## Gibbs Relation

8 From the chain rule we can express du dt du\ ds ds ) dt v

dvp dt

substituting into Clausius-Duhem inequality of Eq. 6.66

we obtain ds

but the second principle must be satisfied for all possible evolution and in particular the one for which D = 0, dVp = 0 and V9 = 0 for any value of dt thus the coefficient of dt is zero or

thus and Eq. 7.3 can be rewritten as

du ds

— = » — + dt dt dt and if we adopt the differential notation, we obtain Gibbs relation

For fluid, the Gibbs relation takes the form du du = »ds - pdv; and » = ( —

ds du

where p is the thermodynamic pressure; and the thermodynamic tension conjugate to the specific volume v is —p, just as 0 is conjugate to s.

siorr Tj as

v s ii From the caloric equation of state, Eq. 7.1, and the the definitions of Eq. 7.2 it follows that the temperature and the thermodynamic tensions are functions of the thermodynamic state:

we assume the first one to be invertible s = s(0, v) (7.12)

and substitute this into Eq. 7.1 to obtain an alternative form of the caloric equation of state with corresponding thermal equations of state (obtained by simple substitution).

12 The thermal equations of state resemble stress-strain relations, but some caution is necessary in interpreting the tesnisons as stresses and the Vj as strains.

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