Relationships between partial derivatives Reminder to the chain

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

Recently Viewed Presentations

  • Swaps Drop/Balloon payments Bonds

    Swaps Drop/Balloon payments Bonds

    Commodity Swaps-Example 2. FBI needs 100,000 barrels of oil delivered 1 year from now and another 100,000 barrels 2 years from now . The one year forward price on a barrel of oil is 100 , and the two year...
  • FY 2018 YEAR-END TRAINING Schedule H  Capital Leases

    FY 2018 YEAR-END TRAINING Schedule H Capital Leases

    The ACO has two new leases . which . meet . the . requirement to be recorded as . capital . leases. The beginning fiscal year lease . principal balance . is $1,266,885. Additions of . $531,754 were made during...
  • EDEXCEL GCSE English Literature Walking-Talking Mock Exam Paper

    EDEXCEL GCSE English Literature Walking-Talking Mock Exam Paper

    "all mixed up together like bees in a hive" - Mr Birling. Explore how the theme of social responsibility important in the play. You . must. refer to the context f the play in your answer. (Total for Question 7...
  • 2.5: Model Direct Variation 2.6: Draw Scatter Plots & Best ...

    2.5: Model Direct Variation 2.6: Draw Scatter Plots & Best ...

    The table gives the systolic blood pressure ? of patients ? years old. Determine if a correlation exists. If it is a strong correlation, find the line of best fit.
  • Cax2002 - UDC

    Cax2002 - UDC

    La antigüedad: años 60 Ivan Sutherland, estudiante del MIT, realiza su tesis doctoral en la aplicación de los gráficos por computador al diseño en la ingeniería: el proyecto se llamó Sketchpad, y se considera el inicio de la industria del...
  • Parametric Shape Analysis via 3-Valued Logic

    Parametric Shape Analysis via 3-Valued Logic

    Admin. Compiler Project 50%. 4-4.5 practical exercises. Groups of 3. Final exam 50% . must pass
  • The United States of America Presented by Ms.

    The United States of America Presented by Ms.

    The United States of America Presented by Ms. Hedstrom Location Places to see: The Grand Canyon -Located in Northern Arizona Mount Rushmore - Located in South Dakota Location Capi
  • Macbeth - Grade

    Macbeth - Grade

    Represents the path Macbeth chose NOT to take, the path in which ambition does not lead to betrayal and murder. Fearful of the witches' prophecy that Banquo's heirs will seize the throne, Macbeth hires a group of murderers to kill...