## 3. TEMPERATURE

Microscopic interpretation of temperature, and relationship with other thermodynamic quantities (P,U).
The microscopic interpretation of temperature makes easier the understanding of many phenomena involving the pressure in a gas and dilatation in solids.

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.

 EXPERIENCE: Microscopic interpretation of temperature 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:  Stop the simulation.  Select 'see velocities', which will give an idea of the speed of each particle  When the temperature is lowered the modulus of the velocity vectors decreases.  By pressing "On" in the "Control Panel" the simulation goes on. 03.Q1. QUESTION: Select one of the following options: In an ideal gas, the temperature is directly related to the particle speed. Increasing the particle speed results in a larger energy of the system and a lower temperature. Temperature increases when particle speed decreases
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.

 EXPERIENCE: Searching for the relation between P and T. Modify the value of the temperature with the arrows at the sides of the label TEMPERATURE.  Write down the values of pressure and temperature, and try to find some relationship between these quantities. 03.Q2. QUESTION: Looking at the results obtained in the experience choose one of the following options: P/T = constant. P*T = constant. P does not depend on T Check your conclusions by clicking here.
THE IDEAL GAS LAW

From this experience and that one in Section 2. 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 arrÙve at the conclusion that

The temperature of an ideal gas is a measure of the average translational kinetic energy of its molecules.
We shall now carry out an experience to study the relation between the internal energy U and the temperature T in an ideal gas.

 EXPERIENCE: Internal energy and Temperature. Select 100 particles  Record the temperature and internal energy values  Lower the number of particles to 50  Modify the temperature until the initial energy value is reached. 03.Q3. QUESTION: Looking at the results obtained in this experience, carried out with a monoatomic gas, choose one of the following options For a given value of the internal energy, a larger number of particles means a higher temperature. A larger number of particles implies a larger internal energy. For a given value of the internal energy, halving the number of particles results in doubling the temperature.
Internal Energy.

 Index 1.Introduction 2.Pressure 4.Internal Energy 5.Heat 6.First Law 7.First Law 8.Entropy 9.Velocity Distribution 10.Specific Heat