chapter 5

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5-2 ®ðÅ骺¤º¯à

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5-3 ²z·Q®ðÅé

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5-4 ¼ö®e¶qªº¹êÅç´ú¶q

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§Q¥Î¤Uµ¥¦¡ «h¥i¥H¨DCV £Gt¬°¥Î¹q¥[¼ö®É¶¡¡A£G£c¬°¦b¦¹®É¶¡¤º·Å«×ªºÅܤÆ,m¬°®ðÅ骺½è¶q¡A«hcv ¬°¬Û¹ïÀ³¤§©w®e¤ñ¼ö¡A CPªº´ú¶q¤èªk»PCVÃþ¦ü¡F¦ý®ðÅé«O«ùµ¥À£¤O©Ò¥H¥Î®ðÅé«O«ùµ¥À£¤O¬y¹L¤@¼ö¶q­p¡A´úª¾®ðÅé³q¹L«á¤§·Å®t¡A®ðÅ骺¬y¶q«h¥i¥H­pºâ©wÀ£¼ö®e¶q¡F¦]¬° ¦b§CÀ£ªº±¡ªp¤U¡A¹êÅ窺µ²ªG»P²z·Q®ðÅé¬Ûªñ¡A(©w¸qcV©McP¬°²ö¦Õ¼ö®e¶q)²z·Q®ðÅ馳¤U¨Ò©Ê½è¡G(¨£½Ò¥»²Ä110­¶) 1¡GcV is a function of £c only 2¡GcP is a function of £c only 3¡GcP¡ÐcV = const. = R 4¡G£^=cP/cV = a function of £c only and >1 §CÀ£®Éªº³æ­ì¤l®ðÅé¡G¦pHe,Ne©MAr¤Î¤j¦h¼Æª÷ÄÝ»]®ð¦pNa,Cd©MHg¡A¦b«Ü¤jªº¤@­Ó·Å«×½d³ò¦³¤U¨Ò©Ê½è¡G 1¡G 2¡G 3¡G §CÀ£ªºÂù­ì¤l®ð Åé¤À¤lH2,D2,O2,N2,NO,COµ¥ 1¡G¡A¦b±`·Å®É¬O±`¼Æ¡AÀH·Å«×¼W¥[¦ÓÅܤj¡C 2¡G¡A¦b±`·Å®É¬O±`¼Æ¡AÀH·Å«×¼W¥[¦ÓÅܤj¡C 3¡G¡A¦b±`·Å®É¬O±`¼Æ¡AÀH·Å«×¼W¥[¦ÓÅܤp¡C¹ï©ó¦h­ì¤l®ðÅé¤À¤l¡A©Î¬O¬¡©Ê¸û¤jªº®ðÅé¤À¤l¡A¦pCO2, NH3, CH4, Cl2, Br2 µ¥¡A¨äCv¡A Cp©MgÀH·Å«×¤§ÅܤƦӦ³©Ò§ïÅÜ¡A¥B¨äÅܤÆÀH®ðÅé¦Ó¦U¦³¤£¦P¡C¹Ï5-5¬OH2ªºÀH·Å«×Åܤƪº´ú¶q¡A¦b§C·Å®É¡A±`·Å®É¡A¦Ó°ª·Å®É¬°9/2¡C¬°¤°»ò¡H¡H¡H¦]¬°§C·Å®É²¾°Ê®ÄÀ³©úÅã¡A¦Ó¶È¦³¤T­Ó¦Û¥Ñ«×¡C±`·Å®É¬O»P§Ú­Ì©Ò¹w´ú¬Û¦P¡A¦h¤F¨â­ÓÂà°Ê¦Û¥Ñ«×¡C°ª·Å®É®¶°Êªº®ÄÀ³¥çÅã²{¡C ³oºØÅܤơA¬O¥i¥H¥Î¯à¶q§¡¤À­ì²z¸ÑÄÀªº¡C(the principle of the equipartition of energy)¯à¶q§¡¤À­ì²z¬O«ü¨C¤@¦Û¥Ñ«×¨ÑÄm(1/2)nRqªº¤º¯à¡C¨ä¾lªºÂù­ì¤l¤À¤l¥i¥H¦³¼g¦¨

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5-6 Ruchhardt's Method ´ú g

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¦p¦¹«h¥i¥H¨D¥Xg¡C§Q¥Î³oºØ¤èªk¶q´úªº»~®t¨Ó·½¦³¤@) ®ðÅé«D²z·Q®ðÅé ¤G) ¼¯À¿¤´¦s¦b ¤T) The volume change is strictly adiabatic, but it is not absolutely adiabatic. The modified Ruchhard's experiment to get more accurate data can be found in the text book.

5-7 Speed of a Longitudinal wave

In this section we want to how g paly the role in the longitudinal wave. ·í¤@ª«Åé(©Î¨t²Î)¤§¬Y¤@³¡¤À³QÀ£ÁY«á¡A¦¹À£ÁYªº®ÄÀ³±N·|µ¥³t²v¡Aw¡A¶Ç»¼¦Ü¨ä¥¦³¡¤À¡A¦¹³t²vªº¤j¤p»P²Õ¦¨ª«½èªº¥»½è¦³Ãö¡C¤U¹Ï¤¤¡A¦pªG¬¡¶ë³Q¬I¤O P+DP ¦V¥k¥Hµ¥³t«×¡Aw0¡A²¾°Ê®É¡A·|²£¥ÍÀ£ÁY(compression)¥Hµ¥³t«×w¹B°Ê¡C·íÀ£ÁY¦æ¶iwt¶ZÂ÷®É¡A¬¡¶ë²¾°Êw0tªº¶ZÂ÷¡C

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¥B¾ã­ÓÀ£ÁY°Ï¤¶½èªº³t«×¬°w0¡AÀ£ÁY°Ï°Ê¶qÀH®É¶¡ªº¼W¥[¶q(Dp/Dt)¬°¡C¥t¤@ºØ¬Ýªk¬OÀ£ÁY°Ï½è¶q¼W¥[ªº³t²v¬°

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In general, the typical wavelength is from a few centimeters to several hundred centimeters. For gas at 0¢J, we have

therefore

which is much smaller than the wavelength. In the case of metal, K would be larger, but this will be compensated by the much larger values of w and q, hence it will be still smaller than 155 nm. Therefore the adiabatic condition is fulfilled. The volume changes which take place under the influence of a longitudinal wave at ordinary frequencies are adiabatic, not isothermal. Let

- = kS,

as the adiabatic compressibility. So

For ideal gas

and

where M is the molar mass and v is the molar volume. Hence

or

For example, the speed of sound in are at 0¢J is about 331 m/s, M for air is 28.96 kg/kmol, therefore g is 1.40. The speed of sound wave in a gas can be measured roughly by means of Kundt's tube.

5-8 Acoustical Thermometry(¥i¥H¥Î¨Ó¶q´ú²z·Q®ðÅé·Å)

In previous section, we discuss the longitudinal wave in a tube. For more accurate measurement, the instrument equiped with an acoustic interferometer, one end of the tube is a source of waves such as a piezoelectric crystal and at the other a receiver. The distance of source and receiver is kept constant, the frequency of wave can be varied, the various resonances are reached. We can also use this equipment to determine the ideal gas temperature £c by plotting the square of the speed of sound as a function of pressure. Then

where is the extrapolation of the square of the speed of sound to zero pressure, which assures ideal conditions.(¨£¹Ï5-11)

5-9 The Microscopic point of view

¥j¨å¼ö¤O¾Ç¬O«Ø¥ß¦b¨t²Î¥¨Æ[ªº¦æ¬°¤W¡A¥Î¥H´y­z¨t²Îªºª«²z¶q¤]¬O¥¨Æ[ªº¥­§¡­È¡C²Ä¤@©w«ß¥H¯à¶q¦uùÚªºªk«h¡A±N¼ö¡B¥\©M¤º¯à³sô¦b¤@°_¡C·í²Ä¤@©w«ßÀ³¥Î¨ì¦P¤@ºØÃþªº¨t²Î(¦pÀR¬yÅé)©Ò±o¤§Ãö«Y¦¡¹ï©ó¦¹ºØÃþ¤¤ªº¥ô¤@¨t²Î§¡¦¨¥ß¡A¤]´N¬O»¡¤£¦Ò¼{(©Î»¡µLªk¦Ò¼{)¦U¦PÃþ¨t²Î¤¤·LÆ[©Ê½èªº®t²§¡C¨Ò¦p¹ï©Ò¦³ÀR¬yÅé¨t²Î¤¤ªº©TÅé¡B²GÅé©M®ðÅ駡¦¨¥ß¡C¦pªGª¾¹DU©MV¡B£cªºÃö«Y¡A§Ú­Ì¥i¥H­pºâCV¡C¦bµ¥Åé¿n¹Lµ{¤¤¡A¦pªGª¾CV©M£cªºÃö«Y¡A¼ö¶q¥i¥H¥Ñ¤U¦¡ºâ¥X: ¦ý¬OÃö©óCV©MUªºÃö«Yªº¸Ô²ÓÃö³s¡A¥j¨å¼ö¤O¾Ç«oµLªk¥J²Óªº´£¨Ñ¡A¦ÓU ¥i¥H¥Î·LÆ[ªº¤èªk­pºâ¦Ó±o¡C¥j¨å¼ö¤O¾Çªº¥t¥~¤@­Ó·¥­­¬OµLªk§ä¥X©Î²z½×¾É¥Xª¬ºA¤èµ{¦¡¡C¨t²Îªºª¬ºA¤èµ{¦¡¡A¦h¥b¥Ñ¹êÅç¤èªk¨M©w¡C¦Ò¼{¨ì¨t²Î·LÆ[©Ê½èªº²z½×¦³¤G¡G¤@¬O®ðÅé°Ê¤O½×(Kinetic Theory)¡A¥t¤@¬O²Î­p¤O¾Ç(Statistical Mechanics)¡C¦¹¨âªÌ¥H¦Ò¼{¤À¤l¹B°Ê¬°µÛ²´¡A¦p¸I¼²¡A§@¥Î¤O¡A¯à¶qµ¥¡C®ðÅé°Ê¤O½×(Kinetic Theory)¥i¥H³B²z¤U­z«D¥­¿Åªºª¬ºA:(¥Ñ¨ä¬O²GºA¨t²Î) ¤@) Effusion¡G¤À¤l¥Ñ®e¾¹¤p¤Õº¯³z¥X¨Óªº¹Lµ{¡C ¤G) Laminar Flow¡GMolecules moving through a pipe under the action of a pressure difference. ¤T) Viscosity¡GMolecules with momentum moving across a plan and mixing with molecules of lesser momentum. ¥|) Heat Conduction¡GMolecules with kinetic energy moving across a plan and mixing with molecules of lesser energy. ¤­) Diffusion¡GMolecules of one sort moving across a plane and mixing with molecules of another sort. ¤») Chemical Kinetics¡G¨âºØ©Î¦hºØ¤Æ¾Ç¤À¤lªº¦³­­³t²vªº¤Æ¾Çµ²¦X ¤C) Brownian Motion¡G ²Î­p¤O¾Ç¥Ñ¤À¤l¯à¶qªºÆ[ÂI¥Xµo¡A¹ï©ó¥­¿Å¨t²Î¥i¥H¸ÑÄÀªº«Ü¦n¡A¦ý¹ï©ó«D¥­¿Å¨t²ÎÅé¨t¤´¦³¬Û·í¤jªºªÅ¶¡»Ý­n¥h¤F¸Ñ¡C 5-10 Kinetic Theory--Equation of state of an ideal gas ¥Î²³æªº®ðÅé°Ê¤O½×¨D²z·Q®ðÅ骺ª¬ºA¤èµ{¦¡¡C¨ä°ò¥»°²³]¦³ ¤@)¥ô¦ó¤@¨t²Î©Ò¥]§tªº²É¤l¼Æ¡AN¡A§¡«D±`¤j¡A¥B¥~Æ[³£¬Û¦PµLªk°Ï§O¡C¨Ò¦p²z·Q®ðÅé¦b¤@¤j®ðÀ£¤U0¢J®ÉÅé¿n¬°2.24¡Ñ104cm3¡A¦ý¦³6.0225¡Ñ1023­Ó²É¤l¡CÅé¿n¬° 1 cm3¡A¦³3¡Ñ1019­Ó²É¤l¡CÅé¿n¬° 1 mm3¡A¦³3¡Ñ1016­Ó²É¤l¡CÅé¿n¬° 1 mm3¡A¦³3¡Ñ107­Ó²É¤l¡C¤G) ²z·Q®ðÅé¤À¤l¥Ñ¤p¦Óµwªº²y©Ò²Õ¦¨¡A¥B¬°ÀH¾÷¹B°Ê¡C¤À¤lªºÅé¿n¥i¥H³Q©¿²¤¡A©Î»¡¤À¤l¤§¶¡ªº¶ZÂ÷¸û¤À¤lªº¤j¤p¤j³\¦h¡C¤T) ²z·Q®ðÅé¤À¤l¶¡¨S¦³¥ô¦óªº§@¥Î¤O¡A¦p§l¤Þ¤O¡B±Æ¥¸¤Oµ¥¡C ¥|) ²z·Q®ðÅé¤À¤l¶¡ªº¸I¼²§¡¬°¼u©Ê¸I¼²¡A»P¾Àªº¸I¼²¥ç¬°¼u©Ê¸I¼²¡C¤­) ²z·Q®ðÅé¤À¤l³B©óµL¥~¤O§@¥Îªº±¡ªp¤U®É¡A¤À³¡¬O§¡¤Ãªº¡C ¤») ²z·Q®ðÅé¤À¤l¹B°Ê³t«×¤è¦V¨S¦³¯S©wªº¨ú¦V¡C(¤]´N¬O¦U¤è¦V¹B°Êªº²É¤l¼Æ¬O¤@¼Ë¦h¡C¤C) ²z·Q®ðÅé¤À¤l¹B°Ê³t«×¬O¤@¤À§G¡A¨C¤@¤À¤lªº¹B°Ê³t«×³£¤£¦P¡C ³t«×¤è¦V¨S¦³¯S©wªº¨ú¦V¡A¦Ò¼{¤@¥ô·Nªº³t«×¦V¶qw¡A¦p¹Ï5-12©Ò¥Ü¡A¥Ñ­ìÂI¨ì­±¿ndA'¡A¦¹­±¿n©Ò¹ïÀ³¤§¥ßÅ騤¬° (³Ì¤j¥ßÅ騤ªº«×¶q¬°4p)³]¬O³t«×¤¶©ów¨ìw + dw¤§¶¡ªºÁ`²É¤l¼Æ¡A¦Ó³t«×¤¶©ów¨ìw + dw¤§¶¡¥B¤è¦V¬O´ÂµÛ¥ßÅ騤ªº²É¤l¼Æ¡A¡A¬° ¤W¦¡¤]»¡©ú²z·Q®ðÅé¤À¤l¹B°Ê³t«×¤è¦V¨S¦³¯S©wªº¨ú¦V¡C¦Ò¼{¤@¶ê¬W«¬ªºÅé¿ndV¡A¦p¹Ï5-13©Ò¥Ü¡A¡C­Y³]V¬OÁ`Åé¿n¡A«h¥u¦³¦b¶ê¬W«¬¤¤¤À¤l¦û©Ò¦³Á`¤À¤l¼Æªº¤ñ¨Ò¬°¡A«h Number of w, £c, f molecules striking dA in time dt = ¤W¦¡ªí¥Ü²z·Q®ðÅé¤À¤l¤À§G¬O§¡¤Ãªº¡C¥t¤@°²³]¬O©Ò¦³ªº¸I¼²³£¬O¼u©Ê¸I¼²¡A«h¥H£c¤J®g¦Ó¸gdA¤Ï¼uªº¤À¤l¨ä°Ê¶qªº§ïÅܬ°¡A¦]¬°¥u¦³««ª½¤À¶q¦³ÅܤơC¬G¥H£c¤J®g¦Ó¸gdA¤Ï¼u¤§©Ò¦³²z·Q®ðÅé¤À¤l©Ò³y¦¨Á`°Ê¶qªº§ïÅܬ°

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kªi¯÷°Ò±`¼Æ¡A¨ä­Èµ¥©ó1.3805¡Ñ10-23 J/K¡C µù¡G¥t¥~¤@ºØ¸Ñªk¬O±N¨t²Î¦Ò¼{¦¨¥¿¤èÅé¡A«h The momentum change for the collision per molecule is

and the average per collision is

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The force due to one collision is

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thus the pressure on the suface is

This is the same result we have discussed above.

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