Page 13 - 4498

1 2
pV  N  (m v rms ) (1.5.11)
3
or

2 m v  2
pV  N (  rms ) (1.5.12)
3 2
Thus, we derived equation that is called the main equation of molecular-

kinetic theory of ideal gas, i.e. a relation between macroscopic quantities
(pressure p, volume V) and microscopic quantities : N - total number of
molecules, m- mass of molecule and v rms - root-mean-square speed .

It is important to notice that the value in round brackets represents
mean kinetic energy of molecule in the equation (5.2) and is denoted as

E  .Therefore,

2
pV  N    E (1.5.13)
3
N
So  n is a concentration of molecules(total number of molecules)
V

 1 3 
per unit of volume n   m
 3 
 m 
2
 np    E (1.5.16)
3
Thus, thepressure of ideal gas is equal to two-thirds of the mean
kinetic energy of molecules per unit of volume .
So, the main equation of molecular-kinetic theory of ideal gas states a

relation between macroscopic quantities and microscopic quantities and,
therefore, it's possible to calculate microscopic quantities by macroscopic
quantities. For example, rewrite equation in the next form

1 N  m 2
p  r rms (1.5.17)
3 V

N m
Where   is density of gas, therefore
V
1 2 3 p
p   r rms  r rms  (1.5.18)
3 
5
So at the normal atmospheric pressure p 10 Pa, the density of air is
kg
equal to  , 1 29 3 , root-mean-square speed of air molecules is equal
m

13