16 ENGINEERING THERMODYNAMICS

dharm
M-therm/th2-1.pm5

According to Newton’s second law of motion the rate of change of ‘momentum is the force’.

If F 1 is the force due to one molecule, then

F 1 mul^1

2

=

Similarly, then force F 2 due to the impact of another molecule having velocity C 2 whose

components are u 2 , v 2 , w 2 is given by

F mu

(^2) l
=^22
Hence total force Fx on the face ABCD due to impact of N molecules is given by
FxN=++ml(uu 12 222 ......u)
Since the pressure (p) is the force per unit area, hence pressure exerted on the wall ABCD is
given by
p
F
l
m
l
x== 23 x (uu 12 + + 222 ......uN)
Similarly, if py and pz represent the pressures on other faces which are perpendicular to the
Y and Z axis respectively, we have
p m
l
y=++ 3 (vv 12 222 ......vN)
and p
m
l
zN=++ 3 (ww^12222 ......w)
Since pressure exerted by the gas is the same in all directions, i.e., px = py = pz the average
pressure p of the gas is given by
p
pppx y z

++
3
or p
m
l
=++++++ ++ 3 [(uvw u v w 12 12 12 ) ( 22 22 22222 ) ......(u v wNN N)]
But Cuvw 12 =++() 12 12 12
Cuvw 22 =++() 22 22 22 and so on
l^3 = V = volume of gas (m^3 )
∴ p
m
v
=^1 CCC+++CN
3 1
2
2
2
3
(^22 ...... )
or p=^13 mvNC^2 ...(2.1)
where C^21 CC CN CN
2
2
2
3
22
= +++
F
HG
I
KJ
...... known as mean square velocity
or C= CCC^1 +++N CN
2
2
2
3
(^22) ......
where C is called the root mean square velocity of the molecules and equal to the square root of
the mean of square of velocities of individual molecules which is evidently not the same as mean of
velocities of different molecules
ie C CCC C
mean N
L.., = +++...... N
NM
O
QP
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