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In classical statistical mechanics, the number of microstates is actually infinite, since the properties of classical systems are continuous. For example, a microstate of a classical ideal gas is specified by the positions and momenta of all the atoms, which range continuously over the real numbers. Therefore, a method of "classifying" the microstates must be invented if we are to define Ω. In the case of the ideal gas, we count two states of an atom as the "same" state if their positions and momenta are within δx and δp of each other. Since the values of δx and δp can be chosen quite arbitrarily, the entropy is not uniquely defined; as before, it is still defined only up to an additive constant. This grouping of microstates is called coarse graining , and has its counterpart in the choice of basis states in quantum mechanics.
This ambiguity is partly resolved with quantum mechanics. The quantum state of a system can be expressed as a superposition of basis states, which are typically chosen to be eigenstates of the unperturbed Hamiltonian. In quantum statistical mechanics, Ω refers to the number of basis states consistent with the thermodynamic properties. Since the set of basis states is generally countable, we can define Ω.
However the choice of the set of basic states is still somehow arbitrary. It corresponds to the choice of coarse graining of microstates, to the distinct macrostates in classical physics.
This leads to Nernst's theorem, sometimes referred to as the third law of thermodynamics, which states that the entropy of a system at zero absolute temperature is a well-defined constant. This is due to the fact that a system at zero temperature exists in its ground state, so that its entropy is determined by the degeneracy of the ground state. Many systems, such as crystal lattices, have a unique ground state, and therefore have zero entropy at absolute zero (since ln(1) = 0).
In real experiments, it is quite difficult to measure the entropy of a system. The techniques for doing so are based on the thermodynamic definition of the entropy, and require extremely careful calorimetry.
For simplicity, we will examine a mechanical system, whose thermodynamic state may be specified by its volume V and pressure P. In order to measure the entropy of a specific state, we must first measure the heat capacity at constant volume and at constant pressure (denoted CV and CP respectively), for a successive set of states intermediate between a reference state and the desired state. The heat capacities are related to the entropy S and the temperature T by
where the X subscript refers to either constant volume or constant pressure. This may be integrated numerically to obtain a change in entropy:
We can thus obtain the entropy of any state (P,V) with respect to a reference state (P0,V0). The exact formula depends on our choice of intermediate states. For example, if the reference state has the same pressure as the final state,
In addition, if the path between the reference and final states lies across any first order phase transition, the latent heat associated with the transition must be taken into account.
The entropy of the reference state must be determined independently. Ideally, one chooses a reference state at an extremely high temperature, at which the system exists as a gas. The entropy in such a state would be that of a classical ideal gas plus contributions from molecular rotations and vibrations, which may be determined spectroscopically. Choosing a low temperature reference state is sometimes problematic since the entropy at low temperatures may behave in unexpected ways. For instance, a calculation of the entropy of ice by the latter method, assuming no entropy at zero temperature, falls short of the value obtained with a high-temperature reference state by 3.41 J/K/mol. This is due to the fact that the molecular crystal lattice of ice exhibits geometrical frustration , and thus possesses a non-vanishing "zero-point" entropy at arbitrarily low temperatures.
Main article: adiabatic process
The following equation can be used to graph entropy on a P-V diagram:
There are a pair of caveats:
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