Appendix 3, Rises and runs, slopes and sums: tools from calculus

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1 Appendi 3, Rises nd runs, slopes nd sums: tools from clculus Sometimes we will wnt to eplore how quntity chnges s condition is vried. Clculus ws invented to do just this. We certinly do not need the full mchinery of clculus, just few of its key ides nd tools, described here. There re two different regimes, ccording to whether we re interested in smll chnge in conditions (in which cse we use slopes) or lrge chnge in conditions (in which cse we use sums). If you re not yet fmilir with clculus, the ides nd tools presented here re ll tht you need (nd bit more!) to pprecite their ppliction in generl chemistry. They my even mke your future study of clculus esier. If you re lredy fmilir with clculus, then the mteril here probbly will be quite fmilir, with one possible eception. The discussion of the nturl logrithm emphsizes its definition in terms of the sum of smll chnges in quntity divided by the vlue of the quntity t ech point in its chnge. If you re like me, on first eposure you my not hve pprecited tht the nturl logrithm rises s this specil kind of sum, nd so tht it cnnot be "derived" from something more fundmentl. Indeed, it is for this reson tht it comes up "nturlly" in chemistry nd so is quite ccessible to us. Slopes If we re interested in the effect,, on property, f, of smll chnge,, in conditions, we cn evlute this s the product of the rte t which f chnges when is chnged times the smll chnge in, = chnge in f due to smll chnge =. The rte of chnge, ê, is the slope of the plot of f versus. It is clled the derivtive. If we hve n epression for the dependence of f on, then it is esy to get n epression for the slope without hving to mke plot of f versus. The process is clled differentition. The key ide of differentition is tht slopes rise over run re computed in the limit tht the run is tiny. Here is wht this sttement mens in generl terms. If chnges from to 2, then the slope of the vrition is defined in the usul wy D f D = f H 2L - f H L Å. 2 - To specify tht the run, 2 -, is tiny, we write nd so 2 = +, 2 - =, with the understnding tht is tiny. Using this lst reltion in the epression for the slope, we get

2 388 Notes on Generl Chemsitry, 2e = lim Ø0 f H + L - f HL ÅÅ, where we hve replce the specific vlue by the generl vlue. This is s the fundmentl definition of slope for tiny runs. To see just wht the definition mens nd how powerful it is, let's pply it to the severl kinds of vritions, f HL, tht rise in chemistry. à Liner vrition, f HL = The simplest emple is liner vritions,. If we plot this we will get stright line etending from the origin with slope. Here is wht we get using the definition of slope. f H + L - f HL = ÅÅ ÅÅ Å H + L - = ÅÅ ÅÅ + - = ÅÅ ÅÅÅ =. Just s we epect, the slope evlutes to. à Qudrtic vrition, f HL = 2 Here is wht we get for qudrtic vrition, 2. f H + L - f HL = ÅÅ ÅÅ Å = H + L2-2 ÅÅ ÅÅ = HL 2-2 ÅÅ ÅÅ Å = 2 + = 2. In the lst step, we use the fct tht we re interested in vlues tht re tiny ("infinitesimlly smll") nd so tht, in the numertor of the second to lst line, the term is negligible compred to 2. Here is plot of f = 3 2 nd the slope, ê = 6 t severl points.

3 Appewndi 3, Rises nd runs, slopes nd sums: tools from clculus Function 3 2 nd its slope, 6, t =. nd = 3. The stright lines re the slope of the function when it hs the vlues indicted by the dots. I find it lmost mgicl tht we cn evlute n nlyticl epression for the slope of the function the line tngent to the plot of the function t ny point. à Generl vrition, f HL = n We cn show in the sme wy tht for positive integers n, the slope f HL = n is = nn-. See if you cn do this. à Inverse liner vrition, f HL = - Here is the slope when property is inversely proportionl to quntity. f H + L - f HL = ÅÅ ÅÅ Å = J ÅÅÅÅ + - Ní H + L = J - H + L H + L Ní - - = J ÅÅ ÅÅÅ Ní = J ÅÅÅÅ 2 - N =-ÅÅ. 2 In the lst step, we use the fct tht we re interested in vlues tht re tiny ("infinitesimlly smll") nd so tht, in the denomintor of the second to lst line, the term is negligible compred to 2. Here is plot of f = 2 - nd the slope, ê =-2-2 t severl points.

4 390 Notes on Generl Chemsitry, 2e Function 2 - nd its slope, -2-2, t =. nd = 3. The stright lines re the slope of the function when it hs the vlues indicted by the dots. à Inverse qudrtic vrition, f HL = -2 Here is the slope when property is inversely proportionl to the squre of quntity. f H + L - f HL = ÅÅ ÅÅ Å = i j k H + L - y ÅÅ 2 2 z ì { = i j k 2 Å H + L 2 - H + L y 2 2 H + L 2 z ì { = i j y ÅÅ ÅÅ ÅÅÅ k z ì { -2 - = ÅÅ Å =-ÅÅÅ In the lst step, we use the fct tht we re interested in vlues tht re tiny ("infinitesimlly smll") nd so tht, in the second to lst line, in the numertor the term is negligible compred to 2, nd in the denomintor the terms re negligible compred to 4. Here is plot of f = 3-2 nd the slope, ê =-6-3 t severl points.

5 Appewndi 3, Rises nd runs, slopes nd sums: tools from clculus Function 3-2 nd its slope, -6-3, t = 0.6 nd =.2. The stright lines re the slope of the function when it hs the vlues indicted by the dots. à Generl vrition, f HL = -n We cn show in the sme wy tht for positive integers n, the slope of f HL = -n is =-n-hn+l. See if you cn do this. à Trigonometric vrition, f HL = sinhl Here is the slope when property is proportionl to the sinhl. f H + L - f HL = ÅÅ ÅÅ Å sinh+ L - sinhl = ÅÅ Å Å sinhl cosh L + coshl sinh dl - sinhl = ÅÅ ÅÅ ÅÅ Å Å sinhl + coshl - sinhl = ÅÅ ÅÅ ÅÅÅ = coshl Here we use the trigonometric identity sinh + bl = sinhl coshbl + coshl sinhbl, nd the fct tht for tiny, sinhl = nd coshl =. Here is plot of f = sinh2 L nd the slope, ê = 2 cosh2 L t severl points.

6 392 Notes on Generl Chemsitry, 2e D Function sinh2 L nd its slope, 2 cosh2 L, t = 0.6 nd =.2. The stright lines re the slope of the function when it hs the vlues indicted by the dots. à Trigonometric vrition, f HL = coshl Here is the slope when property is proportionl to the coshl. f H + L - f HL = ÅÅ ÅÅ Å cosh+ L - coshl = ÅÅ ÅÅ Å coshl cosh L - sinhl sinh dl - coshl = ÅÅ ÅÅ ÅÅ ÅÅ coshl - sinhl - coshl = ÅÅ ÅÅ ÅÅÅ =-sinhl Here we use the trigonometric identity cosh + bl = coshl coshbl - sinhl sinhbl, nd the fct tht for tiny, sinhl = nd coshl =. Here is plot of f = cosh3 L nd the slope, ê =-sinh3 L t severl points. Cos@3 D Function cosh3 L nd its slope, -3 cosh3 L, t = 0.6 nd =.2. The stright lines re the slope of the function when it hs the vlues indicted by the dots.

7 Appewndi 3, Rises nd runs, slopes nd sums: tools from clculus 393 Sums Assume tht quntity represented by function f depends on vrible. We re often interested to know the effect of lrge chnge, D, on the vlue of f. Let's introduce the nottion D f for the effect of lrge chnge in on f HL. A wy to determine D f is to sum up ll of the smll chnges,, tht occur s is chnges throughout the lrge rnge D. This summing up is represented by the symbol Ÿ (for Sum) s follows. f H+DL D f = chnge in f due to lrge chnge D = f HL = f H +DL - f HL The key ide is tht the wy to use this reltion is to interpret the smll chnges,, being summed to be the product of slope, ê, nd smll chnge in conditions,, =, nd then to work bckwrds to see which function f HL hs ê s its slope. You my wnt to red the lst sentence gin. It relly is the key to determining the cumultive effect of successive smll chnges. Here is n emple. In some chemicl rections rectnt, A, disppers t rte proportionl to the squre of its We cn epress this ÅÅÅÅ 2, t where the minus sign tkes cre of the fct tht the concentrtion decreses with time. Tis epression sys tht the smll chnge in tht occurs during the pssge of the smll mount of time, t, is proportionl to the squre of the concentrtion t the time of the chnge. Since [A] thereby chnges s time psses, so to does the rte. We cn rerrnge this epression to 2 = k t. In this form we cn see how to sum the chnge in concentrtion tht tkes plce over lrge chnge in time, sy from t = 0 to t. The sum on the right hnd side is just k times the totl elpsed time, kht - 0L = kt. To evlute the sum on the left hnd side, we interpret it s = Ø f =- Tht is, we interpret - ê@ad 2 s the slope of n (s yet unknown) function of the concentrtion. Tht is, this interprettion mens tht wht we need to know is which function hs s derivtive - ê@ad 2. From our nlysis of derivtives, we know the nswer: - 2 is the derivtive of ê@ad. This mens tht f = ê@ad, nd so the sum on the left hnd side is D f = f H +DL - f HL Ø t t - Å t t = 0 = Å - 0

8 394 Notes on Generl Chemsitry, 2e In this wy we get n eplicit epression for the cumultive chnge in - ê@ad 2 s concentrtion chnges continuously 0 t. In this emple the result is the epression kt= Å - 0 relting the concentrtion of A t ny time t to its initil concentrtion. This emple illustrtes the generl procedure for doing sums. It is less systemtic thn getting epressions for slopes, tht is, thn evluting derivtives. The key ide is lwys the sme, to look t wht is being summed in terms of slope nd then to figure out which function hs tht s its slope. A remrkble specil cse: f HL = lnhl How should we hndle the sum f H+DL f HL = +D? Tht is, wht function hs s its derivtive =? Neither f HL = n nor f HL = -n, for positive integer n will do. This is rel puzzle. In fct, there is no nlytic eplicit functionl form tht hs s its derivtive ê. Wht do we do? Wht we do is to define the nswer in terms of the sum. The sum is known s the nturl logrithm, lnhl ª. with the understnding tht > 0. Such definition is known s n integrl representtion of function. Here is plot tht shows tht the sum is indeed lnhl.

9 Appewndi 3, Rises nd runs, slopes nd sums: tools from clculus 395 lnhl Function loghl nd its slope, ê, t = 0.7 nd = 2. The stright lines re the slope of the function when it hs the vlues indicted by the dots. The plot ws constructed using the vlues of loghl for the curve nd the vlues ê for the slope. In this wy we confirm grphiclly tht the nturl logrithm is function defined in terms of sum wht is clled n integrl representtion. But how do we compute such sum? The nswer is to compute it by breking the sum up into pieces. If the pieces re lrge, our nswer for the sum will not be very ccurte. As we mke the size of ech piece smller, by dding more pieces, the nswer gets better quite quickly. We will see just how to do this during the discussion of work done by n epnding gs, but for now, let's relize tht every time we press the loghl button on our clcultors, the clcultor evlutes the sum for us to get the numericl vlues. à Logrithmic properties of Ÿ H ê L The plce to begin is to recll the just wht logrithm is. The logrithm, y, of number,, to bse, b, is the power to which the bse must be rised to equl the number, = b y Epressed differently, log b HL = y In numericl clcultions, we usully work with logrithms in the bse b = 0, nd often bbrevite log 0 HL simply s loghl, tht is, without writing the vlue of the bse. Bse of the nturl logrithm: The first question tht you my hve bout the nturl logrithm is wht is the bse? We cn determine this by noting tht, from the definition of the logrithm, the logrithm to bse b of the bse b is lwys, log b HbL =, since b = b. This mens we cn determine the numericl vlue of the bse of the nturl logrithm by finding the vlue of for which the sum

10 396 Notes on Generl Chemsitry, 2e is one. This number is clled nd hs the vlue = = Numericl clcultions with nturl logrithms re conveniently done by epressing them in terms of bse 0 logrithms. Here is how to do this. Any number cn be written s = 0 loghl, in terms of its bse 0 logrithm. Tking the nturl ogrithm of both sides, we get lnhl = ln@0 loghl D = loghl lnh0l Since lnh0l = , this epression mens tht lnhl = loghl This is very useful formul in numericl clcultions with logrithms, nd is worth memorizing. Logrithmic properties of the nturl logrithm From the definition of logrithm, two key properties follow: log c HbL = log c HL + log c HbL, since c log c c log c b = c log c +log c b, nd log c H ê L =-log c HL, since ê = -. (These properties re true for ny bse, nd so n unspecified bse c is used in these reltions.) From these properties, we cn get the other properties of logrithms. For emple, we cn show tht log c H ê bl = log c HL + log c H ê bl = log c HL - log c HbL by using the two key properties, nd we cn show tht log c H b L = b log c HL by interpreting b s the product of times itself b times, nd then repetedly using the reltion for the logrithm of product. log c H b L = log c H L = log c HL + log c HL log c HL = b log c HL These key properties re wht we need to confirm tht

11 Appewndi 3, Rises nd runs, slopes nd sums: tools from clculus 397 is in fct logrithm. Tht is, wht we need to do is to show tht the sum stisfies the two key properties of logrithms. The detils given below re not t ll essentil to using our results, but should you wnt to follow long, they do show how powerful the bsic ides of "slopes nd sums, rises nd runs" tht we hve developed here re. Nturl logrithm of product Using the integrl representtion, the logrithm of product is lnhbl ª b. We cn brek the sum up into the piece from to, nd the piece from to b. lnhbl ª + b The first piece is lnhl, by our definition of the nturl logrithm. This mens wht we hve to do is to show tht the second piece is lnhbl. We cn do this by chnging the scle from to y = ê. This chnge mens smll chnge in is equivlent to y, tht = corresponds to y =, nd tht = b corresponds to y = b. With these vlues, we cn write the second piece s b = b ÅÅÅ y y = b y y The lst sum is, by our definition, lnhbl nd so we hve shown tht lnhbl = lnhl + lnhbl Nturl logrithm of reciprocl Now let's consider the logrithm of n inverse, lnh ê L ª ê. The key step in working with this epression is to relize tht if we crry out sum in the opposite direction, the vlue of the sum chnges sign. To see this, note tht f HL f H+DL This mens we cn write f H+DL = f HL - f H +DL =-@ f H +DL - f HLD =- f HL

12 398 Notes on Generl Chemsitry, 2e ê =- ê. To rewrite the right hnd side so the sum strts t we cn chnge the scle from to y =. This chnge mens smll chnge in is equivlent to y ê, tht = ê corresponds to y =, nd tht = corresponds to y =. With these vlues, we cn write ê = Å y ê y ê = y y. The lst sum is, by our definition, lnhl nd so we hve shown tht lnh ê L = ê =- ê =-loghl. à Postscript I must confess tht I did not pprecite t ll just wht the nturl logrithm function ws when I first lerned clculus. I ws not ble to see the forest for the trees! It ws not until I begn teching nd so wnted to eplin just where the nturl logrithm comes from tht I cme to understnd wht we hve seen here. I find the nturl logrithm function quite remrkble. I hope these notes will be helpful in mking esier your journey to understnding this mzing function. Summry Here is summry of the results we hve developed for slopes of different functions. f HL ê 2 2 n, integer n > 0 n n- lnhl -, > n, integer n > sinh L cosh L -n -Hn+L cosh L - sinh L In these epressions, is constnt. We hve lso introduced the nottion tht loghl = log 0 HL nd lnhl = log HL, nd derived the equivlence lnhl = loghl.

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