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The Maxwell-Boltzmann
Distribution
A Chem 101A tutorial
So far, we’ve looked at properties of gases and how they
are related to one another.
We found that all gases obey the relationship PV = nRT,
where R is a constant that does not depend on the
chemical formula or molecular structure of the gas.
He O 2 C 4 H 10 SF 6
All of these gases obey PV = nRT!
Let’s start with some facts about molecular speeds.
A typical sample of a gas contains an enormous number of
particles (atoms or molecules), which have a very wide range
of speeds.
For example, one liter of gaseous nitrogen at 25ºC and 1 atm
contains roughly 24,600,000,000,000,000,000,000 N
molecules.
The average molecular speed is 475 meters per second (about
1,060 miles per hour), but some molecules are going less than
1 m/sec (the speed of a slow walk)… and a few molecules are
going more than 3000 m/sec (over ten times as fast as a
commercial airliner).
In the mid-19th century, James Maxwell and Ludwig Boltzmann
derived an equation for the distribution of molecular speeds in a
gas. Graphing this equation gives us the Maxwell-Boltzmann
distribution.
Here is the Maxwell-Boltzmann distribution for nitrogen
molecules at 25ºC.
Nice graph - but
what does it mean?
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
The vertical scale ( y axis) shows the fraction of
molecules whose speeds lie within ±0.5 m/sec of the
speed shown on the x axis.
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec Uh oh… I wonder what “fraction of molecules” means...
A fraction is a percentage before you multiply by 100.
For instance, if you have 20 molecules and 5 of them are
hydrogen, the fraction of hydrogen molecules is 5 ÷ 20, or
The percentage of hydrogen molecules would be
(5 ÷ 20) × 100%, or 25%.
Another example: instead of saying “8% of doctors hate
coffee,” I could say “the fraction of doctors that hate coffee
is 0.08.”
Okay, I get it.
It’s a bit odd calling something a
fraction when it’s actually a decimal
number, but I’ve noticed that
scientists can be a bit odd at times…
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
The higher the curve at a given speed, the more molecules
travel at that speed.
For example, many molecules have speeds around 500 m/sec,
while far fewer molecules have speeds around 1000 m/sec.
When x = 500 m/sec, the fraction (the y value) is large. When x = 1000 m/sec, the fraction (the y value) is small.
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
The Maxwell-Boltzmann distribution can be thought of as a
smoothed-out bar graph.
To see how this works, let’s “zoom in” on the region between
700 and 730 m/sec.
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
Note that all of the fractions are very small numbers.
This is because the range of speeds we are looking at is
very narrow, just 1 m/sec.
It is not surprising that a very small percentage of the
molecules have speeds in such a narrow range.
Asking for the percentage of molecules that have speeds between 499.5 and 500.5 m/sec is like asking for the percentage of people who have ages between 30 years and 30 years plus one month. The percentage – and therefore the fraction – will be very small. (If you’re curious, about 0.107% of Americans are within this age range – a fraction of 0.00107, or roughly 330,000 out of 308,000, people.)
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
The speed that corresponds to the peak of the curve is called
the most probable speed. More molecules travel at (or close
to) this speed than any other.
For N
at 25ºC, the most probable speed is 421 m/sec.
Most probable speed: 421 For any gas, the most probable speed is given by the formula € v
mp
= 2RT M
where T = the kelvin temperature
R = 8.314 J/mol K
M = the molar mass in kg/mol
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
The root-mean-square speed is the speed that corresponds to
the average kinetic energy of the molecules. It is given by the
formula:
For N
at 25ºC, the root-mean-square speed is 515 m/sec.
€ v
rms
= 3RT M Average speed: 475 Most probable speed: Root-mean-square speed: 515
The root-mean-
square speed is
always the largest
of these three
values. It is 22.5%
higher than the
most probable
speed and 8.5%
higher than the
average speed.
It’s hard to grasp speeds in meters per second, because we
usually don’t express speed this way. Here are the
equivalents in miles/hour and kilometers/hour.
Most probable speed: 421 m/sec = 942 miles/hr = 1516 km/hr Average speed: 475 m/sec = 1063 miles/hr = 1710 km/hr Root-mean-square speed: 515 m/sec = 1152 miles/hr = 1854 km/hr For comparison, a typical commercial jet airplane flies around 550 miles/hr (880 km/hr), and the speed of sound at sea level is about 760 miles/hr (1220 km/hr).
Those molecules are
really moving!
If we wanted to know the fraction of molecules that have speeds
between 500 and 1000 m/sec, we would need to measure and add
up five hundred individual fractions!
Recall that each fraction corresponds to a speed range of 1 m/sec.
We would need to consider…
Speed between 499.5 and 500.5 m/sec: fraction = 0.0018453
Speed between 500.5 and 501.5 m/sec: fraction = 0.0018422
Speed between 501.5 and 502.5 m/sec: fraction = 0.0018391
Speed between 502.5 and 503.5 m/sec: fraction = 0.0018360
Speed between 503.5 and 504.5 m/sec: fraction = 0.0018328
Speed between 504.5 and 505.5 m/sec: fraction = 0.0018297
Speed between 505.5 and 506.5 m/sec: fraction = 0.
etc….
Speed between 999.5 and 1000.5 m/sec: fraction = 0.
YIKES!!!!
Add them all up…
0
0 500 1000 1500 2000 Speed (m/sec) Fraction per m/sec
A far better way to determine the fraction of molecules in a
wide range of speeds is to measure the area of the region under
the Maxwell-Boltzmann curve.
It can be proven that the fraction of molecules in any velocity
range equals the area under the corresponding part of the
curve. This is true because the area under the entire Maxwell-
Boltzmann curve is exactly 1.
Area = 1