Post on 21-Jan-2016
The Kinetic Theory of Gases
Temperature as a measure of average kinetic energy of the particles
ObjectivesState the assumptions of the kinetic model of
an ideal gas.
State that temperature is a measure of the average random kinetic energy of the
molecules of an ideal gas.
Explain the macroscopic behavior of an ideal gas in terms of a molecular model.
Write & understand the equation of state.
The Kinetic Theory of Gases
• The properties of gases can be understood in terms of a simple but effective mechanical model. The gas consists of a very large number of molecules moving randomly about with a range of speeds and colliding with each other and the container walls.
The Kinetic Theory of GasesWe can make a model of this by making certain assumptions and seeing what these lead to. The basic assumptions of the kinetic theory of gases are:1. A gas consists of a very large number of molecules.2. Molecules moves with a range of speeds.3. The volume of the molecules is negligible compared with the
volume of the gas itself.4. The collisions of the molecules with each other and the
container walls are elastic (p and KE are conserved).5. Molecules exert no forces on each other or the container except
when in contact.6. The duration of the collisions is very small compared with the
time between collisions.7. The molecules obey Newton’s Laws of Mechanics.
The Kinetic Theory of Gases
Using these assumptions together with the laws of mechanics and the equation of state allows the
derivation of one of the most important formulae in physics, namely the Boltzmann equation:
.
The Kinetic Theory of Gases
Here, the speed v is defined by . So v is the square root of the average of the squares of the speeds of the molecules of the gas (the average of the velocity
vectors of all the molecules is zero since they move randomly in all directions). We call v the root mean
square speed or rms speed. (It must be realized that v is not the average speed of
the molecules – but it is numerically close to the average, so we are usually excused for calling v the
average molecular speed, even if it is not technically correct.)
The Kinetic Theory of Gases
Appearing in this equation is a new constant of physics, the Boltzmann constant k, the value of which is k=1.38x10-23 J K-1 (it is the ratio of the gas constant R
to the Avogadro constant). The meaning of this important relation is that the average kinetic energy
of molecules is proportional to absolute temperature, that is, the absolute temperature is a measure of the
average kinetic energy of the molecules of a substance.
ExampleFour molecules have speeds of 300 m s-1, 350 m
s-1, 380 m s-1, and 500 m s-1. Find the average speed and the root mean square speed.
ExampleIf the root mean square molecular speed is
doubled, what is the new temperature?
Molecular explanation of pressure
• The pressure of a gas originates from the collisions of the molecules with the walls of its container. At every collision, each molecule has its momentum changed, and so a force acts from the wall onto the molecule. By Newton’s Third Law, the molecule exerts an equal and opposite force on the wall. The total force due to all the colliding molecules, divided by the area over which the force acts gives the pressure of the gas.
Molecular explanation of pressure
• Pressure is the normal force per unit area. The pressure in a gas results from the collisions of the gas molecules with the walls of its container (and not from collisions between molecules).
Molecular explanation of pressure
From the molecular point of view, we may identify two factors that affect the pressure of the gas. The first is the average molecular
speed (the higher the speed, the larger the change in momentum of the molecules and so the higher the force). The second factor is the frequency of collisions. The more frequent the collisions, the higher the pressure. Thus in providing molecular explanations for
pressure it is sufficient to remember that roughly. As an application of this, consider a gas that has been heated under constant volume. The molecules are moving faster on average
(increased temperature) and the frequency with which the collisions take place also increases (the time between collisions is reduced since molecules are moving faster). For both reasons, the
pressure increases.
ExampleA gas is compressed slowly by a piston. Explain why the temperature of the gas will stay the same.
Example
A gas is compressed rapidly by a piston. Explain why the temperature of the gas will increase.
Example
A gas expands isothermally. Explain from a molecular point of view why the pressure decreases.
The Equation of State
– Pressure– Volume– R=8.31 J K-1 mol-1– n = number of moles– T = temperature in
Kelvin
• A gas that obeys this law at all temperatures and pressures is said to be an ideal gas.
• Real gases obey this law only for a range of temperatures, pressures, and volumes.
Example
A gas is heated at constant pressure. Explain why the volume must increase as well.
Example
How many moles of gas are there in a gas of temperature 300 K, volume 0.02 m3 and pressure 2 x 105 Pa?