Modern Engineering Thermodynamics

742 CHAPTER 18: Introduction to Statistical Thermodynamics
EXAMPLE 18.5 (Continued )
Table 18.4 The Total Number of Combinations of Tossing Two Dice
Die 1 Die 2 Die 1 Die 2
1 1 4 1
1 2 4 2
1 3 4 3
1 4 4 4
1 5 4 5
1 6 4 6
2 1 5 1
2 2 5 2
2 3 5 3
2 4 5 4
2 5 5 5
2 6 5 6
3 1 6 1
3 2 6 2
3 3 6 3
3 4 6 4
3 5 6 5
3 6 6 6
Table 18.5 An Event-Frequency Table for Tossing Two Dice
Sum M of Die Values Number of Results Producing Sum M Ways of Obtaining Sum M
0 0
1 0
2 1 1 + 1
3 2 1 + 2, 2 + 1
4 3 2 + 2, 1 + 3, 3 + 1
5 4 2 + 3, 3 + 2, 1 + 5, 4 + 1
6 5 3 + 3, 4 + 2, 2 + 5, 5 + 1, 1 + 5
7 6 6 + 1, 1 + 6, 5 + 2, 2 + 5, 4 + 3, 3 + 4
8 5 4 + 4, 5 + 3, 3 + 6, 6 + 2, 2 + 6
9 4 4 + 5, 5 + 4, 3 + 7, 6 + 3
10 3 6+4,4+6,5+6
11 2 6 + 5, 5 + 6
12 1 6 + 6
This probability concept can be used to further investigate the collision frequency characteristics of the kinetic
theory model of ideal gases. Let the number of molecular collisions occurring in an ideal gas during some time
interval Δt = t 2 − t 1 be δN, and let N(t) be the number of molecules of the gas at time t that have not yet had a
collision. Then, δN = N(t 1 ) − N(t 2 ). Since we define the Δ symbol to be evaluated at time t 2 minus time t 1 , ΔN
becomes
ΔN = Nt ð 2 Þ− Nt ð 1 Þ = − δN
We postulate that δN will be proportional to the product of N(t), Δt, and the average molecular velocity V avg as
follows:
δN = − ΔN∝−NV avg ðΔtÞ