1.2 The First Law for Small Changes

Take-home message: Heat and work are not functions of state

In mathematics, if $f$ is a function, ${\rm d}f$ is the notation used for an infinitesimal change in $f$. Internal energy is a function (of the other state variables) so we can use ${\rm d}E$ for the change in $E$ between two nearly identical states. During this infinitesimal change, tiny amounts of heat have been added to, and work done on, the system. These are not changes in anything, though, so we use a different notation for them, ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q$ and ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}W$:

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle {\rm d}... ...f\symbol{'040}}\llap{d}Q+{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}W $ }}$

Finite changes in state variables are written with a $\Delta$, but finite amounts of heat and work are just written $Q$ and $W$, as in $\Delta E=Q+W$

More on notation here.

References

• Mandl 1.3
• Bowley and Sánchez 1.5
• Adkins 3.4
• Zemansky 4.6