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

2. Let’s calculatewith 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 -Thermodynamic notation: can be well distinguished

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

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

7. Numerical example

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

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

10. We learn: Useful results can be derived from general mathematical relations Are there more such mathematical relations Consider the equation of state:or For Total derivative with respect to temperature ( before we calculated derivative with respect to P @ T=const. now derivative with respect to T @constant P )

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

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