Kinetic Theory of Gases I
Ideal Gas
The number of molecules is large
The average separation between
molecules is large
Molecules moves randomly
Molecules obeys Newton’s Law
Molecules collide elastically with each
other and with the wall
Consists of identical molecules
The Ideal Gas Law
PV  nRT
in K
n: the number of moles in the ideal gas
N
n
NA
total number
of molecules
Avogadro’s number: the number of
atoms, molecules, etc, in a mole of
a substance: NA=6.02 x 1023/mol.
R: the Gas Constant: R = 8.31 J/mol · K
Pressure and Temperature
Pressure: Results from collisions of molecules
on the surface
Pressure:
F
P
A
Force:
dp
F
dt
Force
Area
Rate of momentum
given to the surface
Momentum: momentum given by each collision
times the number of collisions in time dt
Only molecules moving toward the surface hit
the surface. Assuming the surface is normal to
the x axis, half the molecules of speed vx move
toward the surface.
Only those close enough to the surface hit it
in time dt, those within the distance vxdt
The number of collisions hitting an area A in
time dt is
1 N 
 A  vx  dt
2 V 
Average density
The momentum given by each collision to
the surface
2mvx
Momentum in time dt:
1 N 
dp  2mv x  
 A v x dt
2 V 
Force:
dp
1 N 
F
 2mvx  
 A v x


dt
2 V
Pressure:
F N 2
P   mv x
A V
Not all molecules have the same v x  average v2x
N 2
P  mv x
V

2
vx
1 2 1 2
2
2
= v  v x  v y  vz
3
3
2
vx
1 2 1 2
= v  vrms
3
3

vrms is the root-mean-square speed
2
vrms  v 
2
vx
2
+ vy
2
+ vz
3
1N
2 N 1 2 
2
Pressure: P 
mv 
mv




3V
3 V 2
Average Translational Kinetic Energy:
1 2 1 2
K  mv  mvrms
2
2
2 N
P   K
3 V
Pressure:
From
2
PV   N  K
3
Temperature:
and
PV  nRT
3 nRT 3
K 
  k BT
2 N
2
R
23
Boltzmann constant: k B 
 1.38  10
J/K
NA
1
2
PV


N

mv
From
rms
3
and
N
PV  nRT 
RT
NA
vrms
3RT

M
Avogadro’s number
N  nNA
Molar mass
M  mN A
Pressure  Density x Kinetic Energy
Temperature  Kinetic Energy
Internal Energy
For monatomic gas: the internal energy = sum
of the kinetic energy of all molecules:
Eint
3
3
 N  K  nN A  k BT  nRT
2
2
Eint
3
 nRT  T
2
HRW 16P (5th ed.). Consider a given mass of an ideal gas. Compare
curves representing constant-pressure, constant volume, and isothermal
processes on (a) a p-V diagram, (b) a p-T diagram, and (c) a V-T
diagram. (d) How do these curves depend on the mass of gas?
p
p
constant pressure
isothermal V
isothermal
constant pressure
constant volume
isothermal
constant volume
constant volume
V
(d)
constant pressure
T
T
mass of gasn
pV  nRT  n
p nR

n
Constant volume
T V
V
Constant temperature
PV  nRT
Constant pressure
nR

n
T
p
HRW 18P (5th ed.). A sample of an ideal gas is taken through the
cyclic process abca shown in the figure; at point a, T = 200 K. (a)
How many moles of gas are in the sample? What are (b) the
temperature of the gas at point b, (c) the temperature of the gas at
point c, and (d) the net heat added to the gas during the cycle?
PV  nRT
(b)
pBVB
3
TB 
 1.8  10 K
nR
(c)
pC VC
TC 
 6.0  10 2 K
nR
(d) Cyclic process  ∆Eint = 0
Eint  Q  W
Pressure (kN/m2)
(a)
p AVA
n
 1.5 mol.
RTA
b
7.5
2.5
a
c
1.0
3.0
Volume (m3)
Q = W = Enclosed Area= 0.5 x 2m2 x 5x103Pa = 5.0 x 103 J
HRW 30E (5th ed.).(a) Compute the root-mean-square speed of a
nitrogen molecule at 20.0 ˚C. At what temperatures will the rootmean-square speed be (b) half that value and (c) twice that value?
3RT
vrms 
M
(a)
vrms
3RT
38.31 J/mol  K  293 K 


 511 m/s
-3
M
28.0 10 kg/mol
(b) Since
2
vrms

2
vrms
vrms  T
for 0.5 vrms
for 2 vrms
T 

T
2
T  0.5 T  73.3K = -200 C
2
3
T  2 T 1.17 10 K = 899 C
HRW 34E (5th ed.). What is the average translational kinetic energy
of nitrogen molecules at 1600K, (a) in joules and (b) in electronvolts?
(a)


3
3
23
K  kBT  1.38  10
J/K 1600K
2
2
 3.31 10 20 J
(b) 1 eV = 1.60 x 10-19 J
3.31 10 20 J
K
 0.21 eV
19
1.60 10 J/eV
3
K   kB T
2
Kinetic Theory of Gases II
Mean Free Path
Molecules collide elastically with other
molecules
Mean Free Path l: average distance between
two consecutive collisions
l
1
2
2d N / V
the bigger the molecules
the more collisions
the more molecules
the more collisions
Q  cm  T
Eint  Q  W
Molar Specific Heat
3
Eint  nRT
2
Definition:
For constant volume:
Q  nCV T
For constant pressure:
Q  nCp T
The 1st Law of Thermodynamics:
Eint
3
 nRT  Q  W
2
(Monatomic)
Constant Volume
W   PdV  0
3
nRT  nCV T
2
3
CV  R
2
Eint  nCV T
3
nRT  Q  W
2
(Monatomic)
Q  nCV T
Eint
3
 nRT
2
Constant
3
nRT  Q  W
2
Pressure (Monatomic)
W  PV  nRT
3
nRT  nC p T  nRT
2
CV  Cp  R
 
Cp
CV
5
Cp  R
2
5
 
3
Q  nCp T
1st Law
Adiabatic Process
dEint  dQ  dW
Ideal Gas Law
pV  nRT
(Q=0)
dEint  dW   pdV
 nCV dT
Eint  nCV T
Cp  CV  R
 pdV 

pdV  Vdp  nRdT  nR
 nCV 
Divide by pV:
C p  CV dV
dV dp
dV
  (1   )

 
V
p
V
 CV  V
 
Cp
CV
dV dp
dV

 (1   )
V
p
V
dp
dV

0
p
V


ln p  lnV  ln( pV ) = const.

pV = const.
nRT 
(
)V = const.
V
TV
 1
= const.
Ideal Gas Law
pV  nRT
Equipartition of Energy
The internal energy of non-monatomic
molecules includes also vibrational and
rotational energies besides the
translational energy.
Each degree of freedom has associated with
1
it an energy of k B T per molecules.
2
Monatomic Gases
3 translational degrees of freedom:
Eint
3
3
 kBT nN A  nRT
2
2
1 dEint 3
CV  
 R
n dT
2
Eint  nCV T
Eint  nCV T
Diatomic Gases
3 translational degrees of freedom
2 rotational degrees of freedom
2 vibrational degrees of freedom
HOWEVER, different DOFs require different
temperatures to excite. At room temperature,
only the first two kinds are excited:
Eint
5
 nRT
2
5
CV  R
2
HRW 63P (5th ed.). Let 20.9 J of heat be added to a particular ideal
gas. As a result, its volume changes from 50.0 cm3 to 100 cm3 while
the pressure remains constant at 1.00 atm. (a) By how much did the
internal energy of the gas change? If the quantity of gas present is
2.00x10-3 mol, find the molar specific heat at (b) constant pressure
and (c) constant volume.
(a) Constant pressure:
W = p∆V
Eint  Q  W  Q  pV



 20.9  1.0  105 Pa 100 cm3  50cm 3 106 m 3 /cm3

 15.9 J
Eint  Q  W
HRW 63P (5th ed.). Let 20.9 J of heat be added to a particular ideal
gas. As a result, its volume changes from 50.0 cm3 to 100 cm3 while
the pressure remains constant at 1.00 atm. (a) By how much did the
internal energy of the gas change? If the quantity of gas present is
2.00x10-3 mol, find the molar specific heat at (b) constant pressure
and (c) constant volume.
Q  nCp T
Q
Q
R Q


(b) C p 
nT n pV / nR p V

(c)
pV  nRT
8.31 J/mol K  20.9 J 
1.0 10
5

Pa 50  10
6
3
cm
 34.4 J/mol K

CV  Cp  R
 34.4 J/mol K  8.31 J/mol  K = 26.1 J/mol  K
HRW 81P (5th ed.). An ideal gas experiences an adiabatic
compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x
105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or
polyatomic? (b) What is its final temperature? (c) How many moles
of gas are present? (d) What is the total translational kinetic energy
per mole before and after the compression? (e) What is the ratio of
the squares of the rms speeds before and after the compression?
(a) Adiabatic



piVi  p f Vf
pi / p f  Vf / Vi
ln  pi / p f

5


ln V / V  3
f
Monatomic
 
i



Cp
CV
pV = const.
HRW 81P (5th ed.). An ideal gas experiences an adiabatic
compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x
105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or
polyatomic? (b) What is its final temperature? (c) How many moles
of gas are present? (d) What is the total translational kinetic energy
per mole before and after the compression? (e) What is the ratio of
the squares of the rms speeds before and after the compression?
(b)
piVi p f Vf

Ti
Tf
Tf 
pf Vf
piVi

T 


1.0 atm1.0  10 6 L 
1.0  10 5 atm 1.0  103 L 273 K 
i
 2.7  10 4 K
pV  nRT
HRW 81P (5th ed.). An ideal gas experiences an adiabatic
compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x
105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or
polyatomic? (b) What is its final temperature? (c) How many moles
of gas are present? (d) What is the total translational kinetic energy
per mole before and after the compression? (e) What is the ratio of
the squares of the rms speeds before and after the compression?
(c)
piVi
4
n
 4.5  10 mol.
RTi
(d)
For N/n = 1
3
3
Ki  RTi  3.4  10 J
2
3
5
K f  RTf  3.4  10 J
2
(Pay attention to the units)
3 nRT
K 
2 N
pV  nRT
HRW 81P (5th ed.). An ideal gas experiences an adiabatic
compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x
105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or
polyatomic? (b) What is its final temperature? (c) How many moles
of gas are present? (d) What is the total translational kinetic energy
per mole before and after the compression? (e) What is the ratio of
the squares of the rms speeds before and after the compression?
(e)
2
vrms,i
2
vrms,
f
Ti

 0.01
Tf
1 2
3
K  mvrms   kBT
2
2