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Statistical Interpretation of Entropy

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Heat Engines and the Second Law of Thermodynamics
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Entropy and the Second Law of Thermodynamics
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Statistical Interpretation of Entropy

Concept #1: Microstates and Macrostates of a System

Transcript

Hey guys, in this video we're going to introduce the concept of a macrostate and a microstate of a system and how it pertains to the entropy of the system. Alright let's get to it. So far whenever we talk about the state of the system we've been referring to simply the state we've just been saying the word state. That state that we've been talking about is more properly referred to as a macrostate. Macro meaning macroscopic this means pressure, volume, temperature, internal energy, entropy etc. and any of these macroscopic measurements that you can make on a system those define its macrostate. The macrostate is defined by its measurable thermodynamic properties just like I listed. For ideal gases the pressure and the volume and therefore the temperature because they're related by the gas law define system's, this should say, macrostate that is a typo or an autocorrect in word that appeared. Something important to realize is that a particular macrostate is not actually unique to a system. A system can rearrange itself to make the same macrostate multiple times. There can be multiple arrangements of the system internally to make a particular macrostate. It can have many many different microscopic arrangements that produce the same macrostate for instance two samples of a gas can have the same temperature but they could have different positions of the gas particles that make up that gas that lead to the same temperature that's a very very easy one. You can think about all the different positions in fact a gas's particles are constantly changing position even if the gas is in thermal equilibrium and its temperature isn't changing.

A microstate of a system is just a single microscopic arrangement of a system that leads to a particular macrostate. Macrostates typically have multiple microstates but it has to be at least one. At least one microscopic arrangement has to exist for a macrostate otherwise what's the point of even considering that a macrostate because it's not possible for the system to arrange itself in that manner. The number of microstates for a particular macro state is called its multiplicity which is given by the capital Greek letter omega. So a multiplicity is particular to a macrostate so you could have a macrostate of one, a macrostate of two, macrostate three, macrostate four and they could all have different multiplicities, different numbers of microstates available to them. Let's do an example. Consider 4 coins. A particular macrostate for the system could be 2 coins heads up, and this should be 2 because I cannot add, 2 coins heads down. How many different microstates are there for this particular macrostate? So let's just do it. We can have heads up, heads up, heads down, heads down. Heads up, down, up, down. Up, down, down, up. Down, up, down, up. Down, up, up, down. Or down, down, up, up. Those are all the possible arrangements for this macrostate. How many microstates are there? Each of these is a micro state that still results in 2 coins heads up. There is 1, 2, 3, 4, 5, 6. So the multiplicity of this particular macrostate is 6. There are six different microstates available for the system in this particular macrostate.

Mathematically the entropy is best defined in terms of how many microstates are available to a particular macrostate. So in terms of the multiplicity, the entropy is best defined as the Boltzmann constant which we've seen before, 1.38 times 10 to the -23, the natural logarithm LN of the multiplicity. That's the entropy of a particular macrostate. Remember different macrostates have the same, sorry, different microstates with one macrostate have the same entropy. Entropy is a thermodynamic measurable at the macroscopic level. What's the entropy of the system of coins in the previous problem with a two heads up macrostate? Remember that the multiplicity of that macrostate is 6 so the entropy of that macrostate which is KBLN of omega is just going to be 1.38 times 10 to the -23 times LN of 6 which if you plug this into a calculator ends up being 2.47 times 10 to the -23 joules per Kelvin.

Now based on this definition the entropy is never going to be negative. The entropy will always be positive because you can never have fewer microstates than one. Remember all macrostates have to have at least one microstate. So this is mathematically the best possible way to define entropy and as the disorder of a system increases the number of available microstates increases for a particular macrostate and so the entropy also increases as the multiplicity goes up, as the number of available microstates goes up, the entropy goes up. So that's why systems tend to move towards more disorder more available microstates, more disorder, more entropy, that's where the second law of thermodynamics says the system wants to go. Okay guys, that wraps up this discussion on microstates and macrostates and specifically how we can define entropy in terms of them. Thanks for watching guys.

Practice: The macrostate of a set of coins is given by the number of coins that are heads-up. If you have 100 coins, initially with 20 heads-up, what is Δ𝑆 when the system is changed to have 50 heads-up? Note that the multiplicity of k coins which are heads-up, out of N total coins, is Ω = 𝑁!/𝑘!(𝑁−𝑘)! . Does this change in macrostate satisfy the second law of thermodynamics?