Multivariable Calculus

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1 Multivariable Calculus In thermodynamics, we will frequently deal with functions of more than one variable e.g., P PT, V, n, U UT, V, n, U UT, P, n U = energy n = # moles etensive variable: depends on the size of the system intensive variable: independent of the size of the system V, n are etensive, P, T, molar volume V/n are intensive change volume at fied T, n pressure changes P dp V nt, dv these variables are held constant cylinder with piston

2 Suppose we have an ideal gas PV nrt P V P T P n nt, Vn, VT, nrt V nr V RT V subscripts refer to variables that are held constant now suppose we want to see how P changes when both V and T change P P dp dv dt V T Tn, Vn, Note that this is fully consistent with Taylor series epansions If n changes as well need to add a P n TV, dn term

3 In general, y y,, dy n i1 Ideal gas 1 y n ' n d, i ' hold fied all variables ecept i P P P dp dt dv dn T V n nv, T, n TV, nr nrt RT dp dt dv dn V V V for small finite changes nr nrt RT P T V n V V V

4 These equations are most useful when we don't have an analytical function for the quantity of interest. Consider U U( T, P, n) U U U du dt dp dn T P n Pn, Tn, T, P The derivatives are often available eperimentally. We can also write U U T, V, n U U U du dt dv dn T V n Vn, Tn, TV, Note, in general, U U T T P, n V, n

5 Eample 3 z a bycy z a by y Suppose u y then cu z a bu z cu a 3 u Comment: The z, y in the tet should be z z, y Comment: if you work out eample 8.4, note that when it refers to eample 8., it should be 8.3. Also, in the last term of the answer, the u should be u. Above, we saw U U T T P V We may want to know how these two quantities are related (variable change identity)

6 Consider U U T, P, n U U U du dt dp dn T P n Pn, Tn, T, P du U U dp U dn dt T P dt n dt Pn, Tn, T, P U U U P U n T T P T n T Vn, Pn, Tn, Vn, TP, Vn, Note: This essentially follows from a Taylor U series UT, P, n U0 T T0 T Pn, U U U P T T, P, T, Vn, Pn Tn Vn 0 for small changes T T0 dt We have now obtained the relation between the two derivatives of interest.

7 Reciprocal identity y 1 zu, y zu, P 1 e.g., V V nt, P nt, Eample z sin y arcsin z y Show dz 1 in this case d y z Second derivatives E. z z is y y y z z y y U V VT V T y n Vn, Tn, y

8 Euler Reciprocity z y z y It is normal to suppress the info on the variable(s) held constant Mawell relations Consider du TdS PdV U T S V U P V S You will derive this in P Chem. S = entropy, n fied, system assumed to be reversible T U V V S S P U S S V V T P V S Sn, Vn, T P is much easier to measure than V S Sn, Vn,

9 da SdT PdV Helmholtz free energy A U TS da SdT PdV A A S P T V V S A V V T S P P A V T T T V T V T Tn, Vn, Tet describes a rule for finding these derivative relationships but it is confusing. It is easier to start from the definitions of U, H, A, G energies.

10 Cycle rule Show that y z z y z y 1 Start with y y dy d dz z z choose d and dz so that dy = 0 y y z 0 z z y or y z 1 z y z y Chain rule Suppose z = z(u,,y) and = ( u,v,y) z z y y uv, uv, uv,

11 Important thermodynamic quantities C C K K P V T S H S T T T Pn, Pn, U S T T T Vn, Vn, 1 V V P 1 V V P 1 V V T dq ds T rev Pn, Tn, Sn, Note: There are constant volume and constant pressure heat capacities H U PV enthalpy isothermal compressibility adiabatic compressibility coefficient of thermal epansion definition of entropy Eample 8.10 C C P V K K T S S S P S V V T T P T P S P S S V V T V T T V T T P P P T S T T T T V T S S S S S

12 Eact and Ineact Differentials Suppose,, du M y d N y dy This is called an eact differential if M u u and N y y If M and N are as above M u N u y y y y Then we have an eact differential If there is no function u for which this is true, du is an ineact differential

13 Eample dz y d dy y y Note: Typo in tet 9 9 y y y y 3 9 y y 3 Since these are equal, we are dealing with an eact differential More general case For ;,,,,, dum yzdn yzdyp yzdz Note: Typo in tet Eact differential if M N M P, y z z, N P z y y, z, yz, y, yz,

14 Heat + work under reversible conditions heat transferred = dq rev work done on system = dw rev if work is associated only with a volume change dwrev PdV Check to see if dw is an eact differential dw MdT NdV but we have already said that for a reversible process, this is 0, so dw rev is not an eact differential V, T, P. n are state functions w and q are not state functions If only work that is due to volume change du dq dw state function

15 Integrating Factors Sometimes, one can find a factor that, when multiplying an ineact differential, makes it eact Eample: du Md Ndy a by d b cy dy 1 is an integrating factor for du du a byd b cydy y a by b b cy b So du is an eact differential dq rev As it turns out, S is an eact differential, where S is the entropy T

16 Maima and Minima of functions of more than one variable y y Maimum in the direction but minimum in the y direction (saddle point) Maimum in both the and y directions f f Given f, y, we need to search for 0 and 0 y y This tells us that we found a point where the slope = 0, but does not tell us what sort of special point this is.

17 nd derivative matri f f y f y f y f f f D y y f 1 D 0, and 0 local minimum f f 0 0 y f D 0, and 0 3 D 0, local maimum neither maimum nor minimum f f 0 0 y 4 D 0, cannot tell what sort of point it is

18 Constrained optimization Eample:, y f y e subject to constraint g y, y1 y 1 f ( 1) 4 e ( 1) f e e e e e which then gives y 1 1 = is also a solution 1 1 f, The unconstrained maimum of f is 1.

19 An alternative approach Lagrange multipliers find minimum or maimum of f y, subject to the constraint g, y 0 Define a function u such that u, y f, y g, y u f g 0 y y y u u g 0 y y y solve these together with g = 0 Eample: y f e, g y1 y u e y1 u y u y y e 0 y ye 0 y ye y y 0 ye 0 0 y y g y , y Note: this is a simpler derivation of the approach taken in the tet

20 A more general case f f, y, z subject to two constraints g g 1, y, z 0, y, z 0 u f g g u f g g yz, yz, yz, yz, u f g g y y y y z, z, z, z, u f g g 0 z, 1 1 y, z y, z y, z y together with g1 0, g 0

21 Vector operators Gradient ˆ f ˆ f i j f kˆ f y z Note typo in book Gradient of a scalar direction in which function increases most rapidly magnitude is the rate of change in that direction Eample: 3 bz g a ye g 3a iˆe ˆjbye k bz bz ˆ F V, V potential, F = force Gravitational potential, V Gm1m r r y z Gm1m F V iˆ yj ˆkzˆ 3 r 1 1 y z y z 3/

22 Divergence operating on a vector F F F F y z y scalar z ˆ ˆ i j kˆ if ˆ ˆjF kf ˆ y z y z E.g., Continuity Eq. v t = density, v velocity Eample ˆ ˆ ˆ kz F i jyz y z F z y

23 Curl ˆ F F z y F Fz ˆ Fy F F i j k y z z y Laplacian recall C AB iˆab ˆ ˆ y z AB z y j AB z AB z k AB y AB y f f f y z f Spherical coordinates dr e dr e rd e r d ˆ ˆ ˆ sin r z φ θ r y dsr dr ds rd ds r d sin f 1 f 1 f f eˆ ˆ ˆ r e e r r rsin Divergence Curve Laplacian can all be represented in spherical coordinates f r f sin f f r r r r sin r sin

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