Transcript 1 temperature and the gas law - lgh

Microscopic model
of ideal gas
1. The ideal gas law
Microscopic description of an ideal gas includes
the following assumptions:
(1) Ideal gas consists of N identical molecules ;
(2) The motions of molecules obey Newton’s law ;
(3) The average distance between molecules are
much larger than molecular size ;
(4) Collisions between molecules or between
molecules and the walls of container are elastic .
Equilibrium state is a state in which macroscopic
variables have definite values . It is a dynamic
equilibrium .
· To describe equilibrium state ,
P
we introduce state variables
( P, V )
( such as : P , V , T , m … ) .
· We can express an equilibrium
state in a PV diagram .
o
The equation ( ideal gas law ) that describes the
state of an ideal gas can be written as :
PV  NkT
kB=1.38×10–23 JK-1 is called Boltzmann’s constant
V
2. Pressure
P 289 - 292
Imagine the motion of molecules in a closed
container , as shown .
The constant , rapid drumbeat
of molecules exerts a steady
average force on the walls .
Let’s calculate how that force
is related to quantities which
specify the motion of the
molecules , namely , mass and
velocity .
Consider a single collision . A molecule of mass m
approaches the wall with velocity v , and bounces
off like a rubber ball .
The change in momentum of the molecule is :
P  2mvx
According to Newton’s second law the force on
the molecule is equal to the rate of change of
y
momentum .
P  2mvx

t
t

mv '
By Newton’s third law , the force

the molecule exerts on the wall is :
x
mv
2mvx
f 
z
t
Now suppose for a moment that the gas is
extremely rarefied , so a typical molecule travels
back and forth across the box many times before
colliding with other molecule .
y
A molecule travels back and forth
across the box once every t .

mv '
t  2Lx / vx

2
x
mv
2mvx mvx
so f 

z
t
Lx
summing over all the molecules in the box , we
obtain the total force on the wall .
N
mvxi2
F 
i 1 Lx
2
xi
N
mv
1
F 

Lx
i 1 Lx
N
 mv
i 1
2
xi
Since the motion of the molecules is random , we
have :
1 2
2
2
2
vxi  v yi  vzi  vi
3
2 1
so F 
3 Lx
N
1 2 2 K
mvi 

3 Lx
i 1 2
The pressure on the wall is the average force
divided by the area .
F 2 K
2 K 2 NK
P 


A 3 Lx L y Lz 3 V 3 V
3. Temperature and internal energy
The pressure can be written as :
P 297 - 298
2 NK
P
3 V
The ideal gas law is given by : PV  NkT
3
So , we have K  kT
2
It is average kinetic energy for monatomic gases .
The total energy ( called internal energy ) of a
monatomic gas is :
3
U  NkT
2
For ideal monatomic gases :
3
U  NkT
2
For ideal diatomic gases :
5
U  NkT
2
For ideal polyatomic gases :
6
U  NkT
2
For ideal gases : U  qNkT