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First, the missing derivative
can be found from the relation:

(38) 
which is obtained from the equality of the mixed derivatives in
Eq.(20), written as:

(39) 
Then

(40) 
First adiabatic exponent:

(41) 
This relation is obtained by combining Eq.(20) with the identity

(42) 
The adiabatic sound speed
is then obtained as

(43) 
Third adiabatic exponent:

(44) 
This relation is obtained by combining Eq.(20) with the identity

(45) 
and then using Eq.(38).
Adiabatic temperature gradient:

(46) 
since

(47) 
Adiabatic energy changes:

(48) 
or

(49) 
We define the coefficients and through the
relation

(50) 
Entropy change at constant density:

(51) 
This relation is obtained from the equality of the mixed derivatives in
Eq.(20) together with Eq.(44).
Entropy change at constant pressure:

(52) 
This relation is obtained from the equality of the mixed derivatives in
Eq.(21) together with Eq.(46).
Specific heat at constant density:

(53) 
To derive the specific heat at constant pressure, we start from the relation

(54) 
from which we get

(55) 
Using Eqs.(44) and (53), we obtain

(56) 
Now

(57) 
or

(58) 
hence

(59) 
and finally

(60) 
or

(61) 
Using Eqs.(27), (44), (52), (56),
we finally obtain the relation for the specific heat at constant pressure:

(62) 
Alternatively, can be obtained from Eq.(32)

(63) 
or from

(64) 
once and are known (see below).
We can now express the thermodynamic coefficients provided by CO5BOLD
in terms of , , , and
:

(65) 

(66) 

(67) 

(68) 

(69) 

(70) 
We consider again Eq.(39), replacing by

(71) 
so

(72) 
The requirement that the mixed derivatives must be equal then yields

(73) 
or

(74) 
Finally,

(75) 
Comparison with Eq.(69) implies

(76) 
Similarly, replacing by

(77) 
in Eq.(39), we get

(78) 
and the requirement that the mixed derivatives must be equal then yields

(79) 
or

(80) 
or

(81) 
Since

(82) 
we finally obtain, using Eqs.(29), (65) and (70),

(83) 
and

(84) 
The isothermal sound speed
is then obtained as

(85) 
Next: 2.3.5 Ideal gas with
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Previous: 2.3.3 CO5BOLD equation of
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