6.1: Entropy
Let’s return to the definition of efficiency of a Carnot cycle and bring together eqs. 5.3.2 and 5.3.3 :
\[ \varepsilon = 1+\dfrac{Q_3}{Q_1} = 1-\dfrac{T_l}{T_h}. \nonumber \]
Simplifying this equality, we obtain:
\[ \dfrac{Q_3}{T_l} = -\dfrac{Q_1}{T_h}, \label{6.1.2} \]
or alternatively:
\[ \dfrac{Q_3}{T_l} + \dfrac{Q_1}{T_h} = 0. \label{6.1.3} \]
The left hand side of Equation \ref{6.1.3} contains the sum of two quantities around the Carnot cycle, each calculated as \(\dfrac{Q_{\mathrm{REV}}}{T}\), with \(Q_{\mathrm{REV}}\) being the heat exchanged at reversible conditions (recall that according to Definition: Carnot Cycle each transformation in a Carnot cycle is reversible). Equation \ref{6.1.2} can be generalized to a sequence of connected Carnot cycles joining more than two isotherms by taking the summation across different temperatures:
\[ \sum_i \dfrac{Q_{\mathrm{REV}}}{T_i} = 0, \label{6.1.4} \]
where the summation happens across a sequence of Carnot cycles that connects different temperatures. Eqs. \label{6.1.3} and \ref{6.1.4} show that for a Carnot cycle—or a series of connected Carnot cycles—there exists a conserved quantity obtained by dividing the heat associated with each reversible stage and the temperature at which such heat is exchanged. If a quantity is conserved around a cycle, it must be independent on the path, and therefore it is a state function. Looking at similar equations, Clausius introduced in 1865 a new state function in thermodynamics, which he decided to call entropy and indicate with the letter \(S\):
\[ S = \dfrac{Q_{\mathrm{REV}}}{T}. \nonumber \]
We can use the new state function to generalize Equation \ref{6.1.4} to any reversible cycle in a \(PV\)-diagram by using the rules of calculus. First, we will slice \(S\) into an infinitesimal quantity:
\[ dS = \dfrac{đQ_{\mathrm{REV}}}{T}, \nonumber \]
then we can extend the summation across temperatures of Equation \ref{6.1.4} to a sum over infinitesimal quantities—that is the integral—around the cycle:
\[ \oint dS = \oint \dfrac{đQ_{\mathrm{REV}}}{T} = 0. \nonumber \]