Actually, thermodynamics is hard and I don’t understand it. But even without totally understanding thermodynamics, it turns out its possible to do a surprising number of useful calculations with just a couple of simple rules about entropy.
The setup is as follows: Imagine that there is some set of states of the world, called the macrostates, that we humans can distinguish. To each of these macrostates is associated some large number of microstates, where a microstate is a complete specification of all information about all the particles in a system.
For example, given a container of gas, different macrostates would correspond to different pressures and temperatures of the gas, since we can determine those with macroscopic measurements. Microstates would correspond to complete information about how all particles in the gas are moving.
Every macrostate has an associated quantity called its entropy, written with an Image may be NSFW.
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. The entropy of a macrostate obeys the following rules:
- The entropy is equal to Boltzmann’s constant, Image may be NSFW.
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, times the logarithm of the number of associated microstates: Image may be NSFW.
Clik here to view.. - If a system is at a temperature Image may be NSFW.
Clik here to view., and you heat it by adding energy Image may be NSFW.
Clik here to view. to it (while keeping it at temperature Image may be NSFW.
Clik here to view., by allowing it to expand, say), then its entropy increases by Image may be NSFW.
Clik here to view.. - The total of the entropy of the universe always increases. (This is the second law of thermodynamics.)
These rules alone let you do a surprising number of useful calculations:
Doubling the Volume of a Gas
Suppose you have a gas occupying a volume Image may be NSFW.
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at temperature Image may be NSFW.
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with total number of molecules Image may be NSFW.
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. How much energy do you have to add to it to double its volume while keeping it at the same temperature?
With a doubled volume, imagine that the gas occupies two spaces of volume Image may be NSFW.
Clik here to view.. Then the number of microstates of the system gets multiplied by Image may be NSFW.
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since, for each microstate of the original gas, there are Image may be NSFW.
Clik here to view. new microstates corresponding to which space of volume Image may be NSFW.
Clik here to view. the molecule is in.
That means that the entropy must have gone up by Image may be NSFW.
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by Rule #1, but by rule #2, that means that to achieve this entropy increase by heating, we must have added Image may be NSFW.
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energy to the system.
Efficient Refrigeration
Generally, when hot things and cold things touch, the hot things get cooler and the cold things get hotter. This is because hot things that are getting cooler are losing entropy, and cold things that are getting hotter are gaining entropy, but the entropy being gained is greater than the entropy being lost.
How much energy does it take to reverse the process? Suppose you have a refrigerator with 10 kg of food that’s currently at room temperature (say 20° C) and you want to lower it to 0°C. Suppose the specific heat of the food is 3.5 kJ/kg°C. That means that a total of Image may be NSFW.
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can be extracted from the food as it cools.
When the food has lost Image may be NSFW.
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energy, that must mean its temperature is:
Image may be NSFW.
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That means the entropy lost by the food as its cooled is
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This is about 2.5 kJ/°C. Whatever process is used for refrigeration, it must obey Rule #3, and thus increase entropy somewhere else.
If you increase entropy by exhausting heat into the room, which is at 20°C = 293K, then you’ll have to exhaust Image may be NSFW.
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of energy. You can get 700kJ of that from the food, but you still need an extra 32.5 kJ of energy, which is why refrigerators have to be plugged in and don’t work on their own.
Deleting a Bit
How much energy does it take to delete a bit of information in a computer? A bit could be in state 0 or state 1, and after deleting it, it will be in state 0 (say). That means that the process of deleting the bit takes all the microstates of the 0 macrostate and all the microstates of the 1 macrostate to the 0 microstate. That’s halves the number of microstates, or subtracts Image may be NSFW.
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In order to obey Rule #3, you therefore have to exhaust a minimum of Image may be NSFW.
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