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The variation of internal energy with volume

المؤلف:  Peter Atkins، Julio de Paula

المصدر:  ATKINS PHYSICAL CHEMISTRY

الجزء والصفحة:  ص104-105

2025-11-09

563

The variation of internal energy with volume

The quantity πT=(∂U/∂V) T, which represents how the internal energy changes as the volume of a system is changed isothermally, played a central role in the manipulation of the First Law, and in Further information 2.2we used the relation

This relation is called a thermodynamic equation of state because it is an expression for pressure in terms of a variety of thermodynamic properties of the system. We are now ready to derive it by using a Maxwell relation.

Justification 3.4

We obtain an expression for the coefficientπT by dividing both sides of eqn 3.43 by dV, imposing the constraint of constant temperature, which gives

Next, we introduce the two relations in eqn 3.45 and the definition of πTto obtain

The third Maxwell relation in Table 3.5 turns (∂S/∂V) T in to (∂p/∂T) V, which completes the proof of eqn 3.48.