2.7 The Fundamental Thermodynamic Relation

Take-home message: Remember this equation!

The first law for infinitesimal changes says ${\rm d}E={}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q+{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}W$. Since it is obviously true for reversible changes, we have ${\rm d}E={}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}+{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}W^{\rm rev}$. We have already found expressions for reversible work for a variety of systems; now we have one for reversible heat transfer too: ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}=T{\rm d}S$. So we can put these together to form an expression for ${\rm d}E$ which only involves functions of state. For a hydrodynamic system, for instance,

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle {\rm d}E=T{\rm d}S-P{\rm d}V.$ }}$

This is called the fundamental thermodynamic relation. It involves only functions of state, so it is true for all changes, not just reversible ones.

For other systems, $-P{\rm d}V$ is replaced by the appropriate expression for ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}W^{\rm rev}$, eg $\Gamma {\rm d}l$ for a stretched string.

The significance of this equation should become clear as the course continues: it is one of the most important half-dozen equations we will meet.

References

• (Mandl 4.1)
• Bowley and Sánchez 2.5
• Adkins 5.4
• Zemansky 8.14