The_Cambridge_Handbook_of_Physics_Formulas

112 Thermodynamics
5.4 Kinetic theory
Monatomic gas
Pressure p = 1 3 nm〈c2 〉 (5.77)
Equation of
state of an ideal
gas
pV = NkT (5.78)
p pressure
n number density = N/V
m particle mass
〈c 2 〉 mean squared particle
velocity
V
k
N
T
volume
Boltzmann constant
number of particles
temperature
Internal energy U = 3 2 NkT = N 2 m〈c2 〉 (5.79) U internal energy
Heat capacities
Entropy
(Sackur–
Tetrode
equation) a
C V = 3 Nk
2
(5.80)
C p = C V +Nk= 5 Nk
2
(5.81)
γ = C p
= 5 C V 3
[ (mkT ) ]
3/2
S = Nkln
2π¯h 2 e 5/2 V N
(5.82)
(5.83)
C V heat capacity, constant V
C p heat capacity, constant p
γ ratio of heat capacities
S entropy
¯h = (Planck constant)/(2π)
e =2.71828...
a For the uncondensed gas. The factor
Broglie wavelength, λ T , approximately equals n −1/3
Q .
( mkT
2π¯h 2 ) 3/2
is the quantum concentration of the particles, nQ . Their thermal de
Maxwell–Boltzmann distribution a
Particle speed
distribution
Particle energy
distribution
( m
pr(c)dc =
2πkT
) 3/2
exp
( −mc
2
2kT
)
4πc 2 dc
(5.84)
( )
pr(E)dE = 2E1/2 −E
π 1/2 (kT) exp dE (5.85)
3/2 kT
pr
m
k
T
c
E
probability density
particle mass
Boltzmann constant
temperature
particle speed
particle kinetic
energy (= mc 2 /2)
( ) 1/2
Mean speed 8kT
〈c〉 =
(5.86) 〈c〉 mean speed
πm
( ) 1/2 ( ) 1/2
rms speed 3kT 3π c rms root mean squared
c rms = = 〈c〉 (5.87) speed
m 8
Most probable
speed
( ) 1/2 2kT
( π
) 1/2
ĉ = = 〈c〉 (5.88) ĉ most probable speed
m 4
a Probability density functions normalised so that ∫ ∞
0 pr(x)dx =1.