Lecture 24 Temperature and Heat Phases and Phase Changes

1. Lecture 24 Temperature and Heat Phases and Phase Changes

2. Heat Transfer Mechanisms Thermal equilibrium is reached by means of thermal contact, which in turn can occur through three different mechanisms conduction : it occurs when objects at different temperature are in physical contact (e.g. when holding a hot potato). Faster moving molecules in the hotter object transfer some of their energy to the colder one convection : this occurs mainly in fluids. In a pot of water on a stove, the liquid at the bottom is heated by conduction. The hot water has lower density and rises to the top, cold water from the top falls to the bottom and gets heated, etc. radiation : any object at non-zero temperature emits radiation (in the form of electromagnetic waves). The effect is more noticeable when standing next to a red-hot coal fire, or in the sun rays

3. The constant k is called the thermal conductivity of the material Conduction Conduction is the flow of heat directly through a physical material • The amount of heat Q that flows through a rod: • increases proportionally to the cross-sectional area A • increases proportionally to ΔT from one end to the other • increases steadily with time • decreases inversely with the length of the rod

4. Some Typical Thermal Conductivities Substances with high thermal conductivities are good conductors of heat; those with low thermal conductivities are good insulators.

5. Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is 0.525 m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C. (a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain. (b) find the temperature at the lead-copper interface. kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m)

6. Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is 0.525 m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C. (a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain. (b) find the temperature at the lead-copper interface. kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m) • Assumptions: • The end points are infinite heat reservoirs... so their temperature doesn’t change for this exercise • The temperature is constant in time at every point. This is not true at moment of thermal connection. We are solving the “steady state” condition, when the temperature at each point doesn’t change.

7. Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is 0.525 m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C. (a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain. (b) find the temperature at the lead-copper interface. kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m) (a) - The heat (per unit time) through the lead must equal that through the copper - The lead has a smaller thermal conductivity than the copper The lead requires a larger temperature difference across it than the copper, to get the same heat flow. So TJ > 54o C

8. Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is 0.525 m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C. (a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain. (b) find the temperature at the lead-copper interface. (b) kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m)

9. Convection Convection is the flow of fluid due to a difference in temperatures, such as warm air rising. The fluid “carries” the heat with it as it moves.

10. Radiation All objects give off energy in the form of radiation, as electromagnetic waves (light) – infrared, visible light, ultraviolet – which, unlike conduction and convection, can transport heat through a vacuum. Objects that are hot enough will glow – first red, then yellow, white, and blue.

11. Radiation The amount of energy radiated by an object due to its temperature is proportional to its surface area and also to the fourth (!) power of its temperature. It also depends on the emissivity, which is a number between 0 and 1 that indicates how effective a radiator the object is; a perfect radiator would have an emissivity of 1. Here, e is the emissivity, and σ is the Stefan-Boltzmann constant:

13. The surface of the Sun has a temperature of 5500 oC. (a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K (b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m.

14. emissivity The surface of the Sun has a temperature of 5500 oC. (a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K (b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m. (a) (b)

15. Heat Conduction a)a rug b)a steel surface c) a concrete floor d) has nothing to do withthermal conductivity Given your experience of what feels colder when you walk on it, which of the surfaces would have the highest thermal conductivity?

16. Heat Conduction a)a rug b)a steel surface c) a concrete floor d) has nothing to do with thermal conductivity Given your experience of what feels colder when you walk on it, which of the surfaces would have the highest thermal conductivity? The heat flow rate is k A (T1− T2)/l. All things being equal, bigger k leads to bigger heat loss. From the book: Steel = 40, Concrete = 0.84, Human tissue = 0.2, Wool = 0.04, in units of J/(s.m.C°).

17. Phases and Phase Changes

18. Ideal Gases Gases are the easiest state of matter to describe, as all ideal gases exhibit similar behavior. An ideal gas is one that is thin enough, and far away enough from condensing, that the interactions between molecules can be ignored.

19. Soda Bottle a) it expands and may burst b) it does not change c) it contracts and the sides collapse inward d) it is too dark in the fridge to tell A plastic soda bottle is empty and sits out in the sun, heating the air inside. Now you put the cap on tightly and put the bottle in the fridge. What happens to the bottle as it cools?

20. Soda Bottle a) it expands and may burst b) it does not change c) it contracts and the sides collapse inward d) it is too dark in the fridge to tell A plastic soda bottle is empty and sits out in the sun, heating the air inside. Now you put the cap on tightly and put the bottle in the fridge. What happens to the bottle as it cools? The air inside the bottle is warm, due to heating by the sun. When the bottle is in the fridge,the air cools. As the temperature drops, thepressure in the bottle also drops. Eventually, the pressure inside is sufficiently lower than the pressure outside (atmosphere) to begin to collapse the bottle.

21. If the volume of an ideal gas is held constant, we find that the pressure increases with temperature: Constant Volume

22. Constant Temperature If the volume and temperature are kept constant, but more gas is added (such as in inflating a tire or basketball), the pressure will increase: If the temperature is constant and the volume decreases, the pressure increases: (fixed volume V) (fixed number N)

23. Equation of State for an Ideal Gas Combining these observations: where k is called the Boltzmann constant:

24. for n moles of gas: Equation of State for an Ideal Gas Instead of counting molecules, we can count moles.

25. Properties of Ideal Gases: Constant Pressure Charles’s law says that the volume of a gas increases with temperature if the pressure is constant.

26. Properties of Ideal Gases: Constant Temperature Boyle’s law says that the pressure varies inversely with volume. These curves of constant temperature are called isotherms.

27. Kinetic Theory The kinetic theory relates microscopic quantities (position, velocity) to macroscopic ones (pressure, temperature). • Assumptions: • N identical molecules of mass m are inside a container of volume V; each acts as a point particle. • Molecules always obey Newton’s laws and are moving randomly. • Collisions with other molecules and with the walls are elastic.

28. Containing an Ideal Gas What is the impulse to turn a molecule around? I = 2(mvx) What is the period to time between such impulses? t = 2L / vx What is the average force on the surface from one molecule? F = I/t = 2(mvx) / (2L/vx) = 2 mvx2 / L

29. Pressure and an Ideal Gas Pressure is the result of collisions between the gas molecules and the walls of the container. It depends on the mass and speed of the molecules, and on the container size:

30. Distribution of Molecular Speeds Not all molecules in a gas will have the same speed; their speeds are represented by the Maxwell distribution, and depend on the temperature and mass of the molecules.

31. Pressure and Kinetic Energy We replace the speed in the previous expression for pressure with the average speed-squared: Including the other two directions, and all N particles: Therefore, the pressure in a gas is proportional to the average kinetic energy of its molecules.

32. The square root of is called the root mean square (rms) speed. Kinetic Energy and Temperature Compare to ideal gas law: average kinetic energy is related to temperature

33. r.m.s. Speed The rms speed is slightly greater than the most probable speed and the average speed.

34. Internal Energy The internal energy of an ideal monotonic gas is the sum of the kinetic energies of all its molecules. In the case where each molecule consists of a single atom, this is all linear kinetic energy of atoms:

35. Distribution of Molecular Speed Some molecules will have speeds exceeding the planetary escape velocity! Lighter molecules will have higher speeds (at the same temperature) and so will leave the planet more quickly. This is why less massive planets have thin, or no, atmosphere... and why earth has little H2 in the atmosphere, but Jupiter has plenty

36. Solids

37. Solids and Elastic Deformation Solids have definite shapes (unlike fluids), but they can be deformed. Pulling on opposite ends of a rod can cause it to stretch:

38. Stretching / Compression of a Solid The amount of stretching will depend on the force; Y is Young’s modulus and is a property of the material: The stretch is proportional to the force, and also to the original length The same formula works for stretching or compression (but sometimes with a different Young’s modulus)

39. Shear Forces Another type of deformation is called a shear deformation, where opposite sides of the object are pulled laterally in opposite directions. The “lean” is proportional to the force, and also to the original height

40. Shear Modulus S is the shear modulus.

41. Uniform Compression Under uniform pressure, an object will shrink in volume Here, the proportionality constant, B, is called the bulk modulus.

42. Stress and Strain The applied force per unit area is called the stress, and the resulting deformation is the strain. They are proportional to each other until the stress becomes too large; permanent deformation will then occur.

43. Phase Changes

44. Evaporation Molecules in a liquid can sometimes escape the binding forces and become vapor (gas)

45. Phase Equilibrium If a liquid is put into a sealed container so that there is a vacuum above it, some of the molecules in the liquid will vaporize. Once a sufficient number have done so, some will begin to condense back into the liquid. Equilibrium is reached when the numbers in each phase remain constant.

46. Vapor Pressure The pressure of the gas when it is in equilibrium with the liquid is called the equilibrium vapor pressure, and will depend on the temperature. A liquid boils at the temperature at which its vapor pressure equals the external pressure.

47. Boiling Potatoes Will boiled potatoes cook faster in Charlottesville or in Denver? a) Charlottesville b) Denver (the “mile high” city) c) the same in both places d) I’ve never cooked in Denver, so I really don’t know e) you can boil potatoes?

48. Boiling Potatoes Will boiled potatoes cook faster in Charlottesville or in Denver? a) Charlottesville b) Denver (the “mile high” city) c) the same in both places d) I’ve never cooked in Denver, so I really don’t know e) you can boil potatoes? The lower air pressure in Denver means that the water will boil at a lower temperature... and your potatoes will take longer to cook.

49. Phase Diagram The vapor pressure curve is only a part of the phase diagram. There are similar curves describing the pressure/temperature of transition from solid to liquid, and solid to gas When the liquid reaches the critical point, there is no longer a distinction between liquid and gas; there is only a “fluid” phase.

50. Fusion Curve Curve 1 Curve 2 The fusion curve is the boundary between the solid and liquid phases; along that curve they exist in equilibrium with each other. One of these two fusion curves has a shape that is typical for most materials, but the other has shape specific to water. Which is which? (a) Curve 1 is the fusion curve for water (b) Curve 2 is the fusion curve for water (c) Trick question: there is no fusion curve for water!