Lecture Notes in Advanced Thermodynamics

2.4.2 Examining the constitutive equations
motivation
– T (e, v) and p(e, v) functions are often given as empirical approximations
– it is useful to check if they fulfil the propertys of thermostatics
– sometimes the propertys can be used to make restrictions to the parameters
of the model
Condition of the potential property of entropy
– if the entropy function is twice differentiable then the mixed second
partial derivatives must be equal
∂ 2 s(e, v)
∂v∂e
= ∂2 s(e, v)
∂e∂v
– using property (B 1 ):
( )
∂
1
∂v ∣ = ∂ ( )
p(e, v)
e
T (e, v) ∂e∣ v
T (e, v)
(25)
– this condition is useful to check the T and p functions against the (B 1 )
property, without calculating the entropy function
– if (25) is not valid, then there exists no s(e, v) function, if the condition
fulfils, then s exists and the gas described by T and p functions is called
entropic
Condition of the increasing property of entropy
– property (B 2 ) is simple, it requires only
∂s
∂e∣ > 0
v
– if the property (B 1 ) is substituted, a very simple condition is obtained:
T (e, v) > 0 (26)
therefore the range of the temperature function must be positive
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