Relationships between partial derivatives

1. You have to introduce a new symbol for this function, also the physical meaning can be the same Relationships between partial derivatives Reminder to the chain rule composite function: Example: Internal energy of an ideal gas

2. Let’s calculate with the help of the chain rule Example: explicit: Now let us build a composite function with: and

3. Composite functions are important in thermodynamics -Advantage of thermodynamic notation: Example: If you don’t care about new Symbol for F(X,Y(X,Z)) wrong conclusion from can be well distinguished -Thermodynamic notation:

4. Inverse functions and their derivatives inverse function function Apart from phase transitions thermodynamic functions are analytic See later consequences for physics (Maxwell’s relations, e.g.) Reminder: defined according to function Example:

6. if is inverse to What to do in case of functions of two independent variables y(x,z) keep one variable fixed (z, for instance) Let’s apply the chain rule to Result from intuitive relation: Thermodynamic notation: y=y(x,z=const.)

7. Numerical example

8. Application of the new relation Definition of isothermal compressibility Remember the Definition of the bulk modulus With or

9. Application of Isothermal compressibility: Volume coefficient of thermal expansion: = =

10. For 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 @ T=const. now derivative with respect to T @constant P) Total derivative with respect to temperature

11. z=0 plane y=0 plane X=0 plane z y z x x y Is a physical counterpart of the general mathematical relation: Let’s verify this relation with the help of an example Z Y Surface of a sphere X

12. for x,y,z  1st quadrant X cyclic permutation Y Z Change in pressure caused by a change in temperature Physical application: