• The molecular motion in gases 1. The kinetic model of gases
a) Pressure and molecular speed Maxwell distribution of speed
• Ch. 21 Molecules in motion Molecular motion in gases Molecular motion in liquids
Diffusion: migration of matter down a concentration gradient
• One of the simplest molecular motions is the random motion of molecules of a perfect gas.
• Here we see that a simple theory (called kinetic model) accounts for the pressure of a gas and the rates at which molecules and energy migrate through gases.
• In the kinetic model of gases, we assume that the only contribution to the energy of the gas is from the kinetic energies of the molecules.
• Potential energy: molecular interaction, bonding energy, atomic energy For ideal gases, potential energy can be ignored
because they have no molecular volume.
• Assumptions:
1. The gas consists of molecules of mass m in ceaseless random motion.
2. The size of the molecules is negligible. Their diameters are much smaller than the average distance travelled between collision.
3. The molecules interact only through brief, infrequent, and elastic collisions*.
* In the elastic collision, the total translational kinetic energy of the
• The pressure and volume of the gas are related by*
2
3
1 nMc pV
* See Justification 21.1.
** The symbol < > denotes an average quantity
where the molecular molar mass M = mNA, and c is the root mean square speed of the molecules.
• The c is the square root of the mean** of the squares of the speeds (v) of the molecules:
v2
c
• When a particle of mass m elastically collides with a wall perpendicular to the x-axis, the x-
component of velocity (vx) is reversed to –vx but the y-and z-components are unchanged.
• Therefore on each elastic collision, the linear momentum change is 2mvx. The y- and z- components are unchanged.
• In an interval t, many molecules collide with the wall.
• The total change of momentum is:
t in wall reach the
that molecules
of
# molecule each
of momentum in
change The
momentum of
change Total
• A molecule (vx) can travel a distance vxt along the x-axis in an interval t.
• So all the molecules within a distance vxt
from the wall (area A) will collide (if they toward the wall).
• Therefore, all the molecules in a volume Avxt will reach the wall in the interval t.
• If the number density of molecules is nNA/V, the number of molecules in the volume Avxt is (nNA/V)Avxt.
• At any instant, half the molecules are moving to the right and half are moving to the left. So the above number should be divided by 2. 1/2(nNA/V)Avxt
A
x
x A x
A x
mN V M
t nMAv
V
t v t nmAN
V Av mv nN
where
2 2 1
t in wall reach the
that molecules
of
# molecule each
of momentum in
change The
momentum of
change Total
2
2
• Now consider the pressure exerted by the molecules.
Time Momentum Force
Area
ime Momentum/T Area
Force Pressure
Newton’s 2nd law of motion
V nMv A
V t t nMAv
p x
x 2
2
V p nMvx
2
• Note that not all the molecules travel with the same velocity, so the detected pressure p is the average of the quantity just calculated:
V v nM
p x
2
• By the definition, c v2 c2 v2 v2x v2y vz2
• Because the molecules are randomly moving, all three averages are the same. 2 3 2
vx
c
2
3
1 nMc pV
V p nMc
3
2
2
3
1 nMc pV
• According to the ideal gas law,
nRT
pV
23
1 nMc pV
m kT M
c 3 RT 3
• The c is proportional to and inversely proportional to . T
M
• Therefore the root mean square speed of the molecules depends only on T.
A
A M mN
kN
R and
• In an actual gas, the speeds of individual molecules span a wide range, and the collisions continuously redistribute the speeds among the molecules.
• The fraction of molecules which have speeds in the range v to v+dv is proportional to the width of the range.
• The fraction of molecules in the range v to v+dv is written by f(v)dv, where f(v) is called the distribution of speeds.
• The f(v) for molecules of a gas at a temperature T was derived by J. C. Maxwell:
RT Mv
e RT v
v M
f 2 2 2
3 2
4 2 )
(
Maxwell distribution of speeds*:
* See Justification 21.2. Correct the printing typos.
RT Mv
e RT v
v M
f 2 2 2
3 2
4 2 )
(
• The f(v) is a decaying exponential function.
• At low T or large M, the fraction of
molecules with high speeds becomes very small.
• When M is large, f(v) goes more rapidly toward zero.
• The fraction of molecules with low speeds is also very small.
• The integration of f(v) over the entire range of speed (0 ~
RT Mv
e RT v
v M
f 2 2 2
3 2
4 2 )
(
• The fraction of molecules (v1 ~ v2)=
• This integral* is the area under the f(v)-v curve.
dv v f
v
v12 ( )* In general, the integral is numerically evaluated by using mathematical software.
• According to some results of probability theory*, the mean value (also called the expectation value) X of a discrete variable X is given by:
N
i
i i p x X
1
where pi is the probability that X can have a discrete value xi.
• Similarly, the mean value of a continuously varying X is given by:
xf x dx
X ( )
• Note that the f(x)dx is a probability.
• For the speed, the overall range is zero to infinity. So the mean speed of gas molecules (symbol, ) is given by: c
M RT M
RT RT
M
dv e
RT v M
dv v vf c
RT Mv
8 2
2 1 4 2
4 2
) (
2 2 3
0 3 2 2
3 0
2
0 2 32 1
2
dx a e
x ax
• ex) For N2 gas, the mean speed at room T is ~ 475 m/s.
M RT 55 .
2
• The most probable speed (symbol c*) can be obtained from the location of the peak in the f(v)-v curve.
M c 2RT
*
M c RT
8 Mean speed
Root mean square speed
M c 3RT Most probable speed
• Note that the order of magnitudes of them is:
c c
c*
• When one molecule approaches another, each molecule feel some relative speeds.
• The relative mean speed ( ) can be also calculated from the Maxwell distribution of speeds.
• For the identical two molecules, crel
c crel 2
Typical mean direction of approach
• For two dissimilar molecules of masses mA and mB:
crel 8kT
B A
B A B
A m m
m μ m
m
m
1 1 1
where is called the reduced mass of the molecules.
• If two molecules are identical,
m m m
m μ m
B A
B A
2
1
M c RT m
kT m
kT
crel kT 8 2
8 2 8 2
2
8
• Next Reading:
8th Ed: p.752 ~ 760 9th Ed: p.751 ~ 758
