Transcript Maxwell distribution of speeds

Maxwell Distribution of Molecular Speeds
Kinetic molecular theory attempts to explain macroscopic
behavior, e.g., pressure, by making the following assumptions
about molecular (microscopic) behavior:
• atoms or molecules occupy negliable volume
• atoms or molecules do not exert any long range through space
forces on each other, but can only interact by redistributing their
kinetic energies in elastic (billard ball) type collisions
Where have you seen these conditions before and what do they imply
about any conclusions we might derive based on kinetic molecular theory?
The theory additionally assumes a large number of atoms or
molecules in random thermal motion and then proceeds to use
statistical arguments to derive, the Maxwell distribution of
speeds:
2
Pc = dNc / N = 4 p [ m / 2 p k T ]3/2 c2 e- m c / 2 k T dc
In this equation Pc is the probability that an atom or molecule will
have a speed in the differential interval of speeds between c and
c+dc. This probability is equal to the number of atoms or
molecules in this speed interval, dNc, divided by the total number
of molecules, N, under consideration. In the expression on the
right m is the mass of an individual atom or molecule, k is
Boltzmann’s constant, T is the Kelvin temperature, and c is the
atomic or molecular speed.
Could you convert the Maxwell Distribution of speeds, Pc, to a
distribution of energies Pe, using the relation between kinetic energy and
speed, e = 1/2 m c2?
8.1
James Clerk Maxwell
James Clerk Maxwell was a Scottish
physicist, who was born on Nov. 13 in 1831, d.
Nov. 5, 1879, did revolutionary work in
electromagnetism and the kinetic theory of gases.
After graduating (1854) with a degree in
mathematics from Trinity
College, Cambridge, he held professorships at Marischal College in
Aberdeen (1856) and King's College in London (1860) and became the first
Cavendish Professor of Physics at Cambridge in 1871.
Maxwell's first major contribution to science was a study of the
planet Saturn's rings, the nature of which was much debated. Maxwell
showed that stability could be achieved only if the rings consisted of
numerous small solid particles, an explanation still accepted.
Maxwell next considered molecules of gases in rapid motion.
By treating them statistically he was able to formulate (1866),
independently of Ludwig Boltzmann, the Maxwell-Boltzmann kinetic
theory of gases. This theory showed that temperatures and heat involved
only molecular movement. Philosophically, this theory meant a change
from a concept of certainty--heat viewed as flowing from hot to cold--to
one of statistics--molecules at high temperature have only a high
probability of moving toward those at low temperature. This new approach
did not reject the earlier studies of thermodynamics; rather, it used a better
theory of the basis of thermodynamics to explain these observations and
experiments.
8.2
Maxwell's most important achievement was his
extension and mathematical formulation of Michael Faraday's
theories of electricity and magnetic lines of force. In his research,
conducted between 1864 and 1873, Maxwell showed that a few
relatively simple mathematical equations could express the
behavior of electric and magnetic fields and their interrelated
nature; that is, an oscillating electric charge produces an
electromagnetic field. These four partial differential equations first
appeared in fully developed form in Electricity and Magnetism
(1873). Since known as Maxwell's equations they are one of the
great achievements of 19th-century physics.
Maxwell also calculated that the speed of propagation of
an electromagnetic field is approximately that of the speed of
light. He proposed that the phenomenon of light is therefore an
electromagnetic phenomenon. Because charges can oscillate with
any frequency, Maxwell concluded that visible light forms only a
small part of the entire spectrum of possible electromagnetic
radiation. Maxwell used the later-abandoned concept of the ether
to explain that electromagnetic radiation did not involve action at
a distance. He proposed that electromagnetic-radiation waves
were carried by the ether and that magnetic lines of force were
disturbances of the ether. Heinrich Hertz discovered such waves in
1888. James Clerk Maxwell died on Nov. 5, 1879.
This material is taken from the WEB site
http://sirius.phy.hr/~dpaar/fizicari/xmaxwell.html maintained by
Dalibor Paar at the Department of Physics, Faculty of Science,
University of Zagreb.
8.3
The Maxwell distribution of speeds is best understood by
plotting the probability density for speeds, Pc / dc, versus
atomic or molecular speed:
Maxwell Distribution of Speeds
2.500E-03
Ar at 25.0 C
2.000E-03
1.500E-03
Pc / dc
N2 at 25.0 C
1.000E-03
N2 at 200.0 C
5.000E-04
0.000E+00
0
200
400
600
800
1000
1200
speed (m /sec)
Why does the distribution go through a maximum?
How does the distribution change as temperature or mass are
changed? Use the link Maxwell.xls to answer this question.
8.4
1400
Single numbers are often used to characterize distributions, e.g.,
the average grade is used to characterize the distribution of
grades in a class. The Maxwell distribution of speeds is typically
characterized by the most probable speed (the speed associated
with the maximum in the distribution), the average or mean
speed, and the root mean squared speed:
Maxwell Distribution of Speeds for N2 at 25.0 C
2.500E-03
most probable
speed
Pc / dc
2.000E-03
mean speed
root mean
squared speed
1.500E-03
1.000E-03
5.000E-04
0.000E+00
0
200
400
600
800
1000
1200
1400
speed (m /sec)
o
What is the average speed of N2 molecules at 25.0 C in miles per hour?
Why are the mean and most probable speeds not the same? What would
the distribution look like if they were?
Could you derive a formula that you could use to calculate the most
probable speed?
The area under the above curve between zero and infinite speeds
represents the sum of the probabilites for all possible speeds. What
should this area be equal to?
8.5
Maxwell Distribution Bonus Problem
This Bonus Problem is worth 10 points and is due, if you decide
to pursue it, one week at 5:00 P.M. from the day that it is assigned and will
only be graded on the answer.
The fraction of molecues with speeds greater than a given speed ,
say c’, is equal to the area under the curve describing the Maxwell
distribution from speed c’ to infinite speed:
Maxwell Distribution of Speeds for N2 at 25.0 C
2.500E-03
Pc / dc
2.000E-03
1.500E-03
fraction of molecules with
speeds greater than c'
1.000E-03
5.000E-04
0.000E+00
0
200
400
600
c'
800
1000
1200
1400
speed (m /sec)
ratioed to the total area under the curve. Since the square of these speeds
is propotional to their kinetic energy through e = 1/2 mc2, this fraction is
also equal to the fraction of molecules with energies greater than a given
energy, e‘ = 1/2 mc’2. This fraction in turn can be viewed as the fraction
of molecules with sufficient energy to react upon collision and is
important in theories of molecular dynamics (kinetics).
o
Calculate the fraction of He atoms at 25.0 C that have speeds
greater than 900 meters/sec. You should find the resulting integral, which
also occurs when considering normal distributions, challenging! As a test
o
case the fraction of He atoms at 25.0 C that have speeds greater than 500
meters/sec is 0.940.
8.6
To see how to calculate the mean or average of the continuously
distributed speeds consider a discrete distribution of speeds of N
total molecules in which N1 molecules have speed c1, N2
molecules have speed c2, etc. The mean speed, <c>, would be
given by:
<c> = (N1c1 +N2c2 + …) / (N1 + N2 + …) = S Nici / S Ni
= S Nici / N
N, which is a constant and does not depend on the summation
index i, can be brought under the summation sign to give:
<c> = S (Ni /N) ci = S Pi ci
Where Pi, the probability that a molecule will have speed ci, is
equal to the fraction , Ni/N, of molecules with speed ci (does this
make sense to you?)
As the number of molecules becomes large and the difference
between successive possible speeds becomes small, the
distribution becomes continuous and the summation used to
calculate the mean speed is replaced by an integral over all
possible speeds:

<c> = 0 Pc c

= 0 [4 p [ m / 2 p k T ]3/2 c2 e- m c2 / 2 k T] c dc
Derive a formula for the mean speed by evaluating this integral.
8.7
The average kinetic energy per molecule of a collection of atoms
or molecules that follow the assumptions of kinetic molecular
theory would be given by:
<e> =
=
0

0

Pe e =
0

Pc (1/2 mc2)
2
[4 p [ m / 2 p k T ]3/2 c2 e- m c / 2 k T] (1/2 mc2) dc
= 3/2 k T
Since Avogadro’s number, No, times Boltzmann’s constant is equal
to the gas constant:
R = No k
the average kinetic energy per mole is given by:
<E> = No e = No (3/2 k T) = 3/2 (No k) T = 3/2 R T
Since the assumptions that underlye kinetic moleculear theory are
essentially the same assumptions that form our notion of an ideal
gas, these results apply to an ideal gas.
What does the average energy of an ideal gas depend on? If the
pressure of a fixed amount of an ideal gas is lowered by expanding
the volume isothermally does the energy change?
8.8