Winter 2013 Chem 254: Introductory Thermodynamics
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
df f()() x h f x lim dxh0 h instantaneous slope f()() x h f x h lim h0 2h
(better if done on a computer with finite step size)
Often used derivatives df fx() dx xa axa1 eax aeax 1
ln(ax ) x ln(ax ) ln a ln x
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Winter 2013 Chem 254: Introductory Thermodynamics
sin(ax ) acos( ax ) cos(ax ) asin( ax )
Rules for Derivatives d df dg f()() x g x dx dx dx d df dg f()()()() x g x g x f x Leibniz dx dx dx df df du ux() Chain rule dx du dx d Eg. sin(x22 ) cos( x )2 x dx
d 22 e2x 3 x e 2 x 3 x (4 x 3) dx df dg g()() x f x d f() x dx dx Quotient Rule dx g() x gx()2
Partial Derivatives
f(,) x y depends on multiple variables eg. T(,,) x y z the thermometer reading in a room f Pronounced: di f , di x at constant y x 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
22 f(,) x y ex y x
f x22 y x ex(2 1) x y
f x22 y x ey(2 ) y x f( x , y ) ln( x2 2 xy )
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Winter 2013 Chem 254: Introductory Thermodynamics
f 1 2 (2xy 2 ) x y x 2 xy f 1 2 (2x ) y x x 2 xy
Higher Partial Derivatives d2 f() x d df For one variable: 2 dx dx dx 2 dd2 12 Eg. 2ln(xx ) 2 2 2 dxdx x x 2 ff Partial derivatives: 2 x xx y y y 2 ff 2 y yy x x x
Mixed Derivatives ff x y y y xyy x
f x2222 xy x xy e e(2 x 2 y ) x y
f x2222 xy x xy e e(2 x ) y x
f x2 2 xy (e )(2 x 2 y ) y x y x y x
22 ex22 xy(2 x 2 y )(2 x ) e x xy (2)
f x2 2 xy (ex )(2 ) x y x x y
22 ex22 xy(2 x 2 y )(2 x ) e x xy (2) Order of derivatives does not matter f 22 f f x y x y y x x y
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Winter 2013 Chem 254: Introductory Thermodynamics
In mathematics one would typically not write out what’s kept constant f f f instead of ; f(,,) x y z instead of x x y x yz,
In Thermodynamics: ALWAYS write what is kept constant V V , T Pn, T H
Total Differentials
when 2 or more variables are not constant
One dimension df11 d23 f d f df dx dx23 dx .... dx2! dx23 3! dx df dx for dx infinitesimal dx
df f()() x dx f x dx x
Two dimensions ff df dx dy .... xyy x f(,)(,) x x y y f x y ff df x y xy y xy x xy
x in arbitrary direction , We can calculate the change in function f (for small x , y ) y
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Winter 2013 Chem 254: Introductory Thermodynamics
Curve f( x , y ) 0
ff df dx dy 0 xyy x Calculate how much does y change if x x dx f f dy x y ff dx y x dx dy dx f xyy dy f x x y x y y 1 So keeping f constant x f x y f
Consider function f( x , y , z ) 0 (surface in 3 dimensions) ff z constant : dx dy 0 xyyz, zx, f x y zx,. y f z x zy, ff y constant: dx dz 0 xz y,, z y x f z x zy,. x y f z xy, ff x constant: dy dz 0 yzxz, xy,
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Winter 2013 Chem 254: Introductory Thermodynamics
f y z xy,. z x f y zx, x z y Cyclic Rule 1 for f( x , y , z ) 0 yz x yx z
Proof:
f f f 3 1 y x z x z y xz, yz, xy, 1 yz x yx z f f f x zy, z xy, y zx, From the cyclic rule y z 1 y z x x yx x z y z Way to remember cyclic rule z z y x yz yx x
Application in Thermodynamics nRT n2 Van der Waals P a nRT() V nb1 an 2 V 2 V nb V 2 P nR TV V nb
P 2 2 3 nRT( 1)( V nb ) an ( 2) V V T V V : first derive V f(,) P T then find (very hard) P T P T
V How do we calculate ? P T
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Winter 2013 Chem 254: Introductory Thermodynamics
V 1 We can use P P T V T VVP ( 1) (cyclic rule) TPT PTV PP Calculate using above results TV VT
Differential Forms
General differential form (Pfaff): df gxydx(,)(,) hxydy f f Question: Is there a function F(,) x y such that g(,) x y and h(,) x y ? x y y x df dF df F F Path final initial If F(,) x y associated with g(,)(,) x y dx h x y dy then: g(,)(,) x y h x y yxx y If this relationship holds: a function does exist If this relationship does not hold: does not exist
df gxydx(,)(,) hxydy is an exact differential if and only if
, otherwise it’s an inexact differential
If df is an exact differential independent of path that runs between initial and final state
dU,,,, dH dS dG dA: exact differential qw, : inexact differential
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Winter 2013 Chem 254: Introductory Thermodynamics
Integrals
b n f( x ) dx lim f ( x1 ) dx (Area under function) a x 0 i1
f() xi x = area for each rectangle
dF b Find function Fx() such that fx()then: f()()() x dx F b F a dx a
Strategy for solving integrals 1) - guess solution, dF - verify the differentiation fx()? dx - if not right tinker with constants 2) - Look them up (books) or use math programs Examples in thermodynamics dF fx() Fx() dx
1 1 aa xaa ( 1) xa1 (a 1) x x a 1 (a 1) b bln x ln xb x
1 ax 1 ax eax e ae a a
Line Integrals
Consider differential df gxydx(,)(,) hxydy consider paths
y()() x P1 x and y()() x P2 x
df I df I in general II Px() 1 Px() 2 12 1 2
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Winter 2013 Chem 254: Introductory Thermodynamics
dP y()() x P x dy 1 dx 1 dx I gxydx(,)(,) hxydy df 1 Path
x2 dP I gxPxdxhxPx ,(),() 1 dx 1x 1 1 1 dx
x2 dP I gxPx ,(),() hxPx 1 dx 1x 1 1 1 dx x I 2 c() x dx reduced to 1d integral 1 x 1 Likewise:
x2 dP I gxPx ,(),() hxPx 2 dx 2x 2 2 1 dx integrals are different But: they are the same if and only df is an exact differential
If df exact differential then I1 I(,)(,) x yfinal I x y initial
Exact vs Inexact Differential
dd Inexact differential: g(,)(,) x y h x y dyx dx y df gives results but depends on path 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
dh hxy dx h dy measure in small steps h(,) x y defines height function of xy,
Real life example of Inexact differential: shoveling snow
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Winter 2013 Chem 254: Introductory Thermodynamics
d d dx d dy snow x y The amount of snow you shovel depends on
the path you take between points A and B Exact differential: df2 xydx x2 dy (2xy ) ( x2 ) yxxy 22xx Therefore, is exact differential
df is independent of path
dF dF 2 2 Fx() such that 2xy ; x F(,) x y x y dx y dy x
dP d() y x Px(): yx 1 1 1 dx dx dP d() y x2 Px(): yx 2 2 2x 2 dx dx
I gxydx(,)(,) hxydy 1 Path x1 dP gxPxdx ,(),() hxPx 1 dx x0 11dx 11 I2 x xdx x22 1 dx 3 x dx 1 00 1 x3 1 0 11 I2 x x2 dx x 2 2 xdx 4 x 3 dx 2 00 1 x4 1 (integral result is the same because exact differential) 0 2 F(,) x y x y IFFn (1,1) (0,0) 1 0 1
Example of inexact differential: df x2 dx2 xydy ;
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Winter 2013 Chem 254: Introductory Thermodynamics
11 I x22 1 dx 2 xxdx 1 3 xdx 1 00 1 x3 1 0 11 I x2 dx 2 x x 2 2 xdx x 2 4 x 4 dx 2 00 1 41 1 4 17 xx35 1 3 50 3 5 15
(x2 ) (2 xy ) ? yxxy
02y not equal, differential is not exact, II12 in general
F 2 F Does F(,) x y exist such that x and 2xy ? x y y x
F 2 1 3 x F x c x y 3
F 2 2xy F xy c Not equal y x
Hence: for inexact differentials, line integrals can be calculated, but results depends on path. True for qw, in thermo or shoveling snow!
Summary of rules from math
y 1 y y z x z x x z z x x y y z Exact differential: df gxydx(,)(,) hxydy g(,)(,) x y f x y yxx y df is independent of path Path
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Winter 2013 Chem 254: Introductory Thermodynamics
Back to Thermodynamics!
Definition of β and κ
1 V -1 Define: in units K 0 usually VT P V V : Volumetric thermal Expansion coefficient T P
1 V -1 in Bar 0 VP T V V : Isothermal compressibility P T
For solids and liquids and are more or less constant
For gases and are not constant nRT Ideal gas: V P 11V nR P V TP P nRT T 11V nRT P () 2 V PT P nRT P P 11 VVV T P T V PPVV T P TVTV VTP V P T P hard to measure obtain as T V PP dP dT dV TV VT
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Winter 2013 Chem 254: Introductory Thermodynamics
1 dV dT V f f1 f dV dP dT Path i i V
1 Vf PT ln Vi
Vf ln TP Vi
Dependence of U on T and V
UU dU dT dV TV VT dU q w : inexact differential
q Pext dV assume constant volume
dU q qVV C dT
UCT V if V constant U lim CV T 0 T V U CV T V
UP TPTP (will be derived later) VT T
Dependence of H on T and P
H for constant pressure H qPP C T HH lim CP T 0 TT PP H HV CP TVVT (1 ) T P PT TP (to be derived later too)
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Winter 2013 Chem 254: Introductory Thermodynamics
For Ideal gases 1 1 ; T P As expected UPP TPTP 0 VTT T HV 1 TVVT (1 ) 0 PTT TP
Derivations involving dH
dH dU d() PV dH dU VdP PdV HHUU dT dP dT dV VdP PdV TPTV PTVT HU CPV dT V dP C dT P dV PV TT UP Use PT (relation stated before) VT TV HP CPV dT V dP C dT T dV PT TV assume T is constant (some process) HP V dP T dV ( constant) PT TV HPV VTlim P 0 PTP TVT HPV VT now use cyclic rule PTP TVT HV VT PT TP
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Winter 2013 Chem 254: Introductory Thermodynamics
Relation between CP and CV (Exact)
HU CPV dT V dP C dT P dV PV TT Assume P constant PV CCTPV TT VP CCTV PV 2 C C TV (exact) PV For ideal gas
P nRT nR V nRT nR ; TTVV VV TTPP PP nR nR nRT CPVV C T C nR V P PV
CPV C nR (used and derived before)
Joule Thompson Experiment
T CJT (Joule Thompson Coefficient, definition) P H HHT CCP JT PTP TPH
PP12 qtotal 0 qqI II H is constant during process
If H is constant:
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Winter 2013 Chem 254: Introductory Thermodynamics
TT21 T lim CJT PP21 PPP21H H H CP C JT V VT P T 1 V CCP JT measures VT P
CJT 0 for ideal gases
Utotal U I U II w P1 V 1 P 2 V 2 q 0
UPVUPVI 1 1 II 2 2 0
Since P1 and P2 are kept constant:
HHHtotal I II 0
Htotal is a constant
Joule-Thompson measurement in practice
Apply PP12 , T1 then measure T2 CJT
CJT as a function of TP,
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