Relationships between partial derivatives

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Relationships between partial derivatives
Reminder to the chain rule
composite function
Example
Internal energy of an ideal gas
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Lets calculate
with the help of the chain rule
Example
explicit
Now let us build a composite function with
and
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Composite functions are important in
thermodynamics
-Advantage of thermodynamic notation
Example
If you dont care about new Symbol for F(X,Y(X,Z))
wrong conclusion from
can be well distinguished
-Thermodynamic notation
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Apart from phase transitions thermodynamic
functions are analytic
See later consequences for physics (Maxwells
relations, e.g.)
Reminder
defined according to
function
Example
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What to do in case of functions of two
independent variables y(x,z)
keep one variable fixed (z, for instance)
Lets apply the chain rule to
Result from intuitive relation
Thermodynamic notation
yy(x,zconst.)
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Application of the new relation
Definition of isothermal compressibility
Remember the
Definition of the bulk modulus
With
or
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Application of
Isothermal compressibility
Volume coefficient of thermal expansion


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We learn
Useful results can be derived from general
mathematical relations
Are there more such mathematical relations
Consider the equation of state
or
(before we calculated derivative with respect to
P _at_ Tconst. now derivative with respect to T
_at_constant P)
Total derivative with respect to temperature
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Is a physical counterpart of the general
mathematical relation
Lets verify this relation with the help of an
example
Z
Y
Surface of a sphere
X
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for x,y,z ? 1st quadrant
X
cyclic
permutation
Y
Z
Physical application
Change in pressure caused by a change in
temperature