6.3: The Second Law of Thermodynamics
- Page ID
- 414057
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
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:
For any spontaneous process, the entropy of the universe is increasing.