How many microstates distinguisahable indistinguishable energy levels?
Answers
Answer:An isolated system will spontaneously transition between states such that the entropy of the system is increased to its maximum value. Why is this? Is there some strange force pushing things to higher entropy states? The simple fact is that if a final state has higher entropy, it is simply more likely to exist from the myriad of possible states. These states contain distributions of molecules and energies that are the most probable. What distributions are the most probable? The ones with the greatest number of microstates. A microstate is a specific way in which we can arrange the energy of the system. Many microstates are indistinguishable from each other. The more indistinguishable microstates, the higher the entropy.
The best way to wrap your head around this idea can be to look at a very small scale example. Often the idea of "order" and "disorder" is invoked when thinking about entropy. Technically this is not correct, but as we look at microstates you might see why this example arises.
Imagine that we have a certain amount of energy that needs to be distributed between three molecules. Each molecule will have "quantized" energy states in which we can put that energy (like the energy levels of atomic or molecular orbitals). For our example, just to keep it relatively simple, the energy levels will be equally spaced. Now imagine the total energy of our system of three molecules is 3 energy units. These energy units must be distributed between the three molecules. Let's look at how many different ways there are to do this.
We can give each molecule one unit of energy (1 + 1 + 1 = 3 total). There is only one way to accomplish this. Each molecule needs to have one unit of energy. Alternatively, we can give all three units to just one molecule, and zero energy to the other two (3 + 0 + 0 = 3 total). There are three ways to accomplish this since we can give all the energy to either molecule a, or molecule b, or molecule c. Since molecules a, b, and c are equivalent we will consider these three microstates to be equivalent (all degenerate in energy). Finally, we can give 1 unit of energy to one of the molecules, and 2 units of energy to another one. It turns out there are six possible ways to accomplish this distribution of energy. The diagram below illustrates each of these distributions that we have mentioned.
microstates
You can see that there are 10 total possible distributions (microstates). We can classify these based on the distributions mentioned in the previous paragraph. Group i has all the energy in one molecule - although it could be in any of the three molecules a, b, or c. So there are three possible microstates. Group iii has one unit of energy in each molecule. This turns out to be quite special since there is only one way to accomplish this and thus it has only one microstate. Group ii is where one molecule has one energy unit, the next has two, and the last has zero. There are six ways to do this.
Now if we imagine all the possibilities (all 10 microstates) and we simply select one at random, it is clear that it most likely to be form "group ii." Since we can't tell the difference between the molecules, all of these microstates in group ii are the same. If we now ask what will happen in general, you'll see that 60% of the time we get group ii, 30% of the time we get group i, and only 10% of the time do we get the "special" group iii. What is most likely to happen? We are most likely to get group ii.
The odds are the best for group ii above (6 out of 10), but not amazingly higher. However, as we move to systems with larger numbers (Avagadro's number is very, very large), what we see is that the "most likely" configurations are overwhelmingly likely. That is, the most likely configurations are essentially the only configurations that will ever be seen. There are many very similar configurations around the average so we observe small fluctuations, but we never see the extreme "special" configurations.
You can also note that configurations in group iii seem "ordered" and configurations in group ii seem "disordered." So to put this in historic perspective, an "ordered" state is one with very little variability, possibly just a handful of microstates possible. In contrast, a "disordered" state is one with a multitude of possibilities (thousands upon thousands of microstates) - so much so that no apparent patterns are present and we perceive "disorder." It is best to avoid the order/disorder arguments though. Order and disorder are value judgments that we humans impose on arrangements based on our perception. Entropy is a quantifiable measure of the dispersion of energy and our personal perceptions have no place here.
Boltzmann invented this idea of microstates and its relation to macroscopic entropy. He defined entropy as proportional to the natural log of the number of microstates, Ω. (The proportionality constant is the Boltzmann constant k.)
S=klnΩ
Explanation:
Answer:
The average distribution of 9 units of energy among 6 identical particles. There are 26 possible distributions of 9 units of energy among 6 particles, and if those particles are assumed to be distinguishable, there are 2002 different specific configurations of particles.