Entropy?An intuitive approach
Entropy?When it was detected that all processes must comply with energy conservation (note 1), it became obvious that many processes never take place even if energy is conserved (note 2). A stone jumping up in the air while cooling down to obey energy conservation was never observed. Thus there must be something additional that governs events from phase transitions to life: entropy.
There is a quantity called entropy that never decreases when a natural process is going on. A wooden block sliding over a table slows down and eventually comes to rest because of friction; its kinetic energy is gradually converted into heat. Never ever the same block was seen to spontaneously accelerate while cooling down. Obviously, there is a unique direction of natural processes.
Now entropy is defined such as to increase whenever a natural process takes place. It stays constant if a system is in equilibrium and it decreases for the hypothetic process that goes the wrong way round.
The following two pictures show two collision processes; can you find out if the images are in the correct or in the time reversed order?
Picture 2: two billard balls hitting each other. Are the pictures in the right or in the time reversed order? One cannot tell. Mechanical processes are at constant entropy.
So for the mentioned block that accelerates spontaneously while cooling down, entropy would decrease. The second law of thermodynamics prohibits that. Overall entropy cannot decrease. This can happen only locally if, and only if, entropy is increased somewhere else by at least, and generally more than, the same amount.
Thus entropy is a quantity that reflects our empirical knowlegde about the direction of natural processes. It decreases for hypothetical processes that go the wrong way round, as if you were watching a movie that goes backwards. Some examples might help to clarify this. You can try to answer them by yourself, if you want (I recommend you do it!); you can get an answer and further comment by clicking on the number.
Problem 1
A system may consist of two blocks of copper, equal mass, one at 13 °C, the other at 79 °C. The entropy of this arrangement might be S(1); what goes on if one brings the blocks in thermal contact? What can be said about the entropy afterwards S(2):
Problem 2
What do you think, does entropy change in the same direction for the two blocks? Hint: keep in mind that temperatures change in opposite direction!
Problem 3
An isolated box with two compartments subdivided by a wall contains two gases, say nitrogen and oxygen, each enclosed in its compartment, at the same temperature and pressure. What process can be observed if the dividing wall is removed and how does entropy change? Comment on the inverse process! How do you think does entropy of the individual gases change?
Problem 4
A vessel containing water is enclosed in an evacuated box at a temperature of, say, 20 °C. Part of the water evaporates (assuming there is enough water); how does entropy of the box (including its contents) change? How does entropy of the liquid water change? Why?
Problem 5
A closed box containing water vapour is cooled to and then maintained at 50 °C; most of the water condensates; how does entropy of the box change? Note: the removed heat is outside the system and is therefore not taken into consideration! Note also: we are looking at an isothermal process, that is we are looking at the conversion of vapour at 50 °C to water at 50 °C. Of course we have to look fast, since water vapour is not stable at 50 °C! :-)
Problem 6
Ethanol (the common alcohol) and water can be mixed in any proportion; which system has higher entropy (at constant temperature and pressure) the one with 47g of ethanol and 58g water unmixed or the one with those same amounts but mixed?
Problem 7
A thermally insulated vessel containing water and ice is at equilibrium. Its temperature is at 0 °C (if pressure is at 101 325 Pa). Net mass of water is constant as well as net mass of ice. Now a tiny amount of heat is supplied. What physical effects can be observed in the vessel and how does entropy change? Please answer the same questions for the case that heat is withdrawn! What do you conclude with respect to relative entropies of ice and water?
Problem 8
A box contains water at -2 °C; how does entropy of the box, including its contents, change when the water is allowed to freeze (at constant temperature)? Note that the heat withdrawn from the box is not taken into consideration (it has left the box, hasn't it?)
Problem 9
Certainly you used to play around with magnets, didn't you? You might then have observed that nails, needles, paper clips and what else became magnetized if you fiddled around long enough? And that objects thus magnetized eventually lost their magnetism again?
So what would you think, when a piece of iron is magnetized, does its entropy rise (at constant temperature) or does it diminish (surroundings are not taken into consideration)?
As a summary, let us draw some conclusions
As an application of these conclusions, let's see if we can explain why there is something like a melting point. Refer to example no. 8; entropy is lowered upon freezing. But isn't freezing a natural process? And didn't we learn that for natural proceses, entropy increases?
It does. Keep in mind that we have excluded from consideration the heat of melting that is released when something freezes. Now we have to look at the entropy effects of this released heat.
Let us place the box of example no. 8 in a heat insulated vessel, containing, say, a very large amount of ethanol (it does not freeze itself at -2 °C). As soon as the water in the box freezes, a large amount of heat is set free; it warms up the freezing water to 0 °C (this is why temperature of freezing water is always 0 °C at normal pressure) and the excess is transferred to its surroundings, the ethanol in this case. Since we have very much ethanol, its temperature does not rise appreciably. So the ice at 0 °C eventually cools down to -2 °C; that is why we can avoid entropy effects of temperature change and focus on those of phase change. Thus total heat of melting is transferred to the ethanol, increasing its entropy.
The increase of entropy of ethanol must balance out the decrease of entropy when the ice is formed. Otherwise the process would not take place. In fact, if you prevent this heat from leaving the system, the substance in it would not freeze, at least not completely.
Now we have learned that the same amount of heat added to a cold body increases its entropy more than when added to a hot body. It follows that there must exist a temperature T(0), above which the entropy increase of the surroundings is not enough to balance out the decrease of the freezing water. Beneath T(0), entropy increase of surroundings is higher, and at T(0) entropy changes of freezing water and of surroundings are exactly equal. This is exactly how we defined entropy at the top of this article:
There exists a quantity called entropy that increases if a natural process takes place. It stays constant if a system is in equilibrium and it decreases for the hypothetic process that goes the wrong way round.
It is "natural" that a substance freezes below its freezing point. As long as there is both liquid and solid of the same substance they are in equilibrium at the freezing temperature. And finally, freezing above the freezing point would be "unnatural"; it does not work, because at too high a temperature, the released heat does not cause entropy to go up enough. As a final remark, it does not matter what substance is in the surroundings, since entropy change due to addition of a given amount of heat does not depend on the type of material.
Entropy is the key term to the definitions of heat, work, temperature. Just have a look at temperature. Bodies at thermal equilibrium have equal temperature (this is actually how temperature is defined by the Zeroth Law). Why? Look at the two copper blocks of example no. 1! At equilibrium, for an infinitesimal heat transfer we expect to find no net entropy change. Suppose the blocks are at equal temperature.
Now a tiny amount of heat is withdrawn form one body; its entropy decreases by a tiny amount -dS. The heat is added to the other body, so its entropy increases by a tiny amount dS. Since both block are at the same temperature, the two entropy changes are equal (except for the sign), thus there is no net entropy change, as expected. That, by definition, means the two blocks are in thermal equilibrium. Thus, temperature is defined as a quantity that is equal between bodies that are in thermal equilibrium.
Again, since entropy change is not dependend on material this is valid for all substances.
Consider a plant growing in a flower-pot. Is this a spontaneous process? Does entropy therefore become larger within the plant (surroundings excluded from consideration)? The process might be called spontaneous, but it is not spontaneous in the same sense as an avalanche coming down from a mountain.
When a plant grows, its molecular machines (the enzymes) consume energy while building up proteins, carbohydrates, fats. These substances do not at all develop without the help of these machines. Putting water and carbon dioxide together, and shining light through it does not result in the formation of sugar. But within certain cells, sugar is synthetized out of water and carbondioxide, with the help of sunlight.
Thus the entropy of the plant diminishes as it grows. For this to be possible, entropy becomes larger somewhere else in the universe, in the sun for instance by the processes that produce the light necessary for the plant.
Last update Sep. 19, 2001 gVa