Page 208 - Physical Principles of Sedimentary Basin Analysis

190 Heat flow

The specific heat capacity at constant pressure is defined as

∂s

c p = T (6.326)
∂T p
which gives
∂s c p

= . (6.327)
∂T T
p
The thermal expansibility is defined as

1 ∂ν
α = (6.328)
ν ∂T p

and a Maxwell relation of thermodynamics gives

∂s ∂ν
=− (6.329)
∂p T ∂T p
and we have
∂s α

=−να =− . (6.330)
∂p
T
Recall that the specific volume is the inverse density, ν = 1/ . Inserting the partial deriva-
tives (6.327) and (6.330) into the differential (6.325) gives expression (6.321). These
relationships are a part of thermodynamics and they are thoroughly explained in any
standard text book on the subject.

Note 6.16 The temperature equation introduced in section 6.1 takes specific heat capacity
at constant volume, while the temperature equation above takes specific heat capacity at
constant pressure. Using thermodynamics it is possible to show that the difference between
these two heat capacities is
T α 2
c p − c v = (6.331)
γ
where α is the thermal expansibility and γ is the compressibility. This difference (6.331)
is normally of no practical importance. Typical parameters for a rock, α = 3 · 10 −5 K −1 ,
3
γ = 3 · 10 −10 Pa −1 , = 3 · 10 kg m −3 and T = 1000 K give that c p − c v = 1 ·
10 −3 kJ kg −1 K −1 , which is much less than a typical value c p ≈ 1kJ kg −1 K −1 .

Note 6.17 The temperature does not have to be the main variable in an equation for
energy conservation. Alternatives are, for instance, the specific entropy s and the specific