1. Practice questions about the partition function
1.1. Prof. Francl
Would a higher percentage of molecules be in the first excited vibrational state for HCl or DCl? how would Cl2 compare? You should not need to do any calculations, nor do you need to look up the formula for the partition function for this. If you did not have first semester, you should look up what the energy levels are for vibration, that might help!
2. Learn to use the wiki wiki web
Create a subheading with your name on it and answer the question given in my subheading. You may comment on the answers of your colleagues, if you wish!
2.1. Prof. Francl
Suppose you had a system of N particles that has M possible energy states. The energy states are evenly spaced. For example: E0=0, E1=1, E2=2, E3=3, etc. where the highest state is EM=M. When the temperature approaches absolute zero, what is the probability of being in the ground state? of being in state M? When the temperature approaches infinity, what is the probability of being in state M? Does this make sense physically? Explain!OR
Can you ever have a situation where the probability of being in an excited state is greater than that of being in the ground state?
2.2. Leyla's Response
As the temperature approaches absolute zero, the probability of being in the ground state approaches one. As the temperature approaches infinity, the probability of being in energy state M also approaches one. This makes sense physically because as the temperature increases one expects higher energy states to be occupied more often than lower energy states. When the temperature is colder, the lower energy states would be occupied more frequently than the higher energy states.Your results do make qualitative sense. Are they quantitatively consistent with what we derived in class when M=2? mmf
2.3. Dr. Glazier's Response
The answer to both of these questions is the impetus for interesting developments by some types of engineers.Can you give us an example?
I don't know about an answer but what I thought of immediately when I read the second question was a laser, that is based on a population inversion (more atoms or molecules in the excited state thatn in the ground state). Trying to develop new lasing mediums is made more difficult by the problem of damage due to over heating and most lasers have interesting ways to keep things cool. However, after reading Hillary and Dr. Francl's comments, I think a laser is an example of a nonequilibrium system so it is not an example for the partion function that we are considering in this question. It makes me curious to look and see how partition functions are used in quantum electronics where laser action is studied.
2.4. Dr. Chirlian's Response
I had better luck using my head than using Mathematica!!!!
2.5. Darae's Response
As the temperature appproaches 0, the probability of being in the ground state becomes 1, and the probability of being in state M becomes 0. As Leyla explained above, when temperature is low, most of molecules would stay in ground state because they do not have enough energy to be in higher states. As the temperature approaches infinity(apparently the computer does not like the infinity symbol), the probability of being in state M becomes 1/M. Interestingly enough, it can be noted that as temperature approaches infinity, the probability of being in each different state becomes equal!!! When temperature reaches infinity, molecules are free to be in ANY energy state, and no particular energy state is favored over another.The next question...this one is quite mind boggling thing. I have found that when temperature is below absolute 0, the probability of being in excited state is bigger than the probability of being in ground state. Using the P1 and P2 equations from the lecture, it can be deduced mathematically that when the variable T is negative P2 > P1 as long as A is not equal to 0 (in which case, P1 = P2 = 0.5). I am not so confident about this answer because it makes no sense to me in physical sense. Perhaps somebody can help me out here!
Interesting, I hadn't thought to try that! It probably makes no physical sense because there is no physical reality to a negative absolute temperature. mmf
The computer may not have like infinity if you did not capitalize it! Mathematica wants us to use Infinity mmf
2.6. Hillary's Response
I agree with the above answers for the first part of the first question. As the temperature approaches zero the probability of a molecule being in the ground state approaches one and in EM approaches zero. Since temperature is a measure of average kinetic energy it makes sense. As the particles are cooled they have less energy and it becomes harder for the particles to get enough energy to reach any excited state, especially M. As the temperature approaches infinity the particles will get more and more energy and more particles will be able to reach the EM energy level. The probability of a particle being in the EM energy level approaches 1/M, like Darae said.The only way there can be more particles in the excited state than in the ground state is when real molecules are considered and a molecule can become excited, but instead of relaxing back into the ground state they transfer to another energy level with a different spin state. Once in this state it is "forbidden", aka hard, for the molecule to relax back to the ground state, and so more molecules can be in the stuck in the excited state than are left in the ground state. This is due to the molecules relaxing back to ground state at a much slower rate than they become excited at.
You raise an excellent point. Molecules can become kinetically trapped in an excited state, leading to a population inversion. The statistical mechanics we are using is describing the equilibrium distribution. As you note, if you let the system you describe relax (i.e. return to equilibrium) the molecules eventually return to a situation where the majority are in the ground state. Technically what we are doing in this class is equilibrium statistical mechanics or statistical thermodynamics, a more advanced course might consider non-equilibrium statistical mechanics. mmf
2.7. Katie's Response
I also agree with the answers that have been given. As the temperature approaches zero the probability of a molecule being in the ground state approaches one and in EM approaches zero. And, as T goes to infinity, there is no barrier to any molecule's exciting to a higher energy state. This makes all the levels equally likely, and so the likelihood thatWhile we are making the theoretical assumption that T in fact goes to infinity, it is interesting to note that there would be infinitely many energy levels available to the individual molecules. This means that M also goes to infinity. So, in fact, that likelihood that a molecule will have, say, energy E1, goes to zero. This means that, although a given molecule must have a certain amount of energy, the probability of its having any particular energy is always zero, since there are infinitely many possible energies.
Your point about the infinite number of possible energies is well taken. Many systems have an infinte set of possible energies (though not all! NMR is a good example of one that does not), so at very high temperatures, the distribution of molecules among states approaches random.
2.8. Balpreet's Response
Being repetitive, I agree with the above answers to the first question. As the temperature approaches zero, there is no to little thermal energy. Because of this, the particles do not have enough energy to reach the higher states, which causes them to stay in the ground state. So, as temperature approaches zero, the probability that a particle will be in the ground state is 1, while the probability of it being in the M state is 0. When the temperature approaches infinity, there is enough energy for a particle to reach any energy state, and therefore, there is an equal probability for a particle to be in each state. Also, the sum of the all the probabilities for M states is equal to 1. So, the probability of a particle being in M state (and every other state) is 1/M.Unfortunately, I can't think of any original situation for the second question. Sorry!!!
2.9. Michelle's Response
I also agree with the above answer. As the temperature approaches zero, there is little kinetic and thermal energy. As a result of this low energy, the probability of molecules being in ground state approaches 1, implying that the probability of a molecule being in M state will approach zero.For the second part, I agree with Katie. When T goes to infinity, M will also go to infinity. Because there are so many energy levels available (an infinite amount), there are an infinite amount of energy levels available to occupy, the probability of a molecule occupying a particular one will be (1/the number of energy levels available.) (1/infinity) will more or less be zero, therefore the probability of being in M state will also approach zero.
