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Physics 151: Lecture 37 Today’s Agenda

Physics 151: Lecture 37 Today’s Agenda. Topics Temperature and Zeroth Law of Thermodynamics Temperature scales Thermal expansion Ideal gas. Temperature. Temperature : measure of the motion of the individual atoms and molecules in a gas, liquid, or solid.

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Physics 151: Lecture 37 Today’s Agenda

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  1. Physics 151: Lecture 37 Today’s Agenda • Topics • Temperature and Zeroth Law of Thermodynamics • Temperature scales • Thermal expansion • Ideal gas

  2. Temperature • Temperature: measure of the motion of the individual atoms and molecules in a gas, liquid, or solid. • related to average kinetic energy of constituents • High temperature: constituents are moving around energetically • In a gas at high temperature the individual gas molecules are moving about independently at high speeds. • In a solid at high temperature the individual atoms of the solid are vibrating energetically in place. • The converse is true for a "cold" object. • In a gas at low temperature the individual gas molecules are moving about sluggishly. • There is an absolute zero temperature at which the motions of atoms and molecules practically stop. • There is an absolute zero temperature at which the classical motions of atoms and molecules practically stop Animation

  3. Heat • Solids, liquids or gases have internal energy • Kinetic energy from random motion of molecules • translation, rotation, vibration • At equilibrium, it is related to temperature • Heat: transfer of energy from one object to another as a result of their different temperatures • Thermal contact: energy can flow between objects T2 T1 > U2 U1

  4. T2 T1 = If objects A and B are separately in thermal equilibrium with a third object C, then objects A and B are in thermal equilibrium with each other. U2 U1 Zeroth Law of Thermodynamics • Thermal equilibrium: when objects in thermal contact cease heat transfer • same temperature Animation B C A

  5. Thermometers • A thermometer is a device that is used to measure the temperature of a system • Thermometers are based on the principle that some physical property of a system changes as the system’s temperature changes • These properties include: • The volume of a liquid • The dimensions of a solid • The pressure of a gas at a constant volume • The volume of a gas at a constant pressure • The electric resistance of a conductor • The color of an object • A temperature scale can be established on the basis of any of these physical properties

  6. Thermometer, Liquid in Glass • A common type of thermometer is a liquid-in-glass • The material in the capillary tube expands as it is heated • The liquid is usually mercury or alcohol • A thermometer can be calibrated by placing it in contact with some natural systems that remain at constant temperature

  7. Thermometer, Celsius Scale • A thermometer can be calibrated by placing it in contact with some natural systems that remain at constant temperature • Common systems involve water • A mixture of ice and water at atmospheric pressure • Called the ice point of water • A mixture of water and steam in equilibrium • Called the steam point of water Celsius Scale : • The ice point of water is defined to be 0o C • The steam point of water is defined to be 100o C

  8. Constant Volume Gas Thermometer • The physical change exploited is the variation of pressure of a fixed volume gas as its temperature changes • The volume of the gas is kept constant by raising or lowering the reservoir B to keep the mercury level at A constant

  9. Absolute Zero • The thermometer readings are virtually independent of the gas used • If the lines for various gases are extended, the pressure is always zero when the temperature is –273.15o C • This temperature is called absolute zero • Absolute zero is used as the basis of the absolute temperature scale • The size of the degree on the absolute scale is the same as the size of the degree on the Celsius scale • To convert: TC = T – 273.15

  10. Water boils 32 0 273.15 Water freezes -459.67 -273.15 0 Absolute Zero Temperature scales Farenheit Celcius Kelvin • Three main scales 212 100 373.15

  11. T (K) 108 107 106 105 104 103 100 10 1 0.1 Some interesting facts Hydrogen bomb Sun’s interior • 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 96oF) “when placing the thermometer in the mouth of a healthy man”. • On that scale, water boiled at 212. • Later, Fahrenheit moved the freesing 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. Solar corona Sun’s surface Copper melts Water freezes Liquid nitrogen Liquid hydrogen Liquid helium Lowest T ~ 10-9K

  12. L0 L V V + V Thermal expansion • In most liquids or solids, when temperature rises • molecules have more kinetic energy • they are moving faster, on the average • consequently, things tend to expand • amount of expansion depends on… • change in temperature T • original length L • coefficient of thermal expansion • L0 + L = L0 +  L0 T • L =  L0 T (linear expansion) • V =  L0 T (volume expansion)

  13. Thermal expansion

  14. Bimetallic strip • A bimetallic strip is made of two ribbons of dissimilar metals bonded together. • Assume that the strip is originally straight. As they are heated, the metal with the greater average coefficient of expansion (2) expands more than the other, forcing the strip into an arc, with the outer radius having a greater circumference (as in the Fig). A compact spiral bimetallic strip is used in a home thermostat. Find the angle (Q) through which the free end of the strip turns when the temperature changes by 1°C for such bimetalic strip with the initial length L= 21.0 cm and the separation of the centers of the strips (Dr = r2 - r1 = 0.410 mm).The two metals are bronze and invar. Q = (2 - 1 )L (dT/dr) Q = 1.06o

  15. T = 100 C T = 0 C M, V0 r0 = M / V0 M, V100 r100 = M / V100 < r0 Answer: (b) Lecture 37,ACT 1Thermal expansion • As you heat a block of aluminum from 0 oC to 100 oC, its density (a) increases(b) decreases(c) stays the same • Solution • Here  is positive Volume increases Density decreases

  16. Lecture 37: ACT 2Thermal expansion • An aluminum plate has a circular hole cut in it. A copper ball (solid sphere) has exactly the same diameter as the hole when both are at room temperature, and hence can just barely be pushed through it. If both the plate and the ball are now heated up to a few hundred degrees Celsius, how will the ball and the hole fit ? (a) ball won’t fit(b) fits more easily(c) same as before

  17. (b) fits more easily Lecture 37: ACT 2Solution After • Both objects have positive thermal coefficients • all dimensions increase • plate and ball both get larger Before

  18. (kg/m3) T (oC) Special system: Water • Most liquids increase in volume with increasing T • water is special • density increases from 0 to 4 oC ! • ice is less dense than liquid water at 4 oC: hence it floats • water at the bottom of a pond is the denser, i.e. at 4 oC Water has its maximum density at 4 degrees. • Reason: chemical bonds of H20 (see your chemistry courses !)

  19. Lecture 37: ACT 3 • Not being a great athlete, and having lots of money to spend, Gill Bates decides to keep the lake in his back yard at the exact temperature which will maximize the buoyant force on him when he swims. Which of the following would be the best choice? (a) 0 oC(b) 4oC(c) 32 oC (d) 100 oC (e) 212 oC

  20. Water has its maximum density at 4 degrees. (a) 0 oC Lecture 37: ACT 3SOLUTION • The buoyancy force is • since his volume and g are constant, only  is changing FB = rlVg

  21. Lecture 37: Problem 1 • A liquid with a coefficient of volume expansion b just fills a spherical shell of volume Viat a temperature of Ti (see Fig. to the right). The shell is made of a material that has an average coefficient of linear expansion a. The liquid is free to expand into an open capillary of area A projecting from the top of the sphere. • When the temperature increases by DT how high does the liquid rises in the capillary ( Dh ) ? • For a typical system, such as a mercury thermometer, why is it a good approximation to neglect the expansion of the shell ? Dh =(Vi /A) (b - 3a) DT a(for glass) << b(for mercury), so (b -3a) within 5%.

  22. Ideal gas • 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 m=mass M=mass of one mole n = m/M : number of moles One mole contains NA=6.022 X 1023 particles : Avogadro’s number • Equation of state for an ideal gas R is called the universal gas constant PV = nRT In SI units, R =8.315 J / mol·K

  23. PV = N kBT Boltzmann’s constant • In terms of the total number of particles N • P, V, and T are thethermodynamics variables Animation (p,T) PV = nRT = (N/NA) RT Animation (p,V) Animation (p,N) In SI units, with R =8.315 J / mol·K, we get kB = R/NA = 1.38 X 10-23 J/K kBiscalled the Boltzmann’s constant

  24. Lecture 37: Problem 2 • A vertical cylinder of cross-sectional area A=0.001 m2 is fitted with a tight-fitting, frictionless piston of mass m = 20.0 kg (see the Fig. To the right) • If n = 0.200 moles of an ideal gas are in the cylinder at a temperature of T = 350 K, what is the height h at which the piston is in equilibrium under its own weight ? h = n R T/(mg + PA) h = 1.94 m

  25. B = rTo V g rTVg mg Lecture 37: Problem 3 To • The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at 10.0°C and 101 kPa. The volume of the balloon is 400 m3. To what temperature must the air in the balloon be heated before the balloon will lift off ? (Air density at 10.0°C is 1.25 kg/m3.) V, T m T = 472 K !

  26. Lecture 37: Problem 2b • A cylinder is closed by a piston connected to a spring of constant 2.0 x 103 N/m (as on the Fig.). With the spring relaxed, the cylinder is filled with 5.00 L of gas at a pressure of 1.00 atm and a temperature of 20.0°C. • If the piston has a cross-sectional area of 0.010 m2 and a negligible mass, how high will it rise when the temperature is raised to 250°C ? poVo / To = pV/ T p=po +kh / A V=Vo +A h h = 0.169 m

  27. Recap of today’s lecture • Chap. 19: temperature • Temperature and Zeroth Law of Thermodynamics • Temperature scales • Thermal expansion • Ideal gas

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