Phase space (reminder)

1. Phase space (reminder) Statistics involves the counting of states, and the state of a classical particle is completely specified by the measurement of its position and momentum. If we know the six quantities then we know its state. It is often convenient in statistics to imagine a six-dimensional space composed of the six position and momentum coordinates. It is conventionally called "phase space". The counting tasks can then be visualized in a geometrical framework where each point in phase space corresponds to a particular position and momentum. That is, each point in phase space represents a unique state of the particle. The state of a system of particles corresponds to a certain distribution of points in phase space. The counting of the number of states available to a particle amounts to determining the available volume in phase space. One might preclude that for a continuous phase space, any finite volume would contain an infinite number of states. But the uncertainty principle tells us that we cannot simultaneously know both the position and momentum, so we cannot really say that a particle is at a mathematical point in phase space. So when we contemplate an element of "volume" in phase space then the smallest "cell" in phase space which we can consider is constrained by the uncertainty principle to be

2. Entropy of an ideal gas We consider the high-temperature case of q≫N. The Sackur-Tetrode equation (1912) for a monatomic classical ideal gas Free expansion Free expansion is an irreversible process in which a gas expands into an insulated evacuated chamber. This happens quickly, so there is no heat transferred. There is no change in internal energy, so the temperature stays the same. No work is done, because the gas does not displace anything. The gas goes through states of no thermodynamic equilibrium before reaching its final state, which implies that one cannot define thermodynamic parameters as values of the gas as a whole. For example, the pressure changes locally from point to point, and the volume occupied by the gas (which is formed of particles) is not a well defined quantity. Entropy increase is not caused by the input of heat, but the phase space has changed!

3. Entropy of mixing Two cylinders (V = 1 liter each) are connected by a valve. In one of the cylinders – Hydrogen (H2) at P = 105 Pa, T = 200C , in another one – Helium (He) at P = 3·105 Pa, T=1000C. Find the entropy change after mixing and equilibrating. In general, for a gas of polyatomic molecules: The temperature after mixing: H2 : He:

4. For two different monoatomic gases at the same temperature, the entropy of mixing is: But what happens if two gases are of the same kind (indistinguishable)? Gibbs paradox Quantum-mechanical indistinguishability is important! (Even though this equation applies only in the low density limit, which is “classical” in the sense that the distinction between fermions and bosons disappear.) Here!

5. Reversible and Irreversible Processes A reversible process is a process that can be "reversed" by means of infinitesimal changes in some property of the system without entropy production or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic equilibrium throughout the entire process. In a reversible cycle, the system and its surroundings will be exactly the same after each cycle. An alternative definition of a reversible process is a process that, after it has taken place, can be reversed and causes no change in either the system or its surroundings. In an irreversible process, finite changes are made; therefore the system is not at equilibrium throughout the process. At the same point in an irreversible cycle, the system will be in the same state, but the surroundings are permanently changed after each cycle (new entropy is created). An isentropic process or isoentropic process is one in which one may assume that the process takes place from initiation to completion without an increase or decrease in the entropy of the system, i.e., the entropy of the system remains constant. It can be proven that any reversible adiabatic process is an isentropic process. What happens during the slow compression? Each wave function gets squeezed, energies of all levels increase (so do molecular energies), but molecules are not kicked to higher-energy levels. Thus the number of ways of arranging the molecules among the various energy levels still remains the same. Hence, W and Sdo not change.