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Lecture 23

Lecture 23. Goals:. Chapter 16 Use the ideal-gas law. Use pV diagrams for ideal-gas processes. Chapter 17 Employ energy conservation in terms of 1 st law of TD Understand the concept of heat. Relate heat to temperature change

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Lecture 23

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  1. Lecture 23 Goals: • Chapter 16 • Use the ideal-gas law. • Use pV diagrams for ideal-gas processes. • Chapter 17 • Employ energy conservation in terms of 1st law of TD • Understand the concept of heat. • Relate heat to temperature change • Apply heat and energy transfer processes in real situations • Recognize adiabatic processes. • Assignment • HW9, Due Wednesday, Apr. 14th • HW10, Due Wednesday, Apr. 21st(9 AM) Exam 3 on Wednesday, Apr. 21 at 7:15 PM (Statics, Ang. Mom., Ch.14 through 17)

  2. Thermodynamics: A macroscopic description of matter • Recall “3” Phases of matter: Solid, liquid & gas • All 3 phases exist at different p,T conditions  Phase diagram • Triple point of water: p = 0.06 atm T = 0.01°C • Triple point of CO2: p = 5 atm T = -56°C

  3. Modern Definition of Kelvin Scale • Water’s triple point on the Kelvin scale is 273.16 K • One degrees Kelvin is defined to be 1/273.16 of the temperature at the triple point of water Accurate water phase diagram Triple point

  4. Changes due to Heat or Thermal Energy Transfer • Change the temperature (it gets hotter) • Change the state of matter (solidliquid, liquidgas) Accurate water phase diagram Path on heating

  5. Energy Transfer to a solid (ice) 1. Temperature increase or 2. State Change If a gas, then V, p and T are interrelated….equation of state

  6. Water boils 32 0 273.15 Water freezes -459.67 -273.15 0 Absolute Zero Defining a temperature scale • Three main scales Farenheit Celsius Kelvin 212 100 373.15

  7. T (K) 108 107 106 105 104 103 100 10 1 0.1 Some interesting facts • In 1724, Gabriel Fahrenheit made thermometers using mercury. The zero point of his scale is attained by mixing equal parts of water, ice, and salt. A second point was obtained when pure water froze (originally set at 30oF), and a third (set at 96°F) “when placing the thermometer in the mouth of a healthy man”. • On that scale, water boiled at 212. • Later, Fahrenheit moved the freezing point of water to 32 (so that the scale had 180 increments). • In 1745, Carolus Linnaeus of Upsula, Sweden, described a scale in which the freezing point of water was zero, and the boiling point 100, making it a centigrade (one hundred steps) scale. Anders Celsius (1701-1744) used the reverse scale in which 100 represented the freezing point and zero the boiling point of water, still, of course, with 100 degrees between the two defining points. Hydrogen bomb Sun’s interior Solar corona Sun’s surface Copper melts Water freezes Liquid nitrogen Liquid hydrogen Liquid helium Lowest T~ 10-9K

  8. Ideal gas: Macroscopic description • Consider a gas in a container of volume V, at pressure p, and at temperature T • Equation of state • Links these quantities • Generally very complicated: but not for ideal gas

  9. Ideal gas: Macroscopic description • Equation of state for an “ideal gas” • Collection of atoms/molecules moving randomly • No long-range forces • Their size (volume) is negligible • Density is low • Temperature is well above the condensation point pV = nRT R is called the universal gas constant n ≡ number of moles In SI units, R =8.315 J / mol·K

  10. pV = N kBT Boltzmann’s constant m = mass (kg) M= mass of one mole (kg/mol) • Number of moles: n = m/M • One mole contains NA=6.022 X 1023 particles : Avogadro’s number = number of carbon atoms in 12 g of carbon • In terms of the total number of particles N • p, V, and T are thermodynamic variables pV = nRT = (N/NA) RT kB = R/NA = 1.38 X 10-23 J/K kBiscalled the Boltzmann’s constant

  11. The Ideal Gas Law What is the volume of 1 mol of gas at STP ? T = 0 °C = 273 K p = 1 atm = 1.01 x 105 Pa

  12. The Ideal Gas Law There are four things that can vary: p, V, n & T Typically, two of these are held constant and the relationship between the remaining two is studied

  13. Example • A spray can containing a propellant gas at twice atmospheric pressure (202 kPa) and having a volume of 125.00 cm3 is at 27oC. It is then tossed into an open fire. When the temperature of the gas in the can reaches 327oC, what is the pressure inside the can? Assume any change in the volume of the can is negligible. Steps • Convert to Kelvin (From 300 K to 600 K) • Use P/T = nR/V = constant  P1/T1 = P2/T2 • Solve for final pressure  P2 = P1 T2/T1 WD40 foolishness

  14. Example problem: Air bubble rising • A diver produces an air bubble underwater, where the absolute pressure is p1 = 3.5 atm. The bubble rises to the surface, where the pressure is p2 = 1.0 atm. The water temperatures at the bottom and the surface are, respectively, T1 = 4°C, T2 = 23°C • What is the ratio of the volume,V2 , of the bubble just as it reaches the surface to its volume at the bottom, V1? • Is it safe for the diver to ascend while holding his breath? No! Air in the lungs would expand, and the lung could rupture.

  15. Example problem: Air bubble rising • A diver produces an air bubble underwater, where the absolute pressure is p1 = 3.5 atm. The bubble rises to the surface, where the pressure is p2 = 1 atm. The water temperatures at the bottom and the surface are, respectively, T1 = 4°C, T2 = 23°C • What is the ratio of the volume of the bubble as it reaches the surface,V2, to its volume at the bottom, V1? (Ans.V2/V1 = 3.74) • pV=nRTpV/T = const so p1V1/T1 = p2V2/T2 V2/V1 = p1T2/ (T1 p2) = 3.5 296 / (277 1) If thermal transfer is efficient. [More than likely the expansion will be “adiabatic” and, for a diatomic gas, PVg = const. where g = 7/5, see Ch. 17 & 18]

  16. Buoyancy and the Ideal Gas Law • A typical 5 passenger hot air balloon has approximately 700 kg of total mass and the balloon itself can be thought as spherical with a radius of 10.0 m. If the balloon is launched on a day with conditions of 1.0 atm and 273 K, how hot would you have to heat the air inside (assuming the density of the surrounding air is 1.2 kg/m3 and the air behaves and as an ideal gas) in order to keep the balloon at a constant altitude? • Hint: Remember the weight of the air inside the balloon. • p, V and R do not change! Balloon weight = Buoyant force – Weight of hot air Ideal gas law: pV = nRT  nT= pV/R = const. or r T = const. = 1.2 x 273 kg K/m3

  17. Buoyancy and the Ideal Gas Law • mballoon g = rair at 273 K V g – rair at T V g • mballoon = rair at 273 K V – rair at T V • mballoon = (1.2 – 330 / T) V • 700 / 4200 = 1.2 – 330 / T • 330 / T = (1.2 - 0.2) • T = 330 K  57 C

  18. Isochoric Isobaric 2 Isothermal 1 Pressure Pressure Pressure 1 2 1 2 Volume Volume Volume pV diagrams: Important processes • Isochoric process: V = const (aka isovolumetric) • Isobaric process: p = const • Isothermal process: T = const

  19. Isothermal 1 1 Force Pressure 2 2 Position Volume pV vs. Fx diagrams • Work (on the system) remains the area under the curve dW = F dx or dW = F/A (A dx) = P dVworld = -p dVsystem

  20. Work and Energy Transfer (Ch. 17) • K reflects the kinetic energy of the system • ΔK =Wconservative+ Wdissipative + Wexternal • Wconservative= - ΔU (e.g., gravity) • Wdissipative = - ΔEThermal • Wexternal  Typically, work done by contact forces ΔK + ΔU + ΔETh = Wexternal= ΔEsys

  21. Work and Energy Transfer ΔK + ΔU + ΔETh = Wexternal= ΔEsys But we can transfer energy without doing work Q ≡ thermal energy transfer ΔK + ΔU + ΔETh = W+ Q = ΔEsys IfΔK + ΔU = ΔEMech = 0 ΔETh = W+ Q

  22. 1st Law of Thermodynamics • Thermal energy Eth : Microscopic energy of moving molecules and stretching molecular bonds. ΔEth depends on the initial and final states but is independent of the process. • Work W : Energy transferred to the system by forces in a mechanical interaction. • Heat Q : Energy transferred to the system via atomic-level collisions when there is a temperature difference. ΔEth=W + Q W & Q with respect to the system

  23. Lecture 23 • Assignment • HW9, Due Wednesday, Apr. 14th • HW10, Due Wednesday, Apr. 21st(9 AM) Exam 3 on Wednesday, Apr. 21 at 7:15 PM (Statics, Ang. Mom., Ch.14 through 17)

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