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Chapter 13 Temperature & the Ideal Gas

Chapter 13 Temperature & the Ideal Gas. Chapter 13: Temperature & the Ideal Gas Temperature Temperature scales Thermal expansion of solids and liquids Molecular picture of a gas Absolute temperature and the ideal gas law Kinetic theory of the ideal gas Collisions between gas molecules.

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Chapter 13 Temperature & the Ideal Gas

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  1. Chapter 13 Temperature & the Ideal Gas

  2. Chapter 13: Temperature & the Ideal Gas • Temperature • Temperature scales • Thermal expansion of solids and liquids • Molecular picture of a gas • Absolute temperature and the ideal gas law • Kinetic theory of the ideal gas • Collisions between gas molecules Ch13c-Temperature-Revised-4/25/10

  3. Temperature Heat is the flow of energy due to a temperature difference. Heat always flows from objects at high temperature to objects at low temperature. When two objects have the same temperature, they are in thermal equilibrium. Ch13c-Temperature-Revised-4/25/10

  4. The Zeroth Law of Thermodynamics If two objects are each in thermal equilibrium with a third object, then the two objects are in thermal equilibrium with each other. Ch13c-Temperature-Revised-4/25/10

  5. Thermodynamics “Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you any more.” - Arnold Sommerfeld Thermodynamics is the scientific study of work, heat, and the related properties of chemical and mechanical systems. Ch13c-Temperature-Revised-4/25/10

  6. Temperature Measurement Requirement: A substance that exhibits a measureable change in one of its physical properties as the temperature changes. Examples: Volume change with temperature is a common physical property used for the measurement of temperature The change of electrical resistance with temperature is also used to measure temperature. Ch13c-Temperature-Revised-4/25/10

  7. Not a Good Temperature Measurement Instrument Ch13c-Temperature-Revised-4/25/10

  8. Mercury Thermometer A mercury-in-glass thermometer, invented by German physicist Daniel Gabriel Fahrenheit. Bulb – reservoir for the thermometric liquid Stem – glass capillary tube through which the mercury or organic liquid fluctuates with changes in temperature Scale – the scale is graduated in degrees, fractions or multiples of degrees Contraction Chamber– an enlarged capillary bore which serves to reduce a long length of capillary, or prevent contraction of the entire liquid column into the bulb Expansion Chamber– an enlargement of the capillary bore at the top of the thermometer to prevent build up of excessive pressures in gas-filled thermometers http://www.omega.com/Temperature/pdf/INTRO_GLASS_THERM.pdf Ch13c-Temperature-Revised-4/25/10

  9. Temperature Calibration Celsius and Fahrenheit are two different temperature scales but they are calibrated to the same reference points. The two scales have different sized degree units and they label the steam point and the ice point with different temperature values. The “Zero” values on both scales were picked for convenience and can cause problems. Ch13c-Temperature-Revised-4/25/10

  10. Temperature Measurement Calibration This process ensures the accuracy of the measurement. It requires the availability of a repeatable set of states of matter. The boiling point of water and the freezing point of water are two examples. Transferability The calibration needs to be transferable to other measuring instruments. Standards There needs to be agreed upon calibration procedures to allow transferability. Ch13c-Temperature-Revised-4/25/10

  11. Temperature Scale • Anders Celsius devised the Celsius scale. This was published in The origin of the Celsius temperature scale in 1742. • Celsius originally defined his scale "upside-down” • boiling point of pure water at 0 °C and • the freezing point at 100 °C. • One year later Frenchman Jean Pierre Cristin proposed to invert the scale. He named it Centigrade. • In later years, probably when the lower reference temperature was switch to the Triple Point, the name Celsius was restored. Ch13c-Temperature-Revised-4/25/10

  12. Fahrenheit Temperature Scale It was in 1724 that Gabriel Fahrenheit, an instrument maker of Däanzig and Amsterdam, used mercury as the thermometric liquid. Fahrenheit described how he calibrated the scale of his mercury thermometer: "placing the thermometer in a mixture of sal ammoniac or sea salt, ice, and water a point on the scale will be found which is denoted as zero. A second point is obtained if the same mixture is used without salt. Denote this position as 30. A third point, designated as 96, is obtained if the thermometer is placed in the mouth so as to acquire the heat of a healthy man." (D. G. Fahrenheit,Phil. Trans. (London) 33, 78, 1724) On this scale, Fahrenheit measured the boiling point of water to be 212. Later he adjusted the freezing point of water to 32 so that the interval between the boiling and freezing points of water could be represented by the more rational number 180. Temperatures measured on this scale are designated as degrees Fahrenheit (° F). http://www.cartage.org.lb/en/themes/sciences/Physics/Thermodynamics/AboutTemperature/Development/Development.htm Ch13c-Temperature-Revised-4/25/10

  13. Absolute or Kelvin scale Fahrenheit scale Celsius scale Water boils* 373.15 K 212 F 100 C Water freezes* 273.15 K 32 F 0 C Absolute zero 0 K 459.67 F 273.15C Temperature Scales (*) Values given at 1 atmosphere of pressure. • In general, as the temperature of a substance is lowered the atomic motion and vibrations slow down. • Temperature can be related to molecular energy. Negative temperature values imply erroneous negative energy. • The Kelvin temperature scale avoids this problem. Ch13c-Temperature-Revised-4/25/10

  14. Comparison of Temperature Scales The temperature scales are related by: Fahrenheit/Celsius Output Intput Absolute/Celsius Ch13c-Temperature-Revised-4/25/10

  15. Example (text problem 13.3): (a) At what temperature (if any) does the numerical value of Celsius degrees equal the numerical value of Fahrenheit degrees? Require TF = TC (b) At what temperature (if any) does the numerical value of degree kelvins (T) equal the numerical value of Fahrenheit degrees? Require T = TF Ch13c-Temperature-Revised-4/25/10

  16. Comparison of Temperature Scales Ch13c-Temperature-Revised-4/25/10

  17. Thermal Expansion of Solids and Liquids Ch13c-Temperature-Revised-4/25/10

  18. Thermal Expansion of Solids and Liquids Most objects expand in volume when their temperature increases and decrease in volume when their temperatures decrease. Exception: Water expands in volume when it freezes Ch13c-Temperature-Revised-4/25/10

  19. Why Ice Cubes Float Most fluids contract as they cool making the solid phase more dense than the liquid. Water behaves this way until it reaches 4 oC. Below this temperature it begins to expand yielding a solid phase that is less dense than the liquid. Therefore ice cubes float and ice forms on the top of lakes rather than the bottom. The fish say thank you! Ch13c-Temperature-Revised-4/25/10

  20. Why Ice Cubes Float Ch13c-Temperature-Revised-4/25/10

  21. Why Ice Cubes Float Ch13c-Temperature-Revised-4/25/10

  22. Thermal Linear Expansion Ch13c-Temperature-Revised-4/25/10

  23. Thermal Linear Expansion If ΔT is negative then the rod would contract Ch13c-Temperature-Revised-4/25/10

  24. Thermal Linear Expansion An object’s length after its temperature has changed is  is the coefficient of linear expansion where T = TT0 and L0 is the length of the object at a temperature T0. Ch13c-Temperature-Revised-4/25/10

  25. Thermal Expansion in Multiple Rods Ch13c-Temperature-Revised-4/25/10

  26. Example (text problem 13.84): An iron bridge girder (Y = 2.01011 N/m2) is constrained between two rock faces whose spacing doesn’t change. At 20.0 C the girder is relaxed. How large a stress develops in the iron if the sun heats the girder to 40.0 C?  = 12106 K1 (from Table 13.2) Using Hooke’s Law: Ch13c-Temperature-Revised-4/25/10

  27. L0 L0+ L How does the area of an object change when its temperature changes? The blue square has an area of L02. With a temperature change T each side of the square will have a length change of L = TL0. Ch13c-Temperature-Revised-4/25/10

  28. The fractional change in area is: Ch13c-Temperature-Revised-4/25/10

  29. The fractional change in volume due to a temperature change is: For solids  = 3 Ch13c-Temperature-Revised-4/25/10

  30. Does the Hole Grow or Shrink With Higher Temperatures? Ch13c-Temperature-Revised-4/25/10

  31. The Hole Grows With Higher Temperatures Ch13c-Temperature-Revised-4/25/10

  32. The Hole Acts Like It Is Made of the Same Material Ch13c-Temperature-Revised-4/25/10

  33. Molecular Picture of a Gas Ch13c-Temperature-Revised-4/25/10

  34. Molecular Picture of a Gas The number density of particles is N/V where N is the total number of particles contained in a volume V. If a sample contains a single element, the number of particles in the sample is N = M/m. M is the total mass of the sample divided by the mass per particle (m). Ch13c-Temperature-Revised-4/25/10

  35. Avogadro’s Number - NA One mole of a substance contains the same number of particles as there are atoms in 12 grams of 12C. The number of atoms in 12 grams of 12C is Avogadro’s number. Ch13c-Temperature-Revised-4/25/10

  36. A carbon-12 atom by definition has a mass of exactly 12 atomic mass units (12 u). x 12 12 u = This is the conversion factor between the atomic mass unit and kg (1 u = 1.661027 kg). NA and the mole are defined so that a 1 gram sample of a substance with an atomic mass of 1 u contains exactly NA particles. The atomic mass MA (from the periodic table) is the gram weight of a mole of that substance. “u” is the average nucleon mass. Ch13c-Temperature-Revised-4/25/10

  37. Example (text problem 13.37): Air at room temperature and atmospheric pressure has a mass density of 1.2 kg/m3. The average molecular mass of air is 29.0 u. How many air molecules are there in 1.0 cm3 of air? The total mass of air in the given volume is: Ch13c-Temperature-Revised-4/25/10

  38. Example continued: Ch13c-Temperature-Revised-4/25/10

  39. Absolute Temperature and the Ideal Gas Law Experiments done on dilute gases (a gas where interactions between molecules can be ignored) show that: For constant pressure Charles’ Law For constant volume Gay-Lussac’s Law Ch13c-Temperature-Revised-4/25/10

  40. For constant temperature Boyle’s Law Avogadro’s Law For constant pressure and temperature Ch13c-Temperature-Revised-4/25/10

  41. The Ideal Gas Equation of State Putting all of these statements together gives the ideal gas law (microscopic form): k = 1.381023 J/K is Boltzmann’s constant The ideal gas law can also be written as (macroscopic form): R = NAk = 8.31 J/K/mole is the universal gas constant and n is the number of moles. Ch13c-Temperature-Revised-4/25/10

  42. Example (text problem 13.41): A cylinder in a car engine takes Vi = 4.50102 m3 of air into the chamber at 30 C and at atmospheric pressure. The piston then compresses the air to one-ninth of the original volume and to 20.0 times the original pressure. What is the new temperature of the air? Here, Vf = Vi/9, Pf = 20.0Pi, and Ti = 30 C = 303 K. The ideal gas law holds for each set of parameters (before compression and after compression). Ch13c-Temperature-Revised-4/25/10

  43. Example continued: Take the ratio: The final temperature is The final temperature is 673 K = 400 C. Ch13c-Temperature-Revised-4/25/10

  44. Constant Volume Gas Thermometer Ch13c-Temperature-Revised-4/25/10

  45. Constant Volume Gas Thermometer negative-pressure Ch13c-Temperature-Revised-4/25/10

  46. Kinetic Theory of the Ideal Gas Ch13c-Temperature-Revised-4/25/10

  47. Kinetic Theory of the Ideal Gas An ideal gas is a dilute gas where the particles act as point particles with no interactions except for elastic collisions. Ch13c-Temperature-Revised-4/25/10

  48. Gas particles have random motions. Each time a particle collides with the walls of its container there is a force exerted on the wall. The force per unit area on the wall is equal to the pressure in the gas. • The pressure will depend on: • The number of gas particles • Frequency of collisions with the walls • Amount of momentum transferred during each collision Ch13c-Temperature-Revised-4/25/10

  49. The pressure in the gas is Where <Ktr> is the root mean square (RMS) average of the translational kinetic energy of the gas particles; it depends on the temperature of the gas. Ch13c-Temperature-Revised-4/25/10

  50. Root Mean Squared The root mean squared (rms) method of averaging is used when a variable will average to zero but its effect will not average to zero. • Procedure • Square it (make the negative values positive) • Take the average (mean) • Take the square root (undo the squaring opertion) Ch13c-Temperature-Revised-4/25/10

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