Chapter 17 (2)Kinetic Theory of Gases
Ideal Gas Law
PVR
nT
The law
is approximately true for real gases at lowpressure and density.
An ideal gas is a gas that obeys this law.
R = 0.08206 L.atm/(mol.K)
PV Diagram - Isotherms
V
nRTP
Example 17 - 3
( )(0.0821L atm /[mol K] 2
1at
1mol )( )
22.4L
73K
m
T
PV
nR
What is the volume of 1 mol of an ideal gasat T = 0oC and P = 1 atm?
0 273T
Example 17 – 4 (1)
21
2
1 2
1
nRPV P V
TT
Initial Volume of gas V1 = 2 L Initial Temperature T1 = 30o CInitial Pressure P1 = 1 atmFinal Volume of gas V2 = 1.5 L Final Temperature T2 = 60o CFinal Pressure PP22 = ?= ?
11
1
22
2
VP
T
TP
V
Example 17 – 4 (2)
Remember: must use absolute temperaturein gas law
11
1
22
2
2L1atm
(30 273
((60 273)K)( )( ) 1.47atm
( )(1.5L))K
VTPP
VT
Molar Mass
The mass per mole of a substance is called its molar mass (units g/mol)
Example – Molar mass of CO2
= molar mass(C) + molar mass(O2)= 12 g/mol + (16 g/mol + 16 g/mol)= 44 g/mol
Example 17 - 6
100 g of CO2 occupies a volume of 55 L at P = 1 atm. What is its temperature?
Number of moles: n = m / M = 100 g / 44g/mol = 2.27 mol
But T = PV / n R, so T = 295 K
17-5The Kinetic Theory of Gases
Kinetic Theory of Gases (1)
The kinetic theory of gases is a theory that tries to derive the properties of an ideal gas from first principles, in particular, from Newton’s laws.
Kinetic Theory of Gases (2)
In a time t, on average, ½ the moleculesin volume (vx t)A are moving right.Therefore, thenumber ofmolecules thathit the wall is
A
1
2xNv A
V
t
Kinetic Theory of Gases (3)
The total change in the x component of the momentumof these molecules is
A2
1
2
x
xv t A
p mv
NV
Kinetic Theory of Gases (4)
The force exerted on the wall is F = t / t and the pressure P = F/A
A
2x
FP
ANmv
V
Kinetic Theory of Gases (5)
From the kinetic theoryone gets
A
2
2122 x
xPV N mv
N mv
Kinetic Theory of Gases (6)
But since the molecules do not all have thesame speed we must replace
2xv
2avxv
By the its average value
Kinetic Theory of Gases (7)
We now compare the law found experimentally
PV NkT
212 av
2 xPV N mvwith the formula from the kinetic theory
and deduce that 21 12 2avxmv kT
Kinetic Theory of Gases (8)
But, given that the average squared speed is
2 2 2 2 2
av av av av av3x y z xv v v v v
we find that
2 31av 2 2avK mv kT
What is Temperature ?
2 312 2av
Tmv k
From the kinetic theory of gases we havediscovered that temperature is a measureof the average kinetic energy of a molecule:
Root Mean Square Speed
2 312 2av
Tmv kFrom
2avrms
3 3kT Rv
Tv
m M
we can calculate the rms speed of a molecule
Example 17 – 7 (1)
The molar mass of O2 is about 32 g/molThe molar mass of H2 is about 2 g /mol
What is the rms speed of O2 and H2 at T = 300 K?
rms
3v
M
RT
Example 17 – 7 (2)
rms speed for O2 is
rms
3
3
3( )(300K)
32
8.31J
1
/[mol
0
48
kg/mol
s
K
3
]
m/
vT
M
R
Example 17 – 7 (3)
then the rms speed for H2 is given by
2
2
Orms 2
rms 2 H
(H )
(O )
Mv
v M
Since
rms
3v
M
RT
Example 17 – 7 (4)
So the rms speed for H2 is
2
2
Orms 2 rms 2
H
(H ) (O )
32g/mol(483m
1930m/
/s)2g/mol
s
Mv v
M
