chapter 3: the math of thermodynamics

Winter 2013 Chem 254: Introductory Thermodynamics

Chapter 3: The Math of Thermodynamics 25

Chapter 3: The Math of Thermodynamics .................................................................................... 25

Derivatives of functions of a single variable ............................................................................. 25

Partial Derivatives ..................................................................................................................... 26

Total Differentials ..................................................................................................................... 28

Differential Forms ..................................................................................................................... 31

Integrals .................................................................................................................................... 32

Line Integrals ............................................................................................................................. 32

Exact vs Inexact Differential ...................................................................................................... 33

Definition of β and κ ................................................................................................................. 36

Dependence of U on T and V .................................................................................................... 37

Dependence of H on T and P .................................................................................................... 37

Derivations involving dH ........................................................................................................... 38

Relation between CP and CV (Exact) ......................................................................................... 39

Joule Thompson Experiment .................................................................................................... 39

Chapter 3: The Math of Thermodynamics

Derivatives of functions of a single variable

0

( ) ( )limh

df f x h f x

dx h

instantaneous slope

0

( ) ( )lim

2h

f x h f x h

h

(better if done on a computer with finite step size)

Often used derivatives

( )f x df

dx

ax 1aax axe axae

ln( )ax

1

x

ln( ) ln lnax a x


Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 26

sin( )ax cos( )a ax

cos( )ax sin( )a ax

Rules for Derivatives

( ) ( )d df dg

f x g xdx dx dx

( ) ( ) ( ) ( )d df dg

f x g x g x f xdx dx dx

Leibniz

( )df df du

u xdx du dx

Chain rule

Eg. 2 2sin( ) cos( )2d

x x xdx

2 22 3 2 3 (4 3)x x x xd

e e xdx

2

( ) ( )( )

( ) ( )

df dgg x f x

d f x dx dx

dx g x g x

Quotient Rule

Partial Derivatives

( , )f x y depends on multiple variables eg. ( , , )T x y z the thermometer reading in a

room

y

f

x

Pronounced: di f , di x at constant y

The change in function x -direction keeping y constant

Rules to take partial derivatives

same as in 1 dimension, treat remaining variable as constants

2 2

( , ) x y xf x y e

2 2

(2 1)x y x

y

fe x

x

2 2

(2 )x y x

x

fe y

y

2( , ) ln( 2 )f x y x xy



2

1(2 2 )

2y

fx y

x x xy

2

1(2 )

2x

fx

y x xy

Higher Partial Derivatives

For one variable: 2

2

( )d f x d df

dx dxdx

Eg. 2

2

2 2 2

1 2ln( ) 2

d dx x

dxdx x x

Partial derivatives: 2

2

yy y

f f

x xx

2

2

xx x

f f

y yy

Mixed Derivatives

x yy x

f f

x y y y

2 22 2 (2 2 )x xy x xy

y

fe e x y

x

2 22 2 (2 )x xy x xy

x

fe e x

y

2 2( )(2 2 )x xy

xy x

fe x y

y x y

2 22 2(2 2 )(2 ) (2)x xy x xye x y x e

2 2( )(2 )x xy

x y

fe x

x y x

2 22 2(2 2 )(2 ) (2)x xy x xye x y x e

Order of derivatives does not matter

2 2

x y

f f f

x y x y y x



In mathematics

one would typically not write out what’s kept constant

f

x

instead of

y

f

x

; ( , , )f x y z f

x

instead of

,y z

f

x

In Thermodynamics: ALWAYS write what is kept constant

,P n

V

T

, H

V

T

Total Differentials

when 2 or more variables are not constant

One dimension

2 3

2 3

2 3

1 1....

2! 3!

df d f d fdf dx dx dx

dx dx dx

dfdx

dx

for dx infinitesimal

( ) ( )x

dff x dx f x

dx

Two dimensions

....y x

f fdf dx dy

x y

( , ) ( , )f x x y y f x y

y x xyxy

f fdf x y

x y

in arbitrary direction x

y

, We can calculate the change in function f (for small x , y )



Curve ( , ) 0f x y

0y x

f fdf dx dy

x y

Calculate how much does y change if x x dx

y

x

f

xdy

dx f

y

y x

f fdx dy

x y

x

y

f

ydx

fdy

x

So 1

f

f

y

x x

y

keeping f constant

Consider function ( , , ) 0f x y z (surface in 3 dimensions)

z constant : , ,

0y z z x

f fdx dy

x y

,.

,

z x

z

z y

f

yx

fy

x

y constant: , ,

0y z y x

f fdx dz

x z

,.

,

z y

y

x y

f

xz

fx

z

x constant: ,,

0x yx z

f fdy dz

y z



,.

,

x y

x

z x

f

zy

z f

y

Cyclic Rule 1y xz

x z y

y x z

for ( , , ) 0f x y z

Proof:

,x z

y xz

f

yx z y

y x z

,y z

f

x

,x y

f

z

3

,

1

z y

f

x

,x y

f

z

,z x

f

y

1

From the cyclic rule

1

x y z

z

y z y

z x xx

y

Way to remember cyclic rule

y zx

z z y

x y x

Application in Thermodynamics

Van der Waals 2

1 2 2

2( )

nRT nP a nRT V nb an V

V nb V

V

P nR

T V nb

2 2 3( 1)( ) ( 2)T

PnRT V nb an V

V

T

V

P

: first derive ( , )V f P T then find T

V

P

(very hard)

How do we calculate T

V

P

?



We can use 1

T

T

V

PP

V

( 1)P T V

V V P

T P T

(cyclic rule)

V T

P P

T V

Calculate using above results

Differential Forms

General differential form (Pfaff): ( , ) ( , )df g x y dx h x y dy

Question: Is there a function ( , )F x y such that ( , )y

fg x y

x

and ( , )

x

fh x y

y

?

df dF final initialPath

df F F

If ( , )F x y associated with ( , ) ( , )g x y dx h x y dy then:

( , ) ( , )yx

g x y h x yy x

If this relationship holds: a function ( , )F x y does exist

If this relationship does not hold: ( , )F x y does not exist

( , ) ( , )df g x y dx h x y dy is an exact differential if and only if

( , ) ( , )yx

g x y h x yy x

, otherwise it’s an inexact differential

If df is an exact differential final initialPath

df F F

independent of path that runs between initial and final state

, , , ,dU dH dS dG dA : exact differential

,q w: inexact differential



Integrals

10

1

( ) lim ( )nb

a xi

f x dx f x dx

(Area under function)

( )if x x = area for each rectangle

Find function ( )F x such that ( )dF

f xdx

then: ( ) ( ) ( )b

af x dx F b F a

Strategy for solving integrals

1) - guess solution,

- verify the differentiation ( )dF

f xdx

?

- if not right tinker with constants

2) - Look them up (books) or use math programs

Examples in thermodynamics

Line Integrals

Consider differential ( , ) ( , )df g x y dx h x y dy consider paths

1( ) ( )y x P x and 2( ) ( )y x P x

11

( )P xdf I

22

( )P xdf I in general

1 2I I

( )f x ( )F x dF

dx

( 1)ax a 11

1

axa

1( 1)

( 1)

a aa x xa

b

x ln ln bb x x

b

x

axe 1 axea

1 axaea



1( ) ( )y x P x 1dP

dy dxdx

1 ( , ) ( , )Path

I g x y dx h x y dy df

2

1

1

1 1 1, ( ) , ( )x

x

dPI g x P x dx h x P x dx

dx

2

1

1

1 1 1, ( ) , ( )x

x

dPI g x P x h x P x dx

dx

2

11 ( )

x

xI c x dx reduced to 1d integral

Likewise:

2

1

2

2 2 2, ( ) , ( )x

x

dPI g x P x h x P x dx

dx

integrals are different

But: they are the same if and only df is an exact differential

If df exact differential then 1 ( , ) ( , )final initialI I x y I x y

Exact vs Inexact Differential

Inexact differential: ( , ) ( , )yx

d dg x y h x y

dy dx

Path

df gives results but depends on path

Real life example of exact differential: height differences on a mountain

It is clear the height difference is independent of how

you get there

How do you get contours for map? Measure the height differences between

neighbouring points

x ydh h dx h dy measure in small steps

( , )h x y defines height function of ,x y

Real life example of Inexact differential: shoveling snow



snow x yd d dx d dy

The amount of snow you shovel depends on

the path you take between points A and B

Exact differential:

22df xydx x dy

2(2 ) ( )x yxy xy x

2 2x x Therefore, is exact differential

df is independent of path

( )F x such that 2y

dFxy

dx

; 2

x

dFx

dy

2( , )F x y x y

1( ) :P x y x 1 ( )1

dP d y x

dx dx

2 ( ) :P x 2y x 2

2 ( )2

dP d y xx

dx dx

1 ( , ) ( , )Path

I g x y dx h x y dy

1

1

1 10

, ( ) , ( )x

x

dPg x P x dx h x P x dx

dx

1 1

2 2

10 02 1 3I x xdx x dx x dx

1

3

01x

1 1

2 2 3

20 02 2 4I x x dx x xdx x dx

1

4

01x (integral result is the same because exact differential)

2( , )F x y x y (1,1) (0,0) 1 0 1nI F F

Example of inexact differential:

2 2df x dx xydy 1( ) :P x y x ; 2 ( ) :P x 2y x



1 1

2 2

10 0

1 2 1 3I x dx x x dx x dx

1

3

01x

1 1

2 2 2 4

20 0

2 2 4I x dx x x xdx x x dx

1

3 5

0

1 4 1 4 171

3 5 3 5 15x x

2( ) (2 )x yx xyy x

?

0 2y not equal, differential is not exact, 1 2I I in general

Does ( , )F x y exist such that 2

y

Fx

x

and 2

x

Fxy

y

?

2

y

Fx

x

31

3F x c

2x

Fxy

y

2F xy c Not equal

Hence: for inexact differentials, line integrals can be calculated, but results depends on

path. True for ,q w in thermo or shoveling snow!

Summary of rules from math

1

z

z

y

x x

y

z x y

y y z

x z x

Exact differential: ( , ) ( , )df g x y dx h x y dy

( , ) ( , )yx

g x y f x yy x

Path

df is independent of path



Back to Thermodynamics!

Definition of β and κ

Define: 1

P

V

V T

in units K-1 0 usually

P

VV

T

: Volumetric thermal Expansion coefficient

1

T

V

V P

in Bar-1 0

T

VV

P

: Isothermal compressibility

For solids and liquids and are more or less constant

For gases and are not constant

Ideal gas: nRT

VP

1 1

P

V nR P

V T P nRT T

2

1 1( )

T

V nRT P

V P nRT PP

1 1

T

T

P

VV V

P

P

V T P

T

V

TP P V V

VT V T V

P

V

P

T

hard to measure obtain as

V T

P PdP dT dV

T V



1 dV

dTV

1f f f

Path i i

dVdP dT

V

1

lnf

i

VP T

V

lnf

i

VT P

V

Dependence of U on T and V

V T

U UdU dT dV

T V

dU q w : inexact differential

extq P dV assume constant volume

V VdU q q C dT

VU C T if V constant

0

lim VT

V

UC

T

V

V

UC

T

T

U PT P T P

V T

(will be derived later)

Dependence of H on T and P

H for constant pressure P PH q C T

0

lim PT

P P

H HC

T T

P

P

HC

T

(1 )

T P

H VT V V T

P T

(to be derived later too)



For Ideal gases

1

T ;

1

P

As expected

0T

U P PT P T P

V T T

1

(1 ) 0T P

H VT V V T

P T T

Derivations involving dH

( )dH dU d PV

dH dU VdP PdV

P T V T

H H U UdT dP dT dV VdP PdV

T P T V

P V

T T

H UC dT V dP C dT P dV

P V

Use T V

U PP T

V T

(relation stated before)

P V

T V

H PC dT V dP C dT T dV

P T

assume T is constant (some process)

T V

H PV dP T dV

P T

(T constant)

0limP

T V T

H P VV T

P T P

T V T

H P VV T

P T P

now use cyclic rule

T P

H VV T

P T



Relation between CP and CV (Exact)

P V

T T

H UC dT V dP C dT P dV

P V

Assume P constant

P V

V P

P VC C T

T T

P VC C T V

2

P VC C TV

(exact)

For ideal gas

P V

V P

P VC C T

T T

V V

P nRT nR

T T V V

;

P P

V nRT nR

T T P P

P V V

nR nR nRTC C T C nR

V P PV

P VC C nR (used and derived before)

Joule Thompson Experiment

JT

H

TC

P

(Joule Thompson Coefficient, definition)

P JT

T P H

H H TC C

P T P

1 2P P 0totalq I IIq q

H is constant during process

If H is constant:



2 1

2 1

2 1

lim JTP P

HH

T T TC

P P P

P JT

T

HC C V VT

P

P JTC C measures

1

P

V

V T

0JTC for ideal gases

1 1 2 2total I IIU U U w P V P V 0q

1 1 2 2 0I IIU P V U P V

Since 1P and

2P are kept constant:

0total I IIH H H

totalH is a constant

Joule-Thompson measurement in practice

Apply 1 2P P ,

1T then measure 2T JTC

JTC as a function of ,T P

chapter 3: the math of thermodynamics

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