The microscopic definition unavoidably led to assumptions as to what matter is made out of, namely what is the microscopic composition of matter. For example, in applying the equation for entropy to the case of an ideal gas, we had to take into account the microscopic nature of the gas, in particular, that it is made out of an enormous collection of microscopic objects that we identified with atoms. The entire field of statistical mechanics was founded by Boltzmann in the late nineteenth century. As a historical aside, it is worth recording that it was in order to understand the concept of entropy from a microscopic point of view, that Boltzmann had postulated the existence of atoms well before their discovery in the twentieth century. In sum, the challenge posed by thermodynamics was the following: how can we reconcile ideas such as temperature, entropy and so on with the ideas of (Newtonian) mechanics? In particular, if any sample of matter that we observe in daily life is made out of an inordinately large number of atoms, approximately , how can we apply the laws of mechanics to this large collection of particles? Clearly, it is hopeless to try and describe how every single particle is moving, as this would involve specifying, at each instant, number of positions and velocities. So what is the way out of this impasse?

where is a constant which depends on charge, and is the Lennard-Jones (LJ) radius, and is shown in Figure 10.1. Note that there is a minimum value in the inter-atomic potential at a distance of from the atom. As long as the atoms are moving slowly, and are farther away from each other than distance , they can be treated as hard spheres of radius that behave as classical particles. For typical atoms and molecules the LJ-radius is around 3 to 5A (A= Angstrom =). For example, for the argon atom, the LJ radius is 3.5A, and is 5A for a large molecule such as propane. However, in some cases the LJ radius is not suitable for determining the effective classical size of an atom. For example the molecule has an LJ radius of 0.7A, and is too small a distance to be taken as the classical radius of the molecule. As long as the object being analyzed is at temperatures and densities that are not very high or very low, the atoms are not squeezed together closer than the distance of the LJ radius, and we can treat the atoms as classical billiard balls. However at very low temperatures and high densities, this is not true and the classical analysis needs to replaced by quantum mechanics. At very high temperatures, the inner structure of the atoms, composed as it is out of a nucleus and electrons, needs to be taken into account, and requires an analysis which goes beyond classical mechanics.

Recall that by equilibrium we mean that the gas has attained a state of maximum entropy, or equivalently, that there are no more changes of temperature and other state variable taking place. By the statement that the gas is at temperature , we mean that the gas in question is in contact with a heat bath which is at a temperature . The very fact that we have introduced the physical idea of temperature already implies that the gas is not an isolated system, but rather is part of a larger system which includes the heat bath and the object at a given temperature. How do we describe a gas, shown in Figure 10.2, composed out of particles, occupying a volume and at temperature ? There are simply too many particles to keep track of. To provide a mechanical description of the gas, we need to know the exact position and velocity of each and every particle, and which in general, is called a microstate of the system. A description of the microstate of any large object, containing about Avogardo's number of atoms, is in practice too difficult. And even more importantly, there is

(10.3) | |||

(10.4) |

In other words, in colliding off the piston, the particle's velocity changed from to , and hence the momentum imparted to the piston is

(10.5) |

(10.6) | |||

(10.7) |

Since the force on the piston is nothing but the rate at which momentum changes on the piston due to collisions of the gas atoms, we have

(10.8) | |||

(10.9) |

Recall pressure is defined to be force per unit area, and hence the pressure on the piston due to the gas is

From the ensemble point of view, the velocity of the atom is a random variable, and what the piston really experiences is the

The reason we have dropped the factor of in going from (10.12) to (10.13) is that we need to perform the average over only those particles which are heading

We finally have, from eq.(10.13), the following

The total energy, of the gas is solely composed of kinetic energy. Hence we have

From (10.13) and (10.15) and (10.17) we have

(10.18) |

Are the densities of the two gases the same on two sides of the piston, that is equal to ? The answer is yes, although to prove this is quite difficult. The intuitive proof that the two densities are equal is that if there was a difference in the densities, there would be a net ``osmotic'' pressure on the piston forcing it to move, and consequently the system would not be in equilibrium. Hence, in equilibrium, we have

(10.20) |

We see that the equation above is simply a statement that the average kinetic of the atoms in two gases which are in equilibrium is the same. Hence temperature is

Temperature is a measure of how fast, on the average, that the atoms of a gas are moving. At room temperature eV. The faster the atoms move, the hotter the temperature. The sensation of burning that we have on putting our hands into, say a fire, is because fast moving atoms from the fire impart high amounts of kinetic energy to our hands, causing atoms in our hand to move very fast and result in the sensation of burning. Hence, from eqns.(10.2) and (10.22)we have

(10.23) | |||

(10.24) |

Combining our results, from (10.16) and (10.22) we finally obtain the ideal gas law

The result above has the remarkable implication that no matter what the gas is composed of, for example, be it nitrogen, helium and so on, equal volumes of the various gases at the same pressure and temperature have the