Entropy

3.2 Entropy

The entropy is defined via its variation as (the absolute value of \(S\) will be discussed later):

\begin{equation*} \begin{split} dS =&\;\; \delta q_{rev} / T \quad \mbox{for reversible process} \\ dS \gt &\;\; \delta q_{rev} / T \quad \mbox{for irreversible process} \\ \end{split} \label{eq:dS_definition} \end{equation*}

(3.1)

So for the same heat transfer a larger increase of entropy is found when heating a cold compared to a warm bath. BUT we have a larger entropy in the warm bath.
Why is it impossible for an isolated system to find \(\Delta S_{total} \lt 0\)? We will discuss a system with warm subsystem 1 at \(T_1\) and cold subsystem 2 at \(T_2\). So heat \(\delta q\) will flow from subsystem 1 into subsystem 2, i.e.

\begin{equation*} \Delta S_{total} = \frac{\delta q}{T_2} - \frac{\delta q}{T_1} \gt 0 \quad \mbox{since} \quad T_1 \gt T_2 \label{eq:} \end{equation*}

(3.2)

The opposite sign \(\Delta S_{total} \lt 0\) would be against all experience: no heat will flow spontaneously from cold to hot!

The general problem about entropy deals with the fact that we need the equal sign (”=”) in Eq. (3.1) to do calculations. But since \(S\) is a state function one can find always an alternative reversible path for describing irreversible changes.
The first law is of course applied to express \(\delta q\) by expansion work and \(C_v\), respectively.