We have introduced Thermodynamics using a statistical, quantum-based approach and have not relied on postulates. However, historically Thermodynamics was analyzed in terms of four separate unverified statements known as the Laws of Thermodynamics. We have more tools to verify the statements, though, and you may be surprised at the simplicity of the laws.
The Zeroth Law supposes that we have three systems in which the first two are each in thermal equilibrium with the third. Then the Law claims that the first two are likewise in thermal equilibrium with each other. Recall that the equilibrium condition was that the temperatures be equal. Then we have: If τ1 = τ3 and τ2 = τ3 then τ1 = τ2. It isn't hard to see why this is so.
The First Law has many formulations. Historically, the Law is stated as such: the work done in taking an isolated system from one state to another is independent of the path taken. We know from previous study of mechanics that energy behaves the same way. It turns out that this work can be called heat, and therefore a sleeker definition of the First Law is: Heat is a form of energy. The path independence follows from this simple statement.
The Second Law has an overwhelming number of formulations. We shall present two here, one that makes sense given the statistical origins we've focused on, and one that has historical value and will be useful later when we deal with engines.
Statistically, we say that: if a closed system is not in equilibrium, then the most probable future is that the entropy will increase with each passing bit of time, and will not decrease. The more foreign formulation, useful later (see Heat, Work, and Engines), known as the Kelvin-Planck formulation, is: it is impossible for any cyclic process to occur whose sole effect is the extraction of heat from any reservoir and the performance of an equivalent amount of work. The popularized version of the second law looks more like the first explanation and has been recently challenged by considerations of the physics of black holes.
Qualitatively, the Third Law claims that as a system approaches absolute zero, or T = 0, it becomes increasingly ordered, and thus exhibits a low entropy. Strictly, we say that: the entropy of a system approaches a constant value as the temperature approaches zero. This constant value is near or at zero, usually. Consider a system with a non-degenerate (i.e. having a multiplicity function value of one) ground state. Then the entropy of that state is zero. As the temperature decreases, the system becomes more and more likely to be found in the ground state, as we shall see in Statistics and Partition Function. Thus the entropy will approach a small, near zero value.
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