Our model of an ideal gas consists of a number of particles moving with a given energy, as explained before. The internal energy U of a system is defined as the sum of the energies of its individual particles, while the temperature T is related to the average energy per particle.
Collisions are the mechanism that allows random exchanges of energy between the particles. All the degrees of freedom of the molecules of the gas have to be considered concerning the energy exchanges. For the simplest case of a monoatomic gas all the energy is translational kinetic energy, which is the one directly related to the temperature, as we shall see below.
The simulation shows an ideal gas with 150 particles moving randomly. The values of the total energy and the temperature are shown in two different windows, and can be changed with the arrows. In order to look at the microscopic effect of temperature, without changing the number of particles:
Let us now analyze the relation between pressure P and temperature T in an ideal gas when the volume and the number of particles are kept constant.
Check your conclusions by clicking here.
To change the temperature means to change the speed of the particles so when they impact the walls the force that the wall needs to change the momentum of the particles will change then the pressure P will vary.
Make three or four measurements of the pressure value for each temperature and get their average value. You will get a table similar to this one:
Pressure Temperature -------- ----------- 4 950 994 4 500 906 3 000 606 2 500 505 |
With this experience we will have tested that there is a proportionality between pressure P and temperature T, if we reduce the temperature to their half value the pressure will be reduce also to their half value.
P/T = Constant |
From this experience and that one in Section PRESSURE we are led to the conclusion that the following relationship holds for an ideal gas made up of N molecules
PV/T = constant
In a more quantitative way the experimentally found relationship reads
PV = NkT
where P is the pressure, V the volume, N the number of particles, k the Boltzmann's constant and T the temperature.
The pressure on the walls arises from the change in momentum of the particles colliding with the walls. A particle with x-component Px of momentum before the collision with the wall emerges from the collision with reversed sign in this component, so the change in momentum is
The particles that collide with the wall within a given time interval dt are those placed at a distance from the wall less than
Therefore, the total change of momentum of the gas is
The force exerted by the gas on the wall is obtained as the change in momentum per unit time, and divided by the area of the wall gives the value of the pressure. Taking into account that the x-component of the velocity is not the same for all the particles in the gas, the average value will be used
where the experimental relationship between P, V and T has been employed. From this expression we deduce that the average kinetic energy per particle corresponding to motion along the X-axis is kT/2. As all directions in space are equivalent, the energy corresponding to motion along the Y- and Z-axis must be the same, and finally we arrive at the conclusion that
We shall now carry out an experience to study the relation between the internal energy U and the temperature T in an ideal gas.