6.3: The Second Law of Thermodynamics

Now we can consider an isolated system undergoing a cycle composed of an irreversible forward transformation (1 \(\rightarrow\) 2) and a reversible backward transformation (2 \(\rightarrow\) 1), as in Figure \(\PageIndex{1}\).

This cycle is similar to the cycle depicted in Figure \(\PageIndex{1}\) for the Joule’s expansion experiment. In this case, we have an intuitive understanding of the spontaneity of the irreversible expansion process, while the non-spontaneity of the backward compression. Since the cycle has at least one irreversible step, it is overall irreversible, and we can calculate:

\[ \oint \dfrac{đQ_{\mathrm{IRR}}}{T} = \int_1^2 \dfrac{đQ_{\mathrm{IRR}}}{T} + \int_2^1 \dfrac{đQ_{\mathrm{REV}}}{T}. \nonumber \]

We can then use Clausius inequality (Equation 6.2.4) to write:

\[\begin{equation} \begin{aligned} \int_1^2 \dfrac{đQ_{\mathrm{IRR}}}{T} + \int_2^1 \dfrac{đQ_{\mathrm{REV}}}{T} < 0, \end{aligned} \end{equation} \nonumber \]

which can be rearranged as:

\[ \underbrace{\int_1^2 \dfrac{đQ_{\mathrm{REV}}}{T}}_{\int_1^2 dS = \Delta S} > \underbrace{\int_1^2 \dfrac{đQ_{\mathrm{IRR}}}{T}}_{=0}, \label{6.3.3} \]

where we have used the fact that, for an isolated system (the universe), \(đQ_{\mathrm{IRR}}=0\). Equation \ref{6.3.3} can be rewritten as:

\[ \Delta S > 0, \label{6.3.4} \]

which proves that for any irreversible process in an isolated system, the entropy is increasing. Using Equation \ref{6.3.4} and considering that the only system that is truly isolated is the universe, we can write a concise statement for a new fundamental law of thermodynamics:

Definition: Second Law of Thermodynamics

For any spontaneous process, the entropy of the universe is increasing.

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