# Relationships between partial derivatives Reminder to the chain Relationships between partial derivatives Reminder to the chain rule composite function: F( u, v, ....) u ( x , y,...), v( x , y,...) ,... F( x , y,...) F(u ( x , y,...), v( x , y,...),...) 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 U (T ) n u 0 n c V T U ( P , V ) n u 0 c V PV R T ( P, V ) PV nR F( x , y,...) F(u ( x , y,...), v( x , y,...),...) Lets calculate F( x , y,...) x with the help of the chain rule F( x, y,...) F(u, v,...) u ( x, y,...) F(u, v,...) v( x, y,...) ... x u x v x Example: 2 F( x , y,...) x y 2 3/ 2 sin xy F( x, y,...) 3 explicit: x 2 y 2 2x sin xy x 2 y 2 x 2

2 2 Now let us build a composite function with: u ( x , y) x y and F(u , v) u 3/ 2 sin v F(u , v) 3 1 / 2 u sin v u 2 F(u, v) u 3 / 2 cos v v 3/ 2 y cos xy v( x , y) xy u ( x, y) 2 x x v( x , y) y x F( x , y) F(u , v) u ( x, y) F(u , v) v( x , y) x u x v x 3/ 2 F( x, y) 3 2 2 2 2 u ( xx, y) y 2x sinxy y cos F(u , v) 3 F(u, vx) 3y/ 2 v( xxy , y) 2 x v u sin u cos v y y x

2 x u 2 v x Composite functions are important in thermodynamics -Advantage of thermodynamic notation: Example: F(X, Z) F(X, Y(X, Z)) If you dont care about new Symbol for F(X,Y(X,Z)) wrong conclusion from F F F Y X X Y X F Y 0 Y X can be well distinguished F F F Y -Thermodynamic notation: X X Y X Z Y X Z Apart from phase transitions thermodynamic functions are analytic F( x , y) F( x , y) y x x y 2 F( x , y) 2 F( x , y) yx xy See later consequences for physics (Maxwells relations, e.g.) Inverse functions and their derivatives

Reminder: function y( x ) inverse function x ( y) defined according to y( x ( y)) y 1 ( x 1) y 1 1 y xy xy y 1 x 1 1 y 1 y y( x ( y)) 1 y x ( y) x ( y) 1 1 y (1 y) y y 1 y Example: function y( x ) 10 X 8 Y 6 4 2 0 0 2 4 6 X Y 8 10 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 y( x ( y, z), z) y dy( x ( y), z) dx 1 dx

dy Result from intuitive relation: y x 1 x y Thermodynamic notation: y=y(x,z=const.) y( x , z) is inverse to x ( y, z) if y( x ( y, z), z) y Y X 1 X Z Y Z 10 6 Numerical example 8 4 4 2 0 0 2 4 6 8 10 2 0 0 2 X 4 6 8 X 10 6

4 dX/dY 6 x 2 1 0 0 1 2 X 8 4 2 0 0 3 dY/dX*dX/dY dY/dX Y 6 2 4 6 Y 8 10 2 0 0 2 4 Y 6 8 3

Application of the new relation Y X 1 X Z Y Z Definition of isothermal compressibility T Remember the P B T V V T Definition of the bulk modulus V With P T 1 V V P T P 1 V T V T T B T 1 or BT 1 V T 1 / B T Application of 2 F( x, y) 2 F( x, y) yx xy Isothermal compressibility:

1 V V P T T Volume coefficient of thermal expansion: V V V T P T V V V T P 1 V V T P ( V 2 V 2 V (VVV) T ) V T TP TP PT P PT PT P T P T T V V = T T P T P T V V T V T P

T T = V V V V P T P T V T V V V P V P P T T We learn: Useful results can be derived from general mathematical relations Are there more such mathematical relations Consider the equation of state: P P( V, T ) or V V ( P, T ) P P( V(P, T), T) For P const. P(V(P, T), T) const. (before we calculated derivative with respect to P @ T=const. now derivative with respect to T @constant P) Total derivative with respect to temperature P V P 0 V T T P T V P V T P T

0 V T T P P V T V P V 1 T P V P V T 1 V T T P P V Is a physical counterpart of the general mathematical relation: P V T 1 V T T P P V X Y Z 1 Y Z Z X X Y Lets verify this relation with the help of an example Z X=0 plane z=0 plane y=0 plane z z y Y X x 2 y 2 z 2 R 2 y x x Surface of a sphere 2 2 2

x y z R 2 x R y z 2 x y y x y z R 2 y2 z2 2 2 2 z z y y z x R 2 x 2 z2 2 2 2 z R x y X Y 2 y R x z for x,y,z 1st quadrant cyclic 2 permutation Z x x z z

x y R 2 x 2 y2 y z x x y z x y z y z z x x y 1 Physical application: Change in pressure caused by a change in temperature P V T 1 V T T P P V P P V V T T V 1 P V P T V T T P T V P V 1 V BT V V T P