The Maxwell-Boltzmann Distribution, Study notes of Calculus

Download The Maxwell-Boltzmann Distribution and more Study notes Calculus in PDF only on Docsity!

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