The fundamentals of thermodynamics can be encapsulated in four postulates. We list them all here, but in the class we will go in a stepwise manner.
Postulate 1: Existence of Equilibrium States
There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy $U$, the volume $V$, and the mole numbers $N_{1}$, $N_{2}$, … , $N_{r}$, of the chemical components.
Do you think this set of variables make sense from a physical angle?
Postulate 2: Definition of Entropy
There exists a function (called the entropy, $S$) of the extensive parameters $(U, V, N_{1}, N_{2},\dots)$ of any composite system, defined for all equilibrium states such that the values assumed by the extensive parameters in the absence of an internal constraint are those that maximize $S$ over the manifold of constrained equilibrium states.
Notice that unlike traditional developments, we are not starting with the internal energy or $U$ as a state function. We are defining the entropy as a state function instead. Think about why!
Postulate 3: Properties of Entropy
The entropy of a composite system is additive over the constituent subsystems. $S$ is continuous and differentiable, and a monotonically increasing homogenous function of the energy of the first order. $$S = \sum_{j} S_{j}$$ $$S(\lambda U, \lambda V, \lambda N) = \lambda S(U,V,N)$$ $$\left( \frac{\partial S}{\partial U} \right)_{V, N} > 0$$
Postulate 4: Nernst Postulate
The entropy of any system vanishes at $T=0$ K, or when $$\left( \frac{\partial U}{\partial S} \right)_{V, N} = 0$$
This implies that the entropy has a unique zero.