the next most important system to consider is reversible and non reversible processes. A reversible process is one that can be undone without resulting in any change to the system or its surroundings, now this is unfortunately not possible and is an idealized process as for instance you cant un-bake a cake, or un-pop a balloon. But even on a much smaller level, if you wheel your chair across a room a breeze is created and when you wheel it back another breeze is created. the previous breeze is not sucked back towards the chair. However, in general the main thing that cannot be undone is the entropy of the system because as we will find out. Because in nature, entropy always increases.

Take for instance the graph shown in figure 1 , if the process was reversible then the area under the graph would be zero and so nothing would change, However, since there are no reversible processes the path back to the starting point will be not be equal to zero and so will result in some amount of work done. But, if this happens the change in internal energy will be zero as it has returned back to the starting point so if you look at the first law of thermodynamics this implies that the heat energy is equal to the negative of the work done. In other words, heat in = work out, and work in = heat out.

Figure 1. Area under the graph of pressure against volume is the work done. In more mathematical terms dW = -p dV where p is the pressure and dV is the change in volume.

I would then go on to talk about the Carnot cycle. The Carnot cycle is a very important process that describes and idealised heat engine, one which in reality is not achievable but is important in the fundamental understanding of thermodynamics. It is best explain with use of a diagram shown in figure 2.

Figure 2. Diagram of the Carnot cycle which runs from point A-B, B-C and so on back to point A.

From figure 2 it should be said that the green lines pointing up and down are adiabatic processes which is a process that occurs without loss or gain of heat, and the lines marked T1 and T2 are isothermal which simply means they occur at constant temperatures. This is a process that uses an ideal gas which again is not attainable in reality but helps to define what is happening. So the gas is expanded from A-B and B-C and then compressed from C-D and D-A taking it back to its original point. Q1 is the heat into the system at a higher temperature and Q2 is the heat going out of the system at a lower temperature.

From this the thermal efficiency can be calculated by dividing the work out by the heat energy in and it is found that this must be less than one due to Clausius's statement which we shall come back to later. This shows that no system, apart from an idealized one, can be 100% efficient. This is important to know apart from the implications in to creating efficient engines and such in real life but it also implies that no matter what you do energy will be lost. In a sense, entropy is the measure of the amount of energy that is not available to be converted to work in the system. E.g. how much energy is lost and this, is the second law of thermodynamics.

I would then go on to explain more details of the second law. The second law of thermodynamics can be quite hard to define and understand conceptually. There are two statements of the second law of thermodynamics that are vital in our understanding of it. The first is Clausius's statement which is “No process is possible where the sole result is the transfer of heat from a colder to a hotter body.”. The second is Kelvins statement which is “No process is possible whose sole result is the complete conversion of heat into work” and this relates back to what was defined with the Carnot cycle.
These two statements are of paramount importance even though they do seem somewhat obvious it is important to consider that if they did not explicitly define these facts then they would be up for interpretation and could have changed the way many systems were considered and not for the better.
Together they can be used to disprove any other form of engine apart from the one which we know and use today in which heat goes into the engine, work is put out but there is also an exhaust to vent any excess heat.

Mathematically entropy, S, can be defined as dS = dQrev/T where dS is the change in entropy, dQrev is the change in heat energy in an reversible system, and T is the temperature. This results in the unit of entropy being J/K or Joules per Kelvin which makes sense when entropy is the measure of energy lost of a system at a given temperature.

Finally we must look at the fundamental equation of thermodynamics. First remember the 1st law of thermodynamics: dU=dQ+dW, and we also now that dW = -pdV and finally we know that the entropy change of a reversible process is: dS = dQrev/T. If we put all these equations together we get the final equation:

dU=TdS-pdV in a reversible process.

But, it should be noted that all these terms are functions of state and so do not depend on the path they took which means that this equation must also hold true for irreversible processes. This is the fundamental or central equation of thermodynamics as it allows entropy to be meausred as a physical value of all systems.