Answers to the questions
We see that the two sides have about the same temperature and volume, but can have different pressures. How is this possible?
This is due to different numbers of particles on each level. The astute
user will notice more particles at the lower potential. Also, the ideal gas
law PV = nKT applies to this situation. So, a higher number of particles
gives a higher pressure when other quantities are kept constant.
Why is the total energy slightly higher than the temperature?
The total energy takes into account the PE when the particles cross the
barrier going from the lower to higher potentials. Some of the molecules <
also have potential as well as kinetic energy. This is taken into account
in the total energy, but the temperature only records the kinetic energy.
The Boltzmann ratio is the number of particles on the left divided by the number of particles on the right. Plot this ratio versus the potential difference between the two sides. What is the trend? How can you plot the data to obtain a linear relationship? Explain why the trend occurs.
The user should see that the ratio rises very quickly as the PE rises...one
might say almost exponentially. There's a good reason to believe that the
curve would produce a straight line when the natural log is taken of the
ratio and plotted against the potential difference. This also makes sense
due to the fact it takes more energy and therefore more velocity the higher
the jump gets. Therefore, less molecules are able to make the jump since
fewer molecules have the necessary velocity.
Next do an experiment where you keep the same potential difference for all trials, but move the divider. Plot the ratio of area on the left to area on the right versus the Boltzmann ratio. What is the trend? How can you obtain a linear relationship? Why does the trend occur?
This plot is already a straight line, so the user should just plot the areas
on the x-axis and the ratio on the y-axis. The slope of the line should
correspond to Boltzmann's equation. e^(-PE/kT) Since we keep the PE
constant and the temperature constant as well. An analogy to this question
is the act of throwing darts. A person aiming for a large area will be more
likely to hit his target than a person aiming for a small area.
Suppose the molecules represent air at room temperature. How fast are the air molecules traveling? A typical value of mgh is 10. What height would this correspond to if we were simulating real molecules? To see a dramatic effect in our simulation, we set mgh = kT. (T = 300 K, k = 1.38E-23 J/K, air is 80% diatomic nitrogen, and 20% diatomic oxygen)
What is entropy?
The entropy of a system is the number of possible configurations that the
system has. The common belief is that entropy is a quantity that describes
the "randomness" of the system, however this is not always the case. For
example, the entropy of a 6-sided-die is higher than the entropy of a coin.
Temperature of a group of molecules is related the average kinetic energy of
the molecules. The precise equation for kinetic energy is equal to
0.5*k*Temperture (where k = Boltzmann's Constant) for every degree of
freedom of the system. The degrees of freedom are the number of coordinate
directions a molecule can move. For instance in a one dimensional system, a
molecule can only move front to back so it therefore has 1 degree of freedom
(a 2-dimensional system has 2 degrees, etc). Therefore, since we are
dealing with a 2 dimensional system (molecules can move vertically and
horzantaly) the kinetic energy is equal to 1*k*Temperature.
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