Lecture Notes in Advanced Thermodynamics

transformation of the state functions
– the (E, V, M) state space is simplified to (e, v)
– considering an extensive state function:
˜f(e, v) := f (e, v, 1) = f
specially:
( E
M , V M , 1 )
= Mf ( E
M , V M , 1)
M
Ẽ = e;
Ṽ = v
=
f(E, V, M)
M
– considering an intensive state function:
( E
˜g(e, v) := g (e, v, 1) = g
M , V )
M , 1 = g(E, V, M)
– the functions over the state space (e, v) are neither extensive not intensives
potential relation for specific quantities:
– let us divide (8) by M:
– using the definitions:
E
M = T S M − p V M + µ
e = T s − pv + µ (20)
Gibbs relation for specific quantities:
– let us substitute the specific quantities to (6):
d(eM) = T d(sM) − pd(vM) + µdM
– using the derivation rule:
Mde + edM = T vds + T sdM − pvdM − pMdv + µdM
dM(ε − T s + pv − µ) + M(de − T ds + pdv) = 0
– from (20) the first bracket equals zero, thus the Gibbs relation:
de = T ds − pdv (21)
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