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\begin{document}
\section{Ideal gas}
The pressure of an \crossout{real} ideal gas rises with temperature at fixed
volume.\cacographynote*[right]{check units!} For one mole we have
\begin{equation}
  pV = \crossout[scribble]{nT} RT,
\end{equation}
where $R$ is the \crossout[double]{Boltzmann} gas constant. Doubling the
temperature \crossout[line]{halves} doubles the pressure.\cacographynote*{see lecture 3}

\section{Work done}
Expanding \underline{slowly} at constant temperature, the gas does the work
\[
  \boxed{W = nRT \ln\frac{V_2}{V_1}}
\]
which is \crossout[scribble]{negative} positive when the gas expands.\cacographynote[right]{sign!}
Note that \crossout{heat and work are the same} heat flows in to keep $T$ fixed.
\end{document}