The aim of this problem is to make speed distribution plots similar to Fig. 19-7 p.517.
According to the textbook: "The quantity P(v) in Eq. 19-27 and Fig. 19-7 is a probability
distribution function: For any speed v, the product P(v) dv (a dimensionless quantity) is
the fraction of molecules whose speeds lie in the interval dv centered on the speed v."
Thus in order to calculate P(v) for some particular speed v we select an interval dv (often
called the bin size) and find the fraction of molecules that have speeds in the range (bin):
[v-dv/2,v+dv/2]. P(v) is then that fraction divided by dv:
P(v)= (fraction of molecules in bin)/(bin size)
Note: the larger bins will capture more molecule speeds, but the ratio: fraction over
bin-size should be (approximately) independent of bin size. The bin size is something you
select. What bin size should you use? If the bin size is made quite small, the bins
will just capture 0 or 1 speed...this is like calculating a density using a volume that
just includes 0 or 1 nucleus. For the continuum approximation we want to capture lots
of speeds in each bin. If you select a huge bin, that includes every molecule's speed,
you've washed over the differences in distribution we'd like to measure. You can get
an idea of proper bin size by remembering error analysis: the error in a count of something
is given by the square root of that number. Thus if a bin collected 121 speeds in this
experiment it should be expected that, on a repeat of the experiment, the bin
would capture somewhere between 121-11 and 121+11 speeds.
If a larger bin is used, more molecule speeds would be in the bin so both the count
and the error in the count would be larger. However the relative (percentage) error
in the count sqrt(N)/N=1/sqrt(N), gets smaller with larger count. If you get bins with
100 or so counts, you'll be measuring P(v) with an accuracy of about 10%.
Homework: On this web site in the 2006 directory you'll find four data files with
molecule speeds (in units of m/s).
MB.speed2.dat contains 1000 speeds which have been sorted from high to low.
MB.Vx2.dat contains the x velocity of those same molecular speeds, again sorted from high to low.
These files contain 1000 rows of data in one column. You may prefer to work with the wrapped version
which contains exactly the same data in 10 columns of 100 rows. These are named:
MB.speed2w.dat and MB.Vx2w.dat
If you want to print the data, I've made pdf files that will print on one page:
MB.speed2w.pdf and MB.Vx2w.pdf
(A) Plot P(v) vs v for both data sets. (For both data sets: calculate P(v) at 8 different v, plot
those 8 data points, and sketch in a smooth curve.) The result should be a bell shaped curve.
It may be helpful to compare your y-axis results (both in terms of units and magnitude)
with text's plots on p.517.
(B) Using MB.speed2.dat estimate the most probable speed V(P), and note that I've given you the
mean V(mean) and rms speed V(rms). According to the equations of p. 517-8, the ratio of these speeds is
expected to be:
V(mean)/V(rms)=sqrt(8/(3 pi))
V(P)/V(rms)=sqrt(2/3)
Record your ratios next to the theoretical ratios...are they as expected?
(C) Using MB.Vx2.dat estimate the most probable x velocity, and note that I've given you the
mean and rms speed. It is expected that Vx(mean) and Vx(P) are zero...are they as expected?
Vx(rms)=sqrt(k T/m)
where k is Boltzmann's constant and m is the mass of the molecule. Assuming that the
molecule is argon, determine the temperature of the gas.
(D)In class we've assumed that:
Vx(rms)^2 = (1/3) V(rms)^2
Is this equation satisfied?