Transcript 14 The Ideal Gas Law and Kinetic Theory

C H A P T E R 14
The Ideal Gas Law and
Kinetic Theory
14.1 The Mole, Avogadro's
Number, and Molecular
Mass
Atomic Mass Unit, U
By international agreement, the reference element is chosen to be
the most abundant type of carbon, called carbon-12, and its
atomic mass is defined to be exactly twelve atomic mass units, or
12 u.
Molecular Mass
The molecular mass of a molecule is the sum of the atomic
masses of its atoms.
For instance, hydrogen and oxygen have atomic masses of
1.007 94 u and 15.9994 u, respectively.
The molecular mass of a water molecule (H2O) is:
2(1.007 94 u) + 15.9994 u = 18.0153 u.
Avogadro's Number NA
The number of atoms per mole is known as Avogadro's number
NA, after the Italian scientist Amedeo Avogadro (1776–1856):
Number of Moles, n
The number of moles n contained in any sample is the number
of particles N in the sample divided by the number of particles
per mole NA (Avogadro's number):
The number of moles contained in a sample can also be found from
its mass.
14.2 The Ideal Gas Law
An ideal gas is an idealized model for real gases that have
sufficiently low densities.
The Ideal Gas Law
An ideal gas is an idealized model for real gases that have
sufficiently low densities.
The condition of low density means that the molecules of the
gas are so far apart that they do not interact (except during
collisions that are effectively elastic).
The Ideal Gas Law
An ideal gas is an idealized model for real gases that have
sufficiently low densities.
The condition of low density means that the molecules of the
gas are so far apart that they do not interact (except during
collisions that are effectively elastic).
The ideal gas law expresses the relationship between the
absolute pressure (P), the Kelvin temperature (T), the volume
(V), and the number of moles (n) of the gas.
PV  nRT
Where R is the universal gas constant. R = 8.31 J/(mol · K).
The Ideal Gas Law
The constant term R/NA is referred to as Boltzmann's
constant, in honor of the Austrian physicist Ludwig
Boltzmann (1844–1906), and is represented by the symbol k:
PV = NkT
14.3 Kinetic Theory of Gases
Kinetic Theory
of Gases
The pressure that a gas exerts is
caused by the impact of its molecules
on the walls of the container.
Kinetic Theory
of Gases
The pressure that a gas exerts is
caused by the impact of its molecules
on the walls of the container.
It can be shown that the average translational kinetic
energy of a molecule of an ideal gas is given by,
where k is Boltzmann's constant and T is the Kelvin temperature.
Derivation of,
Consider a gas molecule colliding elastically with the right
wall of the container and rebounding from it.
The force on the molecule is obtained using Newton’s second
law as follows,
F 
P
,
t
The force on one of the molecule,
According to Newton's law of action–reaction, the force on
the wall is equal in magnitude to this value, but oppositely
directed.
The force exerted on the wall by one molecule,
mv 2

L
If N is the total number of molecules, since these particles
move randomly in three dimensions, one-third of them on the
average strike the right wall. Therefore, the total force is:
Vrms = root-mean-square velocity.
Pressure is force per unit area, so the pressure P acting on a wall
of area L2 is
Pressure is force per unit area, so the pressure P acting on a wall
of area L2 is
Since the volume of the box is V = L3, the equation above can
be written as,
PV = NkT
EXAMPLE 6 The Speed of
Molecules in Air
Air is primarily a mixture of nitrogen N2 (molecular mass = 28.0 u)
and oxygen O2 (molecular mass = 32.0 u). Assume that each behaves
as an ideal gas and determine the rms speed of the nitrogen and
oxygen molecules when the temperature of the air is 293 K.
The Internal Energy of a
Monatomic Ideal Gas