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Chapter 3: the Math of Thermodynamics 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 Chapter 3: The Math of Thermodynamics 25 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 x22 y x f(,) x y e f x22 y x ex(2 1) x y f x22 y x ey(2 ) y x f( x , y ) ln( x2 2 xy ) Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 26 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 x2222 xy x xy e(2 x 2 y )(2 x ) e (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 Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 27 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 Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 28 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, Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 29 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 Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 30 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 Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 31 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 Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 32 Winter 2013 Chem 254: Introductory Thermodynamics dP dy 1 dx 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 x2 I 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 y()() x P x How do you get contours for map?1 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 Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 33 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 22 I2 x xdx x 1 dx 3 x dx 1 00 3 1 x 1 0 11 2 2 3 I2 x x dx x 2 xdx 4 x dx 2 00 4 1 x 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 ; Chapter 3: The Math of ThermodynamicsChapter 3: The Math of Thermodynamics 34 Winter 2013 Chem 254: Introductory Thermodynamics 11 22 I x 1 dx 2 xxdx 1 3 xdx 1 00 3 1 x 1 0 11 2 2 2 4 I x dx 2 x x 2 xdx x 4 x dx 2 00 1 135 4 1 4 17 xx 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.
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