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2
 P x N  S  m x
 (1.5.4)
t  V
According to Newton's second law, this equals the average force exerted
by the wall area S on the molecules; from the third law, this is the negative

of the force exerted on the wall by the molecules. Finally, pressure p is
force per unit area, and we obtain
2
F N  m x
p   (1.5.5)
S V
In fact, v is not now the same for all molecules. But we could have

x
sorted them into groups having the same v within each group, and added
x
up the resulting contributions to the pressure. The net effect of this is
2
2
simply to replace v in Eq. (20-5) by the average value of v which we
x
x
2
2
denote by v . Furthermore v , is related simply to the speeds of the
x
x
molecules. The speed v (magnitude of velocity) of any molecule is related
to the velocity components v , v and v by
y
x
z
2
2
2
2
v  v  v  v (1.5.6)
x y z
This may be averaged over all molecules which move with different
speeds v , v ... v :
N
1
2
2
2
2
2
v  v  v  v rms (1.5.7)
y
x
z
Where v rms is root-mean-square speed and it is equal to

2
2
v  v  ... v 2
v  1 2 N (1.5.8)
rms
N
v  v ...  v
(N.B. the average or mean speed is equal to v  1 2 N )
N
since the x-, y-, and z-directions are all equivalent,

2 2 2
v  v  v (1.5.9)
z
x
y
Hence
2 1 2
v  v rms (1.5.10)
x
3
and eq. (20-5) becomes

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