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FIGURE $4-24$

$P-U-T$ surface of a substance that expands on freezing $($like water$).$

All the points on the surface represent equilibrium states. All states along the path of a quasi$-$equilibrium process lie on the $P-U-T$ surface since such a process must pass through equilibrium states. The single$-$phase regions appear as curved surfaces on the $P--T$ surface, and the two$-$phase regions as surfaces perpendicular to the $P-T$ plane. This is expected since the projections of two$-$phase regions on the $P-T$ plane are lines.

All the two$-$dimensional diagrams we have discussed so far are merely projections of this three$-$dimensional surface onto the appropriate planes. A $P-U$ diagram is just a projection of the $P-U-T$ surface on the $P-U$ plane, and a $T-U$ diagram is nothing more than the bird's$-$eye view of this surface. The $P-U-T$ surfaces present a great deal of information at once, but in a thermodynamic analysis it is more convenient to work with two$-$dimensional diagrams, such as the $P-U$ and $T-U$ diagrams.