Entropy(# 2): A Simplified Preview of the Second Law of Thermodynamics
Origin of Heat
The sun is approximately 150 million kilometers away from earth and since light travels at a speed of 300,000 kilometers per second, it takes about 8.3 minutes (500 seconds) for light to reach Earth from Sun’s surface.
On the other hand, the radius of sun is about 700,000 km, so it would take light only slightly over two seconds to travel from the core of the sun to it’s surface. But the real scenario is far away from this, it takes approximately 5000 years to travel from the center to surface !! What’s behind this surprising phenomenon? It’s because, on it’s way out, light undergoes innumerable collisions in an irregular fashion with the atoms and electrons in the interior of sun.
The truth is that, collisions is prevalent everywhere on the molecular scale. If you pour a glass of water, put it at rest, then sometime later, you will find the water level to be completely quiet and uniform, without any visible motion. However, if the water is magnified a few millions times, it will be found that, the water molecules are in a state of violent agitation moving around and pushing one another as though they were people in a highly excited crowd. If very smaller organisms like bacteria or plant spores are suspended in the water, they will be incessantly kicked, pushed and tossed around (randomly and irregularly) by the restless molecules. Scottish botanist Robert Brown observed this irregular motion in objects in his microscope (thus it’s known as Brownian motion).
This irregular motion of water molecules, or any other molecules of any other material substance is known as thermal motion and this is what is responsible for the phenomenon of heat.
How Physicists deal with irregular (disordered) motion?:
The irregularity of thermal motion is not outside the scope of physical description, rather physicists handle it quite well with the help of statistical techniques. To explore this matter, let’s look at the example of the famous problem of a “Drunkard’s Walk”
Suppose there is a drunkard who has been leaning against a lamp post and then has suddenly decided to walk off (nobody knows which way, neither he himself). Thus he goes, making a few steps in one direction, then in another, and so on, changing his course every few steps in an entirely unpredictable way. How far will be our drunkard from the lamp post after he has executed a hundred steps of his irregular zigzag journey?
One would at first think that, because of the unpredictability of each turn, there is no way of answering this question. Actually, there is. Although we really cannot tell where the drunkard will be at the end of his walk, we can answer the question about his most probable distance from the lamp post after a given number of turns (N). Considering the fact that, his motion is completely disorderly, exploring this problem from a statistical point of view, it’s possible to show that: the most probable distance of our drunkard is given by:
Where, l= the average length of each straight track that he walks.
The statistical nature of the above example is revealed by the fact that we refer here only to the most probable distance and not to the exact distance. In the case of an individual drunkard it may happen that he does not make any turn at all and thus goes far away from the lamp post along the straight line. It may also happen, that he turns each time by, say, 180 degrees thus returning to the lamp post after every second turn.
The situation becomes more predictable once we have a greater numbers of players in the field. An example of spreading due to irregular motion is given in the figure below, where we consider six walking drunkards. It can be found that, the larger the number of drunkards, and the larger the number of turns they make in their disorderly walk, the more accurately the above rule works.
The same law of motion pertains to each separate molecule in our drop of water. In fact, whereas the statistical law of Drunkard’s Walk can give us only approximate results when applied to a half-dozen drunkards, its application to billions of dye molecules undergoing billions of collisions every second leads to the most rigorous physical law of diffusion.
To clarify the situation, let us consider a room divided into two equal halves by an imaginary vertical plane, and ask ourselves about the most probable distribution of air molecules between the two parts. If we pick up one single molecule, it has equal chances of being in the right or in the left half of the room.
All the other molecules also have equal chances of being in the right or in the left part of the room, however an overall fifty-fifty distribution is in this case by far the most probable one. In fact, with the number of air molecules in our case, the probability at 50 per cent becomes greater and greater, turning practically into a certainty when this number becomes very large. In simplified version, that’ what the second law of thermodynamics: the system tends to evolve towards the state of of maximum probability with increasing molecules(or equivalent other things in corresponding examples )
To improve our understandings, let’s add another example to the list, a common one. When you toss a coin, there is a fifty-fifty chance for heads or tails (it is more customary in mathematics to say that the chances are half and half). If you add the chances of getting heads and getting tails you get ½ + ½ =1.
If you toss for twice and for both cases you get a head, then of course, you can conclude: with two tosses, probability of head is 1, and tail is zero. However, if you keep tossing, you will eventually find that, for a larger number of tosses, say 100 or 1000, the probability curve becomes sharper and sharper and the maximum at fifty-fifty ratio of heads and tails becomes more and more pronounced.
Thus the state of maximum probability becomes more and more evident with increasing number of tosses (equivalent to gas molecules in the previous example; time in the example of Part 3) and that’s what the second law of thermodynamics states (through thermodynamic context)
References and for further Readings
[1] “One Two Three… Infinity: Facts and Speculations of Science”, by George Gamow
A classic book by fun-loving physicist George Gamow, with lucid presentation of complex science topics. Most of the examples in this article are extracted from this book.
[2] “An Introduction to Thermal Physics” by Daniel Schroeder
An excellent textbook to get a holistic idea of Entropy/Second Law
