11 Lecture 11: How to manipulate partial derivatives

Transcription

1 11. LECURE 11: HOW O MANIPULAE PARIAL DERIVAIVE Lecture 11: How to manipulate partial derivatives ummar Understand thermodnamics and microscopic picture of the thermal properties of a rubber band. Review partial derivatives. he Jacobian technique ma be full utilized if ou remember three elementar rules/formulas.s (A, B) (B, A) (Y, X) (B, A) (Z, W ) (A, B) (A, B) (Z, W ). (Y, X) (A, B), and Maxwell s relation for conjugate pairs (X, x) and (Y, ): (X, x) (, Y ) 1. Ke words entropic elasticit, Maxwell s relation, adiabatic cooling, adiabatic demagnetization. What ou should be able to do Practice the Jacobian technique. Intuitivel explain rubber elasticit; be able to get various signs of partial derivatives, and to explain them intuitivel. Explain adiabatic demagnetization. Let us start with a review of how to compute entrop change due to irreversible processes. hermodnamics can stud an change occurring during irreversible processes if the initial and the final states are equilibrium states. o compute we devise a reversible process along which we ma use thermodnamics. ince entrop is a state function, ou can use an path in the thermodnamic space connecting the

2 100 initial and the final states. or example, if and P are given as functions of E and V, we can integrate d 1 de + P dv (11.1) along a convenient path connecting the initial and the final states (ig. 11.1). actual irreversible process (E,V) E igure 11.1: You can integrate the Gibbs relation (11.1) along an path in the (E, V ) space (i.e., the thermodnamic space), because is a state function, BU there are convenient paths such as are piecewise parallel to thermodnamic coordinates (red path). V Let us perform a small experiment of quasistatic adiabatic processes using a rubber band. Prepare a thick rubber band (that is used to bundle, e.g., asparagus). Use our lip as a temperature sensor. Initiall, putting the rubber band to our lip, ou sense the room temperature (cool). Now, ou hold a small portion of the band b our hands and stretch it tightl and quickl. hen, feel its temperature with our lip. It must be warm. You have just demonstrated ( ) > 0, (11.2) where L is the length of the stretched portion of the rubber band. he Gibbs relation for a rubber band reads de d + dl, (11.3) where is force stretching the band. ince the process is adiabatic and quasistatic is constant. Even if ou rapidl pull the band, the process is ver slow from the

3 11. LECURE 11: HOW O MANIPULAE PARIAL DERIVAIVE 101 molecular point of view, so the process is quasistatic. ince the heat conduction is not a ver rapid process, during quick stretch the sstem is virtuall thermall isolated ( adiabatic). hus, is constant. Now, let us tr to understand microscopicall what we have observed. A rubber band is made of a bunch of polmer chains. ake a single chain that is wiggling due to thermal motion (ig. 11.2a). a b igure 11.2: a: A schematic picture of a single polmer chain. Each arrow is called a monomer. b: Polmer-kid analog. he temperature represents how vigorousl kids are moving around. his also includes vibration of individual bodies. he figure is after N. aito, Polmer Phsics (hokabo, 1967) (he original picture was due to. akai s lecture according to aito). Entrop is monotonicall related to the width of the range kids can pla around easil, which becomes smaller if the distance between the flags is increased. tretching the chain corresponds to increasing the space between the flags in the plaing kid analog. If the chain does not break, the maneuverable space is decreased, but since the kids must keep entrop, the restricted dancing motion must find substitute degrees of freedom: shaking bodies. hat is, the temperature of the sstem should go up. his suggests that if the chain is stretched under constant, entrop should go down: ( ) < 0. (11.4) Can ou conclude this from what ou observed (11.2)? Yes, ou can, but ou must know how to manipulate partial derivatives (see toward the end of toda s lecture). In thermodnamics a lot of partial derivatives appear. Let us write down the definition of partial derivatives. Consider a two-variable function f f(x, ). hen,

4 102 partial derivatives are defined as f f f(x + δx, ) f(x, ) x lim, δx 0 δx (11.5) f f f(x, + δ) f(x, ) lim. δ 0 δ (11.6) Partial differentiation is extremel trick in general. or example, even if f/ and f/ exist at a point, f can be discontinuous at the same point. E is once continuousl differentiable with respect to and work coordinates, so we ma stud fairl well-behaved functions. Let f be a function of several variables x (x 1,, x n ). We could understand f as a function of the vector x. We wish to stud its linear response to the change x x + δx: δf(x) f(x + δx) f(x) Df(x)dx + o[δx], (11.7) where o denotes higher order terms that vanishes faster than δx, when the limit δx 0 is taken. Here, Df is a linear operator 80 that can be written as Df(x)dx i f i dx i (11.8) If such a linear map Df is well-defined, we sa that f is (totall) differentiable. If there are onl two variables, Df(x, )(dx, d) f f dx + d. (11.9) Consider f(x + δx, + δ) f(x, ). here are two was to go from (x, ) to (x + δx, + δ), δx first or δ first: f(x + δx, + δ) f(x, ) f(x + δx, + δ) f(x + δx, ) + f(x + δx, ) f(x, ) f (x + δx, )d + f x (x, )dx, (11.10) f(x + δx, + δ) f(x, ) f(x + δx, + δ) f(x, + δ) + f(x, + δ) f(x, ) f x (x, + δ)dx + f (x, )d. (11.11) 80 L is a linear operator acting on a function set is a map from to some other vector space such that where f, g and α, β are numbers. L(αf + βg) αlf + βlg

5 11. LECURE 11: HOW O MANIPULAE PARIAL DERIVAIVE 103 he difference of these two formulas is [f (x + δx, ) f (x, )]d [f x (x, + δ) f x (x, )]dx [f x (x, ) f x (x, )]dxd. (11.12) his must vanish if the surface defined b f is at least twice differentiable. 81 hat is, f x f x. (11.13) or example, for a rubber band de d + dl, so ( ) ( ). (11.14) L uch relations are called Maxwell s relations. o manipulate man partial derivatives, it is ver convenient to use the so-called Jacobian technique. his technique ma not even be taught in graduate courses, but it is eas to memorize, and eas to use. It can greatl reduce the insight and skill required in thermodnamics, especiall with the Jacobian version of Maxwell s relation (11.30). he Jacobian for two independent variables is defined as the following determinant: ( (x, ) X ) ( ) X ( ) ( ) ( ) ( ) x ) ( ) X Y Y X. (11.15) x x ( Y In particular, we observe Y x (X, ) (x, ) ( ) X, (11.16) which is probabl the ke observation of this technique. Obviousl, 1. (11.17) now. 81 We must sa something more careful mathematicall, but let us be contented with this for

6 104 here are onl two or three formulas ou should learn b heart (the are ver eas to memorize). One is straightforwardl obtained from the definition of determinants: exchanging rows or columns switch the sign: (x, ) (, x) (Y, X) (, x) (Y, X) (x, ). (11.18) If we assume that X and Y are functions of a and b, and that a and b are, in turn, functions of x and, we have the following multiplicative relation: (a, b) his is a disguised chain rule: ( ) ( ) X X a (a, b) (x, ) (x, ). (11.19) b ( ) a + ( ) X b a ( ) b, (11.20) etc. Confirm (11.19) b ourself (use det(ab) (deta)(detb)). he technical significance of (11.19) must be obvious; calculus becomes algebra: ou can think (A, B) just as an ordinar numbers: formall we can do 82 (x, ) (A, B) (x, ) (A, B) (A, B) (A, B) (x, ). (11.21) rom (11.19) we get at once / (A, B) A, B) 1. (11.22) In particular, we have ( ) X Y Examples: or a rubber band ( ) (L, ) (, ) (L, ) (, ) /( ) 1 X Y (, ) (, ). (11.23) ( ) ( ), (11.24) 82 hat is, all the derivatives showing up in the calculation are well defined. However, in practical thermodnamics, ou can be maximall formal.

7 11. LECURE 11: HOW O MANIPULAE PARIAL DERIVAIVE 105 which reads ( ) ( ) / ( ) ( ) / C. (11.25) Here, C is the heat capacit under constant force. It is explained as follows: he first law tells us d Q d, so if we differentiate this with respect to under constant, it must be the heat capacit under constant. Generall speaking, the heat capacit under constant X alwas have the following expression: ( ) C X. (11.26) We will learn the consequence of the stabilit of the equilibrium states later, but the stabilit of the equilibrium state implies C X 0 (usuall strictl positive). Imagine the contrar. If ou inject heat into a sstem, its temperature goes down, so it sucks more heat from the surrounding world, and further reduces its temperature. hat is, such a sstem becomes a bottomless heat sink. X Using these relations, we can easil demonstrate ( ) X x ( ) X / ( ) X (11.27) as follows: (X, x) (, x) (11.19) (, X) (X, x) (, x) (, X) (x, X) (, X) (11.18) (X, ) (x, ). (11.28) hen, use (11.22). A concrete example of this formula is ( ) ( ) /( ) P V V P V p, (11.29) which relates thermal expansivit and isothermal compressibilit. All the Maxwell s relations can be unified in the following form (X, x) (Y, ) 1, (11.30)

8 106 where (x, X) and (, Y ) are conjugate pairs. his is the third equalit ou should memorize. When ou use this, do not forget that ( P, V ) (not (P, V )) is the conjugate pair. Let us demonstrate this. rom +xdx+dy + Maxwell s relation reads hat is, ( Y his implies (mere a/b c/d a/c b/d!) ) X ( X ) Y. (11.31) (x, X) (Y, X) (, Y ). (11.32) (x, X) (, Y ) (Y, X) 1. (11.33) Equipped with the machiner, let us stud the rubber band in more detail. he rubber band is elastic because of the thermal motion of the polmer chains. hat is, resistance to reducing entrop is the cause of elastic bouncing. hus, such elasticit is called the entropic elasticit. 83 An important feature is that the elastic force increases with under constant length (which is easil understood from the kid picture): ( ) > 0. (11.34) L Is this related to what we have observed (11.2)? Yes. ollow the following logic (as a practice): 0 < ( ) (, ) (L, ) (L, ) (, L) (, L) (L, ) (, ) (L, ) (L. ) (L, ) (L, ) (L, ) (, L) (, L) / (, L) (, L) ( ) L (11.35) C L. (11.36) rom our microscopic imagination visualized in ig. 11.2, we guessed ( ) < 0. (11.37) 83 In contrast, the usual elasticit is called energetic elasticit, which is cause b opposing increase of energ.

9 entrop 11. LECURE 11: HOW O MANIPULAE PARIAL DERIVAIVE 107 Let us derive this from (11.2). ( ) (, ) (L, ) (, ) ( ) (L, ) (L, ) (L, ) C L < 0. (11.38) If a tightl stretched rubber band is suddenl relaxed ( adiabaticall relaxed) after equilibrating with the room temperature, what do ou observe? he band becomes ver cool. his is not surprising: ( ) > 0; (11.39) Now, L is reduced under constant, so must decrease. his is the principle of adiabatic cooling (see ig. 11.3). increasing L (, L 1 ) (, ) L igure 11.3: Initiall, the sstem is at 1. Isothermall, L is increased as L 1 L 2. his decreases entrop. Now, L is returned to the original smaller value adiabaticall. he entrop is maintained, and the temperature decreases (adiabatic cooling) to 2. he dotted path is the one ou experienced b initial rapid stretching of a rubber band. Unfortunatel, we cannot use a rubber band to cool a sstem to a ver low temperature, since it becomes brittle (chain motion freezes out easil. In actual low temperature experiments, a collection of almost non-interacting magnetic dipoles (i.e., a paramagnetic material) is used. he sstem is closel related to polmer chains as illustrated in ig he Gibbs relation of the magnetic sstem is de d + HdM, (11.40) where H is the magnetic field, and M the magnetization. he correspondences H and M L are almost perfect: M is the sum of small dipole vectors,

10 108 a b igure 11.4: a of ig corresponds to b a paramagnet, a collection of onl weakl interacting magnetic dipoles. and L is also the sum of (the projected components) of steps (monomer orientation vectors). hus, we expect ( ) > 0 (11.41) M and adiabatic cooling can be realized; first appl a strong magnetic field and align all the dipoles, and remove the generated heat. hen, turn off the magnetic field to make M 0 (demagnetization). impl replacing L with M in ig. 11.3, we can understand this adiabatic demagnetization strateg to cool a sstem.