Lecture 6
Kinetic theory of ideal gases.
The kinetic theory of gases (also known as kinetic-molecular theory) is a law that
explains the behavior of a hypothetical ideal gas. According to this theory, gases are
made up of tiny particles in random, straight line motion. They move rapidly and
continuously and make collisions with each other and the walls. This was the first theory
to describe gas pressure in terms of collisions with the walls of the container, rather than
from static forces that push the molecules apart. Kinetic theory also explains how the
different sizes of the particles in a gas can give them different, individual speeds.
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Kinetic theory makes many assumptions in order to explain the reasons gases act the way they
do. According to kinetic theory:
Gases consist of particles in constant, random motion. They continue in a straight line until they
collide with something—usually each other or the walls of their container.
Particles are point masses with no volume. The particles are so small compared to the space
between them, that we do not consider their size in ideal gases.
No molecular forces are at work. This means that there is no attraction or repulsion between the
particles.
Gas pressure is due to the molecules colliding with the walls of the container. All of these
collisions are perfectly elastic, meaning that there is no change in energy of either the particles or
the wall upon collision. No energy is lost or gained from collisions.
The time it takes to collide is negligible compared with the time between collisions.
The kinetic energy of a gas is a measure of its Kelvin temperature. Individual gas molecules have
different speeds, but the temperature and kinetic energy of the gas refer to the average of these
speeds.
The average kinetic energy of a gas particle is directly proportional to the temperature. An
increase in temperature increases the speed in which the gas molecules move.
All gases at a given temperature have the same average kinetic energy.
Lighter gas molecules move faster than heavier molecules.
An ideal gas can be characterized by three state variables: absolute pressure (P),
volume (V), and absolute temperature (T). The relationship between them may be
deduced from kinetic theory and is called the
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n = number of moles
R = universal gas constant = 8.3145 J/mol K
N = number of molecules
k = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 eV/K
k = R/NA
NA = Avogadro's number = 6.0221 x 1023 /mol
The ideal gas law can be viewed as arising from the kinetic pressure of gas
molecules colliding with the walls of a container in accordance with Newton's
laws. But there is also a statistical element in the determination of the average
kinetic energy of those molecules. The temperature is taken to be proportional to
this average kinetic energy; this invokes the idea of kinetic temperature. One mole
of an ideal gas at STP occupies 22.4 liters
Molecular Constants
In the kinetic theory of gases, there are certain constants which constrain the
ceaseless molecular activity.
A given volume V of any ideal gas will have the same number of
molecules. The mass of the gas will then be proportional to the
molecular mass. A convenient standard quantity is the mole, the
mass of gas in grams equal to the molecular mass in amu.
Avogadro's number is the number of molecules in a mole of any
molecular substance.
The average translational kinetic energy of any kind of molecule
in an ideal gas is given by
Ideal Gas Law with Constraints
For the purpose of calculations, it is convenient to place the ideal gas law in the
form:
where the subscripts i and f refer to the initial and final states of some process. If
the temperature is constrained to be constant, this becomes:
which is referred to as Boyle's Law.
If the pressure is constant, then the ideal gas law takes the form
which has been historically called Charles' Law. It is appropriate for experiments
performed in the presence of a constant atmospheric pressure.
All the possible states of an ideal gas can be represented by a PvT surface as
illustrated below. The behavior when any one of the three state variables is held
constant is also shown.
Pressure and kinetic energy
Pressure is explained by kinetic theory as arising from the force exerted by
molecules or atoms impacting on the walls of a container. Consider a gas
of N molecules, each of mass m, enclosed in a cuboidal container of volume V=L3.
When a gas molecule collides with the wall of the container perpendicular to
the x coordinate axis and bounces off in the opposite direction with the same speed
(an elastic collision), then the momentum lost by the particle and gained by the
wall is:
where vx is the x-component of the initial velocity of the particle.
The particle impacts one specific side wall once every
(where
L is the distance between opposite walls).
The force due to this particle is:
The total force on the wall is
where the bar denotes an average over the N particles. Since the assumption is that
the particles move in random directions, we will have to conclude that if we divide
the velocity vectors of all particles in three mutually perpendicular directions, the
average value along each direction must be equal. (This does not mean that each
particle always travel in 45 degrees to the coordinate axes.)
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We can rewrite the force as
This force is exerted on an area L2. Therefore the pressure of the gas is
where V=L3 is the volume of the box. The ratio n=N/V is the number density of the
gas (the mass density ρ=nm is less convenient for theoretical derivations on atomic
level). Usingn, we can rewrite the pressure as
This is a first non-trivial result of the kinetic theory because it relates pressure,
a macroscopic property, to the average (translational) kinetic energy per
molecule
which is a microscopic property.
Temperature and kinetic energy
From the ideal gas law
where
is the Boltzmann constant and
and from the result
the absolute temperature,
, we have
and, thus,
Stefan-Boltzmann Law
When you turn on an electric heater you may observe that it has changed its
color to red or orange. At high temperatures all objects emit visible radiation. But
when the heater is set to work at low power, you will not observed any visible
radiation, however you may check that it is working by simple holding your hand
near the burner. In both cases the energy is emitted by the burner, and it is called
radiation.
As the temperature of an object increases, more radiation is emitted
each second
The Stefan-Boltzmann law, states that the total radiated power per unit surface area
of a black body in unit time (known variously as the black-body irradiance, energy
flux density, radiant flux, or the emissive power), j* is directly proportional to the
fourth power of the black body's absolute temperature T:
The irradiance j* has dimensions of power density (energy per time per square
distance), and the SI units of measure are J/s∙m2 or W/m2.
The constant of proportionality σ, called the Stefan-Boltzmann constant is nonfundamental in the sense that it derives from other known constants of nature. The
value of the constant is:
(W/m2),
where
k is Boltzmann’s constant (J/K);
c is speed of light (m/s);
h is Planck’s constant (J/Hz).
Wien's displacement law
Wien's displacement law is a law that states that there is an inverse relationship
between the wavelength of the peak of the emission of a black body and its
temperature. A black body is an object that absorbs all electromagnetic
radiation that falls onto it. No radiation passes through it and none is reflected.
Despite the name, black bodies are not actually black as they radiate energy as well
since they have a temperature larger than 0 K. The law states:
λmax = b/T,
(1)
where
λmax is the peak wavelength (nm),
T is the temperature of the blackbody (K) and
b is Wien's displacement constant, 2.898∙106 (rounded) (nm∙K).
Wien's law states that the hotter an object is, the shorter the wavelength at
which it will emit most of its radiation. For example, the surface temperature of
the sun is on average 5780 K. Using Wien's law, this temperature corresponds to a
peak emission at a wavelength of 500 nm. Due to a temperature gradient in the
surface boundary layer and local differences the spectrum widens to white visible
light. Due to the Rayleigh scattering (see light scattering) of blue light by the
atmosphere this white light is separated somewhat, resulting in a blue sky and a
yellow sun.