Modern Engineering Thermodynamics

730 CHAPTER 18: Introduction to Statistical Thermodynamics
Path of
a molecule
θ
θ
R
Spherical
shell of
radius R
2R cos θ
θ
θ
R
FIGURE 18.1
Motion of molecules inside a spherical shell of radius R.
In these equations, Δt is the time increment between successive molecular collisions. This can be calculated by
dividing the distance that a molecule travels between successive collisions by its velocity, or
Then, Eq. (18.7) becomes
Δt =
2R cos θ
V i
p = 1 2mV
4πR 2 ∑N i 2 cos θ
= m
i=1
2R cos θ 4πR 3 ∑N Vi 2 (18.8)
i=1
Since the volume of the spherical shell is V = 4 3 πR3 , we can then write Eq. (18.8) as
pV = p 4πR3
3
= m 3 ∑N Vi
2
i=1
= 1 3 NmV2 rms (18.9)
In this equation, the product Nm is equal to the total mass of gas in the shell, m T , and therefore, Eq. (18.9) can
be written as
pV = 1 3 m TV 2
rms
(18.10)
If we now limit our attention to gases that obey the ideal gas equation of state, then Eq. (18.10) becomes
where R is the specific gas constant given by
pV = 1 3 m TV 2
rms = m TRT (18.11)
where
R = R M = N ok
M
= k m
R = universal gas constant, 8314.3 J/(kgmole⋅K) or 1545.35 ft⋅lbf/(lbmole⋅R)
M = molecular mass (kg/kgmole or lbm/lbmole) of the gas
N o = Avogadro’s number 3 (or constant), 6.022 × 10 26 molecules/kgmole
(18.12)
3 In 1909, the French physicist Jean Perrin proposed naming this in honor of the Italian scientist Amedeo Avogadro, who, in 1811,
proposed that the volume of a gas at a given pressure and temperature is proportional to the number of atoms or molecules in the gas
regardless of the type of gas. Perrin won the 1926 Nobel Prize in Physics, mainly for his work in determining the value of the
Avogadro constant by several different methods. In 1971, the name was officially changed from Avogadro’s number to Avogadro constant
(N A ) when the “mole” was introduced as a new fundamental unit in the International System of Units (SI). The change in name from
the possessive form Avogadro’s to the nominative form Avogadro is common usage today for all physical constants.