My book “The Second Law of Life” describes the invention of the concept of entropy in a historical, social and scientific context and all this in layman kind of language. From its early conception and use in situations where heat and work are involved (or perhaps better were we try to get work out of heat), entropy has found a widespread use in many other fields such as economy, communication theory, computer memories, religion and even art.
Let me first give a description of how entropy got originally defined by Rudolf Clausius around 1860. It is all connected with a very simple everyday observation and that is that heat always flows spontaneously from warm to cold, never has the opposite been observed. This looks like a trivial observation but the scientific explanation of this fact was not at all straightforward at that time! (Please realize that atoms were not yet a part of the scientist’s toolkit.) It is a similar observation as that of gravity. When we drop a stone it always will fall back to the earth. To describe that phenomenon the gravitational force was proposed by Isaac Newton. Two bodies of mass m1 and m2 and at a distance d will attract each other with a force F = fxm1xm2/dxd and where f is the gravitational constant.
To understand entropy, imagine that we have a heat reservoir (could be a bucket of water) at temperature Th. If we bring this reservoir in contact with another heat reservoir at slightly lower temperature, Tl, then a certain amount of heat, Q, will flow from the first reservoir to the second. The entropy change, S, now is defined as S = -Q/Th +Q/Tl and this is a number larger than zero . Thus the entropy of the reservoir at Th has been reduced and that of the reservoir at Tl is increased but the net change is positive. The important point to make here is that the entropy of the total system has increased. This is a general valid conclusion for spontaneously occurring processes (also known as the inequality of Clausius for the die-hards among us).
Thus the entropy gives a direction. In the heat exchange described above one could on ground of the First Law of thermodynamics (conservation of energy) not decide a priori what direction the heat would flow because both directions will not violate the First Law. However, the Second Law states that the entropy must increase for spontaneous (natural) processes and that is only possible when the heat goes from warm to cold.
This ever increasing entropy has a very high impact on our society. For instance, this is the reason that car engines or refrigerators can never reach 100% energy efficiency (this is in much detail further explained in the book in Chapter 2).
So far this has been a phenomenological description of entropy. Around 1900 it is Ludwig Boltzmann who links entropy to atomic dimensions and is able to increase insights considerably. More about that soon.
 This is actually a simplification as the given relation between S, Q and T is only true for what is called a reversible heat exchange but we do not need to be concerned with that for our purpose
© Copyright 2007, John Schmitz