Modern Engineering Thermodynamics

736 CHAPTER 18: Introduction to Statistical Thermodynamics
Substituting Eq. (18.19) into Eq. (18.23) and carrying out the integration gives
where
NðV 1 ! V 2 Þ
N
= erfðx 2 Þ − erfðx 1 Þ − p 2 ffiffiffi ðx 2 e −x2 2 − x1 e −x2 1 Þ (18.24)
π
pffiffiffi
Z x
x 1 = V 1 /V mp , x 2 = V 2 /V mp , and erfðxÞ = error function of x = 2/ π e − x2 dx
Representative values for the error function can be found in Table 18.2. Note that erf(0) = 0 and erf(∞) = 1.
0
3
V mp = 395 m/s
f(V) ×1000
2
1
V avg = 445 m/s
V rms = 483 m/s
0 0 100 200 300 400 500 600 700 800 900 1000
Velocity (m/s)
FIGURE 18.5
The Maxwell velocity distribution function f(V ) for oxygen (O 2 ) at 300 K as defined by Eq. (18.19).
Table 18.2 Values of the Error Function
x
erf(x)
0.0 0.0
0.1 0.1125
0.2 0.2227
0.3 0.3286
0.4 0.4284
0.5 0.5205
0.6 0.6039
0.7 0.6778
0.8 0.7421
0.9 0.7969
1.0 0.8427
1.2 0.9103
1.4 0.9523
1.6 0.9764
1.8 0.9891
2.0 0.9953
2.2 0.9981
2.4 0.9993
2.6 0.9998
2.8 0.9999
∞ 1.0
Note: For all x, erfðxÞ = p 2 ffiffi π
x − x3
3ð1!Þ + x5
5ð2!Þ − x7
7ð3!Þ + … , and exp − x 2 = 1 − x 2 /1! + x 4 /2! − x 6 /3! + x 8 /4! − …