Consider a system in two different conditions, for example 1kg of ice at 0 oC, which melts and turns into 1 kg of water at 0 oC. We associate with each condition a quantity called the entropy. The entropy of any substance is a function of the condition of the substance. For an ideal gas it is a function of its temperature and volume, and for a solid and liquid it is a function of its temperature and internal structure. The entropy is independent of the past history of the substance. The entropy of the 1 kg of water at 0 oC is the same if we obtained the water from ice, or if we cooled the water from room temperature down to 0 oC. When a small amount of heat DQ is added to a substance at temperature T, without changing its temperature appreciably, the entropy of the substance changes by DS = DQ/T. When heat is removed, the entropy decreases, when heat is added the entropy increases. Entropy has units of Joules per Kelvin.
To calculate the change in entropy of a system for a finite process, when T changes appreciably, we use
,
where the subscript r denotes a reversible path. To calculate the change in entropy, we find some reversible path that can take the system from its initial to its final state and evaluate the integral along that path. The actual path of the system from the initial to the final state may be very different and not reversible. But the change in entropy depends only on the initial and final state, not on the path.
An ideal frictionless reversible engine removes DQ1 from some substance at T1, does some work, and delivers DQ2 at to some other substance at T2, with DQ1/T1 = DQ2/T2. The entropy of the substance at T1 decreases by DS1 = DQ1/T1 and the entropy of the substance at T2 increases by DS2 = DQ2/T2, i.e. by the same amount. There is no net change in entropy, if we consider the entire system. But a real engine always delivers more heat at T2 than a reversible engine. For a real engine DS2 = DQ2/T2 is always greater than DS1 = DQ1/T1. The entropy of the substance at T1 decreases, but the entropy of the substance at T2 increases by a larger amount. The entropy of the whole system increases.
The total entropy of a closed system is always increasing
is another way of stating the second law of thermodynamics. A closed system is a system that does not interact in any way with its surroundings. In practice there are really no closed systems except, perhaps, the universe as a whole. Therefore we state the second law in the following way: The total entropy of the universe is always increasing.
Are the laws of physics reversible? Evidently not! Where does this irreversibility come from? If you videotape a sequence of events, and you run the tape backwards, it usually does not take a very long time before everybody notices that something is wrong. But when you look on a microscopic scale at any particular interaction, such as a collision between two small particles, you find that no interaction violates Newton's laws. On a microscopic scale each interaction is reversible. Where then does the large-scale irreversibility come from?
Let us look at a simple example of an irreversible process that is completely composed of reversible events. Consider two chambers, separated by a dividing wall. Assume we shoot 25 balls into chamber 1, each with 5 J kinetic energy. The balls will bounce around in the chamber and hit the wall and each other. If the walls of the chamber are perfectly hard and the coefficient of restitution of the balls is 1, then the average kinetic energy of the balls in chamber 1 will stay 5 J, even so some balls will gain and some will loose energy in the collisions. Assume we shoot 25 balls into chamber 2, each with 15 J of kinetic energy. The average kinetic energy of these balls will stay 15 J. So as long as chamber 1 and chamber 2 are separated by a dividing wall, the balls on one side will be "hot" and the balls on the other side will be "cool". If we cut a hole into the dividing wall, big enough for a ball to pass through, and wait long enough, the average kinetic energy of the balls on either side of the wall will be approximately 10 J. There will be 'hot" balls, with energies above 10 J, and "cool" balls, with energies below 10 J, on either side of the wall.
While Newton's laws do not forbid all the hot balls to gather on one side and all the cool ones on the other side, the probability that this will happen is practically zero. There are a very large number of ways to distribute the energy among all the balls. Any one specific way is equally likely or unlikely. It is just as unlikely for each ball to have exactly 10 J of kinetic energy than for one ball to have 500 J and all the others to have 0 J. But there are many more specific ways of distributing the energy so that the average energy is approximately equal on both sides than there are of distributing the energy so that the average kinetic energy is three times as high on side 2 as it is on side 1. There are many more ways of having a disorderly arrangement than of having an orderly arrangement. For a demonstration of the mixing of the molecules with different temperatures click on the link below.
Link:
 | Thermodynamic Equilibrium |
Disorder
is more likely than order. But how do we quantify disorder? The amount of disorder is the number of ways the insides of a system can be arranged so that from the outside things looks the same. It turns out that the logarithm of that number of ways is proportional the entropy. We can define the entropy as the logarithm of the disorder times some constant of proportionality. When we change the entropy of a substance by an amount DS = DQ/T, we change the disorder of the substance. Entropy always increases, because a high amount of disorder is, by definition, is more likely than a low amount of disorder. With our definition of disorder, every condition of a system has a well-defined disorder. If this disorder is small, then in common, everyday language, we say that the system is ordered.Click on the link below to see how entropy increases when a "ball" is dropped and some of its kinetic energy is converted into heat. The applet also shows that Newton's laws allow this process to run backwards, but such a process has practically zero likelihood.
Link:
| The answer is entropy |
We have two ways of figuring out if the entropy of a substance changes. Both ways will lead to the same answer. Sometimes it is easier to get the answer by considering the heat transfer to or from the substance. Sometimes it is hard to follow the heat, but it is easy to decide if the disorder increases or decreases.
Here are some situations in which entropy increases:
| The entropy increases whenever heat flows from a hot object to a cold object. |
| It increases when ice melts, water is heated, water boils, water evaporates. |
| The entropy increases when a gas flows from a container under high pressure into a region of lower pressure. |
| It increases when you spray something out of an aerosol can or you let air out of a tire |
When water freezes its entropy decreases. This does not violate the second law of thermodynamics. The second law does not say that entropy can never decrease anywhere. It just says that the total entropy of the universe can never decrease. Entropy can decrease somewhere, provided it increases somewhere else by at least as much. The entropy of a system decreases only when it interacts with some other system whose entropy increases in the process. That is the law.
Questions:
| Consider the following idea: A ship pumps in warm seawater. It extracts heat from that
water. It concentrates the extracted heat in its boiler and uses it to power its engine.
It pumps the cooled seawater back into the ocean.
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| Water on the top of the Gulf of Mexico is much warmer than water on the bottom.
A ship
heats some gas with the warm water from the top and causes it to expand. It then cools the
gas with the water from the bottom and causes it to contract. The alternate expansion and
contraction drives a piston back and forth and powers an engine.
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Click on the links below for references on ocean thermal energy conversion:
Links:
| Reference 1 |
| Reference 2 |
| An ice tray contains 500 g of water. Calculate the change in entropy
of the water as it freezes completely and slowly at 0 oC.
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| If you toss two dice, what is the total number of ways that you can obtain (a) a 12 and (b) a 7?
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| The surface of the Sun is approximately at 5700 K, and the temperature of
the Earth's surface is approximately 290 K. What entropy changes occur
when 1000 J of thermal energy is transferred from the Sun to the Earth?
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Links to other Web material:
| The second law of thermodynamics |
