CHAPTER 13 Temperature and Kinetic Thery Units Atmic Thery f Matter Temperature and Thermmeters Thermal Equilibrium and the Zerth Law f Thermdynamics Thermal Expansin Thermal Stress The Gas Laws and Abslute Temperature The Ideal Gas Law Prblem Slving with the Ideal Gas Law Ideal Gas Law in Terms f Mlecules: Avgadr s Number Kinetic Thery and the Mlecular Interpretatin f Temperature Distributin f Mlecular Speeds Real Gases and Changes f Phase Vapr Pressure and Humidity Diffusin Atmic Thery f Matter Atmic and mlecular masses are measured in unified atmic mass units (u). This unit is defined s that the carbn-1 atm has a mass f exactly 1.0000 u. Expressed in kilgrams: Brwnian mtin is the jittery mtin f tiny flecks in water; these are the result f cllisins with individual water mlecules. On a micrscpic scale, the arrangements f mlecules in slids (a), liquids (b), and gases (c) are quite different. 1
Temperature and Thermmeters Temperature is a measure f hw ht r cld smething is. Mst materials expand when heated. Thermmeters are instruments designed t measure temperature. In rder t d this, they take advantage f sme prperty f matter that changes with temperature. Cmmn thermmeters used tday include the liquid-in-glass type and the bimetallic strip. Temperature is generally measured using either the Fahrenheit r the Celsius scale. The freezing pint f water is 0 C, r 3 F; the biling pint f water is 100 C, r 1 F. Kelvin Scale When the pressure f a gas ges t zer, its temperature is 73.15º C This temperature is called abslute zer This is the zer pint f the Kelvin scale 73.15º C = 0 K T cnvert: T C = T K 73.15 The size f the degree in the Kelvin scale is the same as the size f a Celsius degree
Cmparing Temperature Scales Cnverting Amng Temperature Scales T C T K 73.15 9 TF TC 5 3 5 TC TF 9 3 9 TF TC 5 Thermal Equilibrium and the Zerth Law f Thermdynamics Tw bjects placed in thermal cntact will eventually cme t the same temperature. When they d, we say they are in thermal equilibrium. The Zerth law f thermdynamics says that if tw bjects are each in equilibrium with a third bject, they are als in thermal equilibrium with each ther. Thermal Expansin The thermal expansin f an bject is a cnsequence f the change in the average separatin between its cnstituent atms r mlecules At rdinary temperatures, mlecules vibrate with a small amplitude As temperature increases, the amplitude increases This causes the verall bject as a whle t expand Linear expansin ccurs when an bject is heated. Here, α is the cefficient f linear expansin. Vlume expansin is similar, except that it is relevant fr liquids and gases as well as slids: Here, β is the cefficient f vlume expansin. Fr unifrm slids, 3
Example 1: Bridge Expansin The steel bed f a suspensin bridge is 00 m lng at 0 C. If the extremes f temperature t which it might be expsed are 30 C t 40 C, hw much will it cntract and expand? 6 L (1 x10 / C )(00 m)(40 C 0 C) 4.8x10 m expand 6 L (1 x10 / C)(00 m)( 50 C ) 1.0x10 m cntract Example : A steel railrad track has a length f 30,000 m when the temperature is 0 C. What is its length n a ht day when the temperature is 40 C? If the track can t mve what is the stress in the track due t the temperature change? 6 1 a) [11 10 ( L L T x C) ](40.0 C) 0.013m L L L 30.013 m b) F L 0.013m A L 30.0m 11 7 Y (.00x10 Pa) 8.67x10 Pa Example 3: Ring n a Rd An irn ring is t fit snugly n a cylindrical irn rd. At 0 C, the diameter f the rd is 6.445 cm and the inside diameter f the ring is 6.40 cm. T slip ver the rd, the ring must be slightly larger than the rd diameter by abut 0.008 cm. T what temperature must the ring be brught if its hle is t be large enugh s it will slip ver the rd? L 6.453cm 6.40cm T 430C 6 1 L (1 x10 C )(6.40 cm) S it must be raised at least t: T (0 C 430 C ) 450 C Example 4: Gas tank in the sun The 70-L steel gas tank f a car is filled t the tp with gasline at 0 C. The car is then left t sit in the sun, and the tank reaches a temperature f 40 C. Hw much gasline d yu expect t verflw frm the tank? The gasline expands: 6 01 V V (950 10 )(70 )(0 T x C L C ) 1.3L The tank increases in vlume by: 6 01 V (36x10 C )(70 L(0 C ) 0.050 L 4
Water behaves differently frm mst ther slids its minimum vlume ccurs when its temperature is 4 C. As it cls further, it expands, as anyne wh has left a bttle in the freezer t cl and then frgets abut it can testify. Example 5: Glbal Warming and Castal Flding Estimate the fractinal change in the vlume f Earth s ceans due t an average temperature change f 1 3 C. Use the fact that the average depth f the cean is 4.00x10 m t estimate the 4 1 change in depth. Nte that water.07x10 ( C) a) Find the fractinal change in vlume V V T Divide the vlume expansin equatin by b) Find the apprximate increase in depth V and substitute V T (.07x10 ( C) x10 V 4 1 4 L LT LT 3 5 1 6.90 10 ( L x C) (4000 m)(1 C) 0.3m Unusual Behavir f Water As the temperature f water increases frm 0ºC t 4 ºC, it cntracts and its density increases Abve 4 ºC, water exhibits the expected expansin with increasing temperature Maximum density f water is 1000 kg/m 3 at 4 ºC Thermal Stresses A material may be fixed at its ends and therefre be unable t expand when the temperature changes. It will then experience large cmpressive r tensile stress thermal stress when its temperature changes. The frce required t keep the material frm expanding is given by: 5
where E is the Yung s mdulus f the material. Therefre, the stress is: F TEA Example 6: Stress in cncrete n a ht day A highway is t be made f blcks f cncrete 10 m lng placed end t end with n space in between them t allw fr expansin. If the blcks were placed at a temperature f 10 C, what frce f cmpressin wuld ccur if the temperature reached 40 C? The cntact area between each blck is 0.0 m. Will fracture ccur? F TEA 6 9 (1 10 / x C )(30 C )(0x10 N / m )(0.0 m ) 6 1.4x10 N The stress, F/A is (1.4 x10 N)/(0.0 m ) 7.0x10 N / m 6 6 This is nt far frm the ultimate strength f cncrete. Ideal Gas A gas des nt have a fixed vlume r pressure In a cntainer, the gas expands t fill the cntainer Mst gases at rm temperature and pressure behave apprximately as an ideal gas Characteristics f an Ideal Gas Mles Cllectin f atms r mlecules that mve randmly Exert n lng-range frce n ne anther Each particle is individually pint-like Occupying a negligible vlume It s cnvenient t express the amunt f gas in a given vlume in terms f the number f mles, n mass n mlar mass One mle is the amunt f the substance that cntains as many particles as there are atms in 1 g f carbn-1 Fr example, the number f mles in 13 g f CO is 13g n 3.0 ml. 44 g / ml This prprtin can be written as an equatin: PV nrt 6
Ideal Gas Law PV = n R T R is the Universal Gas Cnstant R = 8.31 J / mle. K R = 0.081 L. atm / mle. K Is the equatin f state fr an ideal gas The Gas Laws and Abslute Temperature The relatinship between the vlume, pressure, temperature, and mass f a gas is called an equatin f state. We will deal here with gases that are nt t dense. Byle s Law: the vlume f a given amunt f gas is inversely prprtinal t the pressure as lng as the temperature is cnstant. The vlume is linearly prprtinal t the temperature, as lng as the temperature is smewhat abve the cndensatin pint and the pressure is cnstant: Extraplating, the vlume becmes zer at 73.15 C; this temperature is called abslute zer. Prblem Slving with the Ideal Gas Law Useful facts and definitins: Standard temperature and pressure (STP) Vlume f 1 ml f an ideal gas is.4 L If the amunt f gas des nt change: 7
Always measure T in kelvins P must be the abslute pressure Example 7: Vlume f ne ml at STP Determine the vlume f 1.00 ml f any gas at STP, assuming it behaves like an ideal gas. nrt (1.00 ml)(8.315 J / ml K)(73 K) V 5 P (1.013 x10 N / m ) 3 3 3 Since 1 liter is 1000cm 1x10 m, 1 ml f any gas has a vlume f.4 L at STP Example 8: The mass f gas 3 A flexible cntainer f xygen (O, mlecular mass = 3u) at STP has a vlume f 10.0m. What is the mass f gas enclsed? n 3 (10.0 m ) 3 3 (.4x10 m / ml) 446 ml. Since 1 ml has a mass f 0.030 kg, the mass f xygen is.4x10 m (446 ml)(0.030 kg / ml) 14.3kg m 3 3 Ideal Gas Law in Terms f Mlecules: Avgadr s Number Since the gas cnstant is universal, the number f mlecules in ne mle is the same fr all gases. That number is called Avgadr s number: 3 NA 6.0x10 The number f mlecules in a gas is the number f mles times Avgadr s number: Kinetic Thery and the Mlecular Interpretatin f Temperature The mlecules interact nly by shrt-range frces during elastic cllisins The mlecules make elastic cllisins with the walls The gas under cnsideratin is a pure substance, all the mlecules are identical Pressure f an Ideal Gas The pressure is prprtinal t the number f mlecules per unit vlume and t the average translatinal kinetic energy f a mlecule 8
The pressure is prprtinal t the number f mlecules per unit vlume and t the average translatinal kinetic energy f the mlecule Pressure can be increased by Increasing the number f mlecules per unit vlume in the cntainer Increasing the average translatinal kinetic energy f the mlecules Increasing the temperature f the gas Example 9: The ideal gas law t analyze a system f gas. An ideal gas at 0 5 C and a pressure f 1.50x10 Pa is in a cntainer having a vlume f 1.00L. a) Determine the number f mles f gas in the cntainer: PV nrt 5 3 3 PV (1.50 x10 Pa)(1.00 x10 m ) n 6.61x10 ml RT (8.31 J / ml K)(93 K) b) The gas pushes against a pistn, expanding t twice its riginal vlume, while the pressure falls t atmspheric pressure. Find the temperature after the gas expands t.00 L. Divide the ideal gas law fr the final state by the ideal gas law fr the initial state. Pf Vf nrt f Pf Vf Tf PV nrt PV T i i i i i i 5 PV f f (1.01 x10 Pa)(.00 L) Tf Ti 5 (93 K ) 395 K PV (1.50 x10 Pa)(1.00 L) i i Kinetic Thery and the Mlecular Interpretatin f Temperature cnt. Assumptins f kinetic thery: large number f mlecules, mving in randm directins with a variety f speeds mlecules are far apart, n average mlecules bey laws f classical mechanics and interact nly when clliding cllisins are perfectly elastic Temperature is prprtinal t the average kinetic energy f the mlecules 1 3 mv k T B The ttal kinetic energy is prprtinal t the abslute temperature 3 KEttal nrt Example 10: Mlecular KE What is the average translatinal kinetic energy f mlecules in a gas at37 C? 3 KE kt 3 (1.38 10 / )(310 ) 6.4 10 3 1 x J K K x J 9
Internal Energy In a mnatmic gas, the KE is the nly type f energy the mlecules can have U is the internal energy f the gas In a plyatmic gas, additinal pssibilities fr cntributins t the internal energy are rtatinal and vibratinal energy in the mlecules Speed f the Mlecules 3 U nrt Expressed as the rt-mean-square (rms) speed v rms 3 kb T 3 RT m M At a given temperature, lighter mlecules mve faster, n average, than heavier nes Lighter mlecules can mre easily reach escape speed frm the earth Example 11: Speeds f air mlecules What is the rms speed f air mlecules (O and N) at rm temperature (0 C )? Thus fr xygen 7 6 m( O ) (3)1.67 x10 kg) 5.3x10 kg 7 6 m( N) (8)(1.68 x10 kg) 4.7x10 kg v rms 3 3 kt (3)(1.38x10 J / K)(93 K) 6 m (5.3x10 kg) 480 m / s Fr nitrgen v = 510 m/s (abut 1000 mi/hr) Example 1: Armageddn! 4 A cmet half a km in radius cnsisting f ice at 73K hits Earth at a speed f 4x10 m / s (assume that all KE cnverts t thermal energy n impact and all thermal energy ges int warming the cmet. a) Calculate the vlume and mass f the ice. 4 3 4 (3.14)(500 ) 3 5.3 10 8 3 V r m x m 3 3 3 8 3 11 m V (917 kg / m )(5.3x10 m ) 4.80x10 kg b) Use cnservatin f energy t find the final temperature f the cmet material. 10
Qmelt Qwater Qvapr K 0 1 ml f mcwater Twater mlv mcsteamt steam mv 1 mc T K mv steam ( 373 ) 0 The first three terms are negligible cmpared t the KE. (0 ) 0 1 1 v (4.00 x 10 m / s ) c 010 J / Kg K 4 5 T 373K 373K 3.98x10 K steam c) Assuming the steam retains a spherical shape and has the same initial vlume as the cmet, calculate the pressure f the steam using the ideal gas law. This law actually desn t apply t a system at such high pressure and temperature, but can be used t get an estimate. 11 1ml 15 n (4.80x10 kg).67x10 ml 0.018kg 13 5 nrt (.67x10 ml)(8.31 J / ml K)(3.98x10 K) 11 P 8 3 P 1.69x10 Pa V 5.3x10 m several 100 x greater than shear stress f steel Real Gases and Changes f Phase The curves here represent the behavir f the gas at different temperatures. The cler it gets, the farther the gas is frm ideal. In curve D, the gas becmes liquid; it begins cndensing at (b) and is entirely liquid at (a). The pint (c) is called the critical pint. Belw the critical temperature, the gas can liquefy if the pressure is sufficient; abve it, n amunt f pressure will suffice. 11
A PT diagram is called a phase diagram; it shws all three phases f matter. The slid-liquid transitin is melting r freezing; the liquid-vapr ne is biling r cndensing; and the slid-vapr ne is sublimatin. Phase diagram f water The triple pint is the nly pint where all three phases can cexist in equilibrium. Phase diagram f carbn dixide Vapr Pressure and Humidity An pen cntainer f water can evaprate, rather than bil, away. The fastest mlecules are escaping frm the water s surface, s evapratin is a cling prcess as well. The inverse prcess is called cndensatin. When the evapratin and cndensatin prcesses are in equilibrium, the vapr just abve the liquid is said t be saturated, and its pressure is the saturated vapr pressure. The saturated vapr pressure increases with temperature. 1
A liquid bils when its saturated vapr pressure equals the external pressure. Partial pressure is the pressure each cmpnent f a mixture f gases wuld exert if it were the nly gas present. The partial pressure f water in the air can be as lw as zer, and as high as the saturated vapr pressure at that temperature. Relative humidity is a measure f the saturatin f the air. When the humidity is high, it feels muggy; it is hard fr any mre water t evaprate. The dew pint is the temperature at which the air wuld be saturated with water. If the temperature ges belw the dew pint, dew, fg, r even rain may ccur. Diffusin Even withut stirring, a few drps f dye in water will gradually spread thrughut. This prcess is called diffusin. 13
Diffusin ccurs frm a regin f high cncentratin twards a regin f lwer cncentratin. The rate f diffusin is given by: In this equatin, D is the diffusin cnstant. 14