( C) CLASS 10. TEMPERATURE AND ATOMS

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1 CLASS 10. EMPERAURE AND AOMS INRODUCION Boyle s understanding of the pressure-volue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between pressure and teperature were not deterined until the 1800 s, priarily because scientists needed to learn how to separate different gases GOALS Be able to convert teperatures between Celsius, Kelvin and Fahrenheit scales; Understand the physical eaning of the Kelvin teperature scale; Be able to explain and use Charles and Gay-Lussac s Laws; and Understand how teperature relates to the otion and kinetic energy of atos A MACROSCOPIC IEW OF EMPERAURE Definition and Units. he sybol represents teperature. here are three units for teperature. he British unit is included because it is used so coonly in this country. degrees Fahrenheit ( F) degrees Celsius ( C) degrees Kelvin (K) A degree sign is used for Celsius and Fahrenheit scales, but not for the Kelvin scale. Most of the world uses the Celsius scale, with the exception of a few third-world countries and the US, which use the Fahrenheit scale. he Kelvin scale is used priarily by scientists. Figure 10.1 copares the teperature scales. Standard teperature is defined as zero C or 73 K. Converting between the different systes requires the following equations: o convert fro Celsius to Fahrenheit: o convert fro Fahrenheit to Celsius: o convert fro Celsius to Kelvin: o convert fro Kelvin to Celsius: 9 ( C) F (10.3.1) ( F 3) C (10.3.) 9 5 K C+ 73 (10.3.3) C K 73 (10.3.4)

2 Celcius Fahrenheit Kelvin 100 C 1 F 373 K 0 C 0 C 70 F 93 K 3 F 73 K -73 C -459 F 0 K Figure 10.1: Coparison of the Fahrenheit, Celsius and Kelvin Scales CHARLES LAW (OR GAY-LUSSAC S LAW?) here was significant disagreeent about how the volue of a gas changed when the teperature changed because different experients often produced different results. his likely is because different types of gases had different aounts of water vapor in the and the water vapor would have a significant ipact on the teperature-dependence of the gas Gay-Lussac s Law. Joseph Gay-Lussac ( ) was a careful experienter and was able to exclude ost of the water vapor fro his apparatus. His results thus were ore accurate than previous experients. In 180, he showed that constant (10.4.1)a Like Boyle s Law, this relationship requires soe qualification: the volue-teperature law holds only when the pressure is constant and it works only for a subset of gases called ideal gases. We will study ideal gases at the end of this chapter; however, the iportant thing to know right now is that real gases behave like ideal gases at low pressures. We can write the volue-teperature law as a coparison of the volue and the teperature at two different ties: 1 (10.4.1)b 1 Iportant: hese equations only work if you use teperatures in Kelvin units. If you use Celsius units, you WILL get the wrong answer Charles Contribution. Gay-Lussac was not the first to discover this law. Jacques Alexandre César Charles ( ) discovered this volue-teperature relationship in 1787 (fifteen years earlier), but did not published it. Charles, who becae interested in science when he et the Aerican abassador to France, Benjain Franklin, found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand the sae aount over the sae 80-degree interval. Gay-Lussac did

3 iprove on Charles work and was the first to publish. Charles did not easure the coefficient of expansion, and the presence of water in the apparatus and the gases theselves caused Charles to obtain results indicating unequal expansion for watersoluble gases. Gay-Lussac did cite Charles earlier results, and this has resulted in this law being called either Gay-Lussac s Law or Charles Law he Meaning of the Kelvin Scale. When Boyle found the relationship between pressure and volue, he had only air with which to experient. Scientists in the early 1800s had the benefit of being able to study different gases. hey thus could copare and contrast the behavior of different gases. A plot of the volue as a function of teperature (Figure 10.) shows that (until the gas gets so cold that it becoes a liquid) the plot is a straight line. If you extrapolate the line back to zero volue, the lines for any of the gases extrapolate back to the sae intercept with the teperature axis. his special teperature at which the volues go to zero is -73 C, which is 0 K. his is the reason the Kelvin teperature scale is special it is the only one of the three teperature scales not based on arbitrary standards such as water boiling or freezing. 0 K corresponds to the lowest possible teperature possible SOLING PROBLEMS INOLING EMPERAURE AND OLUME Consider a balloon filled with air. At roo teperature, I have one volue. If I put the balloon in the freezer for a little while, the balloon will shrink. If I heat the balloon, it will grow larger. We can use the volue-teperature relationship to calculate how uch the volue will change when we cool a gas. EXAMPLE 10.1: If the teperature of a balloon filled with air doubles, what happens to the volue? olue -73 C Gas eperature Gas 1 Figure 10.: he relationship between volue and teperature. he dashed line is an extrapolation of the solid line and starts at the teperature where the gas changes into a liquid. Draw a picture 1 1 < known: 1 teperature before 1 volue before teperature after 1 need to find: volue after

4 Equation to use: Solve for the unknown Plug in nubers. 1 Answer: he volue will also double. EXAMPLE 10.: A volue of 6.0 L of gas is at a teperature of 30.0 C. If the teperature decreases to 10.0 C, what is the volue of the gas? Draw a picture L 1 30 C 10 C known: 1 teperature before 30.0 C 1 volue before 6.0 L teperature after 10.0 C need to find: volue after Equation to use: Iportant: you have to convert all of the teperatures into Kelvin! Solve for the unknown Plug in nubers K K K ( 6.0 L) 83.0 K L Answer: (3 s.f.) 7.8 L Check that this akes sense. If the teperature increases, we know that the volue should increase. If I had gotten a saller volue, I would know I had done soething wrong GAY-LUSSAC S LAW he origin of this law is not known exactly, but Gay-Lussac found the following to hold true: P constant (10.6.1)a

5 his can be written the sae way the pressure-volue relationship was written: P1 P (10.6.1)b 1 Again, all teperature s ust be in Kelvin, and this is valid only when the volue is held constant and only for ideal gases. EXAMPLE 10.3: A gas at a pressure of 1.30 at is at a teperature of 30.0 C. If the pressure is increased to.50 at, what is the teperature of the gas? Assue that the volue reains constant P 1, 1.30 at C P,.50 at? Draw a picture known: 1 teperature before 30.0 C P 1 pressure before 1.30 at P pressure after.50 at need to find: teperature after Equation to use: P P 1 1 Reeber: you have to convert all of the K teperatures into Kelvin! Solve for the unknown P 1 P1 Plug in nubers..50at 303K 1.30 at K Answer: (3 s.f.) 583 K Check that this akes sense. If the pressure increases, I expect the teperature to increase, so this answer akes sense A MICROSCOPIC HEORY OF EMPERAURE Reeber that Bernoulli s kinetic theory ade specific predictions for the teperature, pressure and volue dependence of gases. he third assuption of Bernoulli s theory, which we delayed discussing until now, was that the teperature of a gas was proportional to the speed at which the gas particle oved.

6 Dependence of Pressure on eperature. his odel can account for why pressure increases when teperature increases: As teperature increases, the speeds of the gas particles increase. he gas particles therefore collide ore frequently with the wall and exert greater force on the vessel wall Dependence of olue on eperature. his odel can account for why volue increases when teperature increases: As teperature increases, the speeds of the gas particles increase. If the walls of the vessel holding the gas can ove, the ore frequent collisions will cause the walls of the vessel to ove further outward Websites. A couple nice websites that show aniations of the olecular properties of ideal gases are: Relationship between Average Kinetic Energy and Average Speed. In Bernoulli s theory, the average kinetic energy KE of a gas is given by KE ( ) N v (10.7.1) 1 av where N is the nuber of gas particles, each gas particle has a ass, and v avg is the average speed of the gas particles. Kinetic energy is energy of otion. he kinetic energy is larger when the gas particles are oving faster. he units of kinetic energy are joules (J), where kg 1J1 1N s Bernoulli s theory predicts that the average kinetic energy of a gas also should be proportional to the teperature. KE Nk (10.7.) where N again is the nuber of gas particles, and k is Boltzann s constant, which has a value of 1.38x10-3 J K he teperature ust be in Kelvin for this equation to work. EXAMPLE 10.4: A gas at a teperature of 0.0 C contains gas particles. What is its average kinetic energy? Draw a picture known: teperature 0.0 C N nuber of atos k 3 J K 3 No picture is necessary need to find: KE average kinetic energy. Equation to use: KE 3 Nk Reeber: you have to convert the teperature into Kelvin! K

7 Plug in nubers. Answer: (3 s.f.) KE 3 3 J ( )( K )( ) K J 3 KE J Relationship between average speed and teperature. We can cobine Equations (10.7.) and (10.7.1): ( avg ) 1 3 v 3 ( vavg ) 3k vavg (10.7.3) Equation (10.7.3) ephasizes the correlation between the speed of the gas particles and the teperature. As the teperature decreases, the average speed of the gas particles also decreases. At the tie, scientists didn t know the ass of the individual gas particles; however, that inforation was about to be deterined. EXAMPLE 10.5: What is the average speed of a hydrogen olecule at 93 K? (he ass of a hydrogen olecule is kg). Draw a picture k k No picture necessary known: 1 teperature 93 K kg need to find: v avg average speed Equation to use: v avg 3k Plug in nubers. v avg 3k 3 J ( K ) K kg J kg he units aren t obvious, so we need to show that they work out to a unit appropriate for velocity. J kg s s kg kg s

8 Answer: (3 s.f.) v avg s

9 10.8. SUMMARIZE Definitions: Define the following in your own words. Write the sybol used to represent the quantity where appropriate. 1. Kinetic Energy. Standard teperature Equations: For each question: a) Write the equation that relates to the quantity b) Define each variable by stating what the variable stands for and the units in which it should be expressed, and c) State whether there are any liitations on using the equation. 1. he equations that allow you to convert between Fahrenheit and Celsius teperatures. he equations that allow you to convert between Kelvin and Celsius teperatures 3. he relationship between the volue and the teperature of a gas. 4. he relationship between the pressure and the teperature of a gas. 5. he relationship between the average kinetic energy and the average speed of gas particles according to kinetic theory. 6. he relationship between average kinetic energy and teperature. 7. he relationship between the average speed of gas particles and their teperature Concepts: Answer the following briefly in your own words. 1. Copare the acroscopic and icroscopic views of teperature. How does teperature relate to olecular otion?. What is the physical significance of the Kelvin teperature scale? 3. Explain how the Kelvin teperature scale relates to the volue vs. teperature relationship. 4. When using either the pressure-teperature or the volue-teperature relationships, you always have to reeber one thing about aking calculations. What is it? 5. Can you ever have a negative teperature on the Kelvin scale? Why not?

10 Your Understanding 1. What are the three ost iportant points in this chapter?. Write three questions you have about the aterial in this chapter Questions to hink About 1. When we studied Boyle s Law, we had to plot P vs. 1/ to get a linear relationship. Why can we plot vs. and have a linear relationship?. Why did we have to extrapolate the volue vs. teperature graphs back to zero volue? 3. Why does water boil at lower teperatures at high altitudes? 4. Is it eaningful to say that an object at a teperature of 00 C is twice as hot as one at 100 C? 5. In the table of densities in class 4, the densities for the gases are specified at a particular teperature. Why is this done? Would the density of a gas change with teperature? 6. You have a Fahrenheit theroeter and a Celsius theroeter in the sae roo. hey show the sae teperature. What is the teperature of the roo? Probles 1. Convert: 61 K to C; -50 C to K; 70 F to C; 0 F to C.. he record low teperature in Lincoln is -33 F. What is this teperature in C and in K? 3. he record high teperature in Lincoln is 115 F. What is this teperature in C and in K? 4. What is noral body teperature in C? 5. A gas is held at a pressure of 4.58 at at a teperature of 5.6 C. If the teperature is changed to C, what is the new pressure? 6. A gas of volue 545 c 3 has a teperature of 34.0 C. If the volue is reduced to 305 c 3, what is the new teperature? 7. What is the average speed of gas particles at roo teperature, assuing that the gas particles have a ass of kg? 8. A gas has a teperature of 40 C. What is the average kinetic energy of one olecule of that gas? 9. What is the kinetic energy of a H olecule at a teperature of 67 C? 10. At constant pressure, the volue of a gas saple is direction proportional to: a) the size of its gas particles, b) its Fahrenheit teperature, c) its Celsius teperature, or d) its teperature in Kelvin. 11. he volue of a gas saple is increased while its teperature is held constant. he gas exerts a lower pressure on the walls of its container because its olecules strike the walls a) less often, b) with lower speed, c) with less energy and d) with less force c 3 of hydrogen is at 0 C and ordinary atospheric pressure. a) If the pressure is tripled while the teperature is held constant, what is the volue of the gas? b) If the teperature then increases to 73 C while the pressure is held constant, what will the volue be? 13. An air tank used for scuba diving has a pressure valve that is set to open if the pressure inside the tank reaches 8 MPa. he noral pressure in the tank when full at 0 C is 0 MPa. If the tank is heated after being filled to 0 MPa, at what teperature will the safety valve open?

11 14. o what teperature in Celsius ust a gas saple initially at 0 C be heated if its volue is to double while its pressure reains the sae? 15. o what teperature ust a gas saple initially at 7 C be raised to double the average kinetic energy of its olecules? 16. he average speed of a hydrogen olecule at 0 C is about 1.6 k s. A carbon dioxide olecule has about ties the ass of a hydrogen olecule. What is the average speed of a carbon dioxide olecule at 0 C?

12 PHYS 61 Spring 007 HW 11 HW Covers Class 10 and is due February nd, ( pts) c 3 of hydrogen gas is at 0 C and 1.00 at pressure. a) If the pressure is tripled while the teperature is held constant, what is the volue of the gas? b) After the change in part a), the teperature increases to 73 C while the pressure is held constant. What is the new volue?. A particular gas at a teperature has an average speed given by Equation (10.7.3). By how uch does the teperature have to change if the average speed is to double?