2. Physics 151: Lecture 37, Pg 1 Physics 151: Lecture 37 Today’s Agenda l Topics çTemperature and Zeroth Law of Thermodynamics çTemperature scales çThermal expansion çIdeal gas

3. Physics 151: Lecture 37, Pg 2 Temperature l Temperature: measure of the motion of the individual atoms and molecules in a gas, liquid, or solid. çrelated to average kinetic energy of constituents l High temperature: constituents are moving around energetically çIn a gas at high temperature the individual gas molecules are moving about independently at high speeds. çIn a solid at high temperature the individual atoms of the solid are vibrating energetically in place. l The converse is true for a "cold" object. çIn a gas at low temperature the individual gas molecules are moving about sluggishly. çThere is an absolute zero temperature at which the motions of atoms and molecules practically stop. l There is an absolute zero temperature at which the classical motions of atoms and molecules practically stop Animation

4. Physics 151: Lecture 37, Pg 3 Heat l Solids, liquids or gases have internal energy çKinetic energy from random motion of molecules n translation, rotation, vibration çAt equilibrium, it is related to temperature l Heat: transfer of energy from one object to another as a result of their different temperatures l Thermal contact: energy can flow between objects T1T1 T2T2 U1U1 U2U2 >

5. Physics 151: Lecture 37, Pg 4 Zeroth Law of Thermodynamics l Thermal equilibrium: when objects in thermal contact cease heat transfer çsame temperature T1T1 T2T2 U1U1 U2U2 = 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. A C B Animation

6. Physics 151: Lecture 37, Pg 5 Thermometers l A thermometer is a device that is used to measure the temperature of a system l Thermometers are based on the principle that some physical property of a system changes as the system’s temperature changes l These properties include: çThe volume of a liquid çThe dimensions of a solid çThe pressure of a gas at a constant volume çThe volume of a gas at a constant pressure çThe electric resistance of a conductor çThe color of an object l A temperature scale can be established on the basis of any of these physical properties

7. Physics 151: Lecture 37, Pg 6 Thermometer, Liquid in Glass l A common type of thermometer is a liquid-in-glass l The material in the capillary tube expands as it is heated l The liquid is usually mercury or alcohol l A thermometer can be calibrated by placing it in contact with some natural systems that remain at constant temperature

8. Physics 151: Lecture 37, Pg 7 Thermometer, Celsius Scale l A thermometer can be calibrated by placing it in contact with some natural systems that remain at constant temperature l Common systems involve water çA mixture of ice and water at atmospheric pressure »Called the ice point of water çA mixture of water and steam in equilibrium »Called the steam point of water Celsius Scale : l The ice point of water is defined to be 0 o C l The steam point of water is defined to be 100 o C

9. Physics 151: Lecture 37, Pg 8 Constant Volume Gas Thermometer l The physical change exploited is the variation of pressure of a fixed volume gas as its temperature changes l The volume of the gas is kept constant by raising or lowering the reservoir B to keep the mercury level at A constant

10. Physics 151: Lecture 37, Pg 9 Absolute Zero l The thermometer readings are virtually independent of the gas used l If the lines for various gases are extended, the pressure is always zero when the temperature is –273.15 o C l This temperature is called absolute zero l Absolute zero is used as the basis of the absolute temperature scale l The size of the degree on the absolute scale is the same as the size of the degree on the Celsius scale l To convert: T C = T – 273.15

11. Physics 151: Lecture 37, Pg 10 Temperature scales l Three main scales 212 Farenheit 100 Celcius 320273.15 373.15 Kelvin Water boils Water freezes 0-273.15-459.67 Absolute Zero

12. Physics 151: Lecture 37, Pg 11 Some interesting facts l 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 30 o F), and a third (set at 96 o F) “when placing the thermometer in the mouth of a healthy man”. çOn that scale, water boiled at 212. çLater, Fahrenheit moved the freesing point of water to 32 (so that the scale had 180 increments). l 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. T (K) 10 8 10 7 10 6 10 5 10 4 10 3 100 10 1 0.1 Hydrogen bomb Sun’s interior Solar corona Sun’s surface Copper melts Water freezes Liquid nitrogen Liquid hydrogen Liquid helium Lowest T ~ 10 -9 K

13. Physics 151: Lecture 37, Pg 12 Thermal expansion l In most liquids or solids, when temperature rises çmolecules have more kinetic energy »they are moving faster, on the average çconsequently, things tend to expand l amount of expansion depends on… çchange in temperature  T çoriginal length  L çcoefficient of thermal expansion »L 0 +  L = L 0 +  L 0  T »  L =  L 0  T (linear expansion) »  V =  L 0  T (volume expansion) L0L0 LL V V +  V

14. Physics 151: Lecture 37, Pg 13 Thermal expansion

15. Physics 151: Lecture 37, Pg 14 Bimetallic strip l 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 (  ) 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 (  r = r 2 - r 1 = 0.410 mm). The two metals are bronze and invar.  (  2 -  1 )L (dT/dr)  1.06 o

16. Physics 151: Lecture 37, Pg 15 Lecture 37, ACT 1 Thermal expansion l As you heat a block of aluminum from 0 o C to 100 o C, its density (a) increases (b) decreases (c) stays the same Solution çHere  is positive Volume increases Density decreases T = 0 C M, V 0  0 = M / V 0 T = 100 C M, V 100  100 = M / V 100 <  0 Answer: (b)

17. Physics 151: Lecture 37, Pg 16 Lecture 37: ACT 2 Thermal expansion l 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

18. Physics 151: Lecture 37, Pg 17 Lecture 37: ACT 2 Solution l Both objects have positive thermal coefficients çall dimensions increase çplate and ball both get larger Before (b) fits more easily After

19. Physics 151: Lecture 37, Pg 18 Special system: Water l Most liquids increase in volume with increasing T çwater is special çdensity increases from 0 to 4 o C ! çice is less dense than liquid water at 4 o C: hence it floats çwater at the bottom of a pond is the denser, i.e. at 4 o C Water has its maximum density at 4 degrees.  (kg/m 3 ) T ( o C) l Reason: chemical bonds of H 2 0 (see your chemistry courses !)

20. Physics 151: Lecture 37, Pg 19 Lecture 37: ACT 3 l 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 o C (b) 4 o C (c) 32 o C(d) 100 o C (e) 212 o C

21. Physics 151: Lecture 37, Pg 20 Lecture 37: ACT 3 SOLUTION l The buoyancy force is çsince his volume and g are constant, only  is changing Water has its maximum density at 4 degrees. F B =  l Vg (a) 0 o C

22. Physics 151: Lecture 37, Pg 21 Lecture 37: Problem 1 For a typical system, such as a mercury thermometer, why is it a good approximation to neglect the expansion of the shell ?  h =( V i /A) (  - 3  )  T  (for glass) <<  (for mercury), so (  -3  within 5%. A liquid with a coefficient of volume expansion  just fills a spherical shell of volume V i at a temperature of T i (see Fig. to the right). The shell is made of a material that has an average coefficient of linear expansion . 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  T how high does the liquid rises in the capillary (  h ) ?

23. Physics 151: Lecture 37, Pg 22 Ideal gas l Consider a gas in a container of volume V, at pressure P, and at temperature T l Equation of state çLinks these quantities çGenerally very complicated: but not for ideal gas n = m/M : number of moles m=mass M=mass of one mole One mole contains N A =6.022 X 10 23 particles : Avogadro’s number PV = nRT Equation of state for an ideal gas R is called the universal gas constant In SI units, R =8.315 J / mol·K

24. Physics 151: Lecture 37, Pg 23 Boltzmann’s constant l In terms of the total number of particles N l P, V, and T are the thermodynamics variables PV = nRT = (N/N A ) RT k B is called the Boltzmann’s constant In SI units, with R =8.315 J / mol·K, we get k B = R/N A = 1.38 X 10 -23 J/K PV = N k B T Animation (p,T) Animation (p,V) Animation (p,N)

25. Physics 151: Lecture 37, Pg 24 Lecture 37: Problem 2 l A vertical cylinder of cross- sectional area A=0.001 m 2 is fitted with a tight-fitting, frictionless piston of mass m = 20.0 kg (see the Fig. To the right) l 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

26. Physics 151: Lecture 37, Pg 25 Lecture 37: Problem 3 l 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 m 3. 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/m 3.) T = 472 K ! m V, T mg B =  To V g  T Vg ToTo

27. Physics 151: Lecture 37, Pg 26 Lecture 37: Problem 2b A cylinder is closed by a piston connected to a spring of constant 2.0 x 10 3 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. l If the piston has a cross- sectional area of 0.010 m 2 and a negligible mass, how high will it rise when the temperature is raised to 250°C ? V=V o +A h h = 0.169 m p=p o +kh / A p o V o / T o = pV / T

28. Physics 151: Lecture 37, Pg 27 Recap of today’s lecture l Chap. 19: temperature çTemperature and Zeroth Law of Thermodynamics çTemperature scales çThermal expansion çIdeal gas