1 Environment to be measured Gas Mercury. Mercury is added to or removed from flask 1 so that height in flask 2 is constant

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1 Lecture 2: Thermal Expansion and Ideal Gases Last lecture, we defined temperature and explored the problem of designing and calibrating a thermometer The liquid thermometer was not too bad, but had several limitations Many of these are overcome by the constant-volume gas thermometer Here s the idea: 2 h 1 Environment to be measured Gas Mercury Mercury is added to or removed from flask 1 so that height in flask 2 is constant meaning that the volume of the gas never changes

2 Start with an environment with a known temperature e.g., ice water, for which the temperature is defined to be 0 o C Record h, which is proportional to the pressure of the gas Now repeat with a 2nd known-temperature environment e.g., boiling water, which is defined to be at 100 o C Plot these two points on a pressure vs. temperature graph: Now we can find the temperature of any environment by measuring the pressure, and finding the corresponding point on the dashed line

3 As long as the gas pressure is kept low, we find with this thermometer that we get the same reading for a given environment no matter: what type of gas is used what pressure the gas initally has We can also measure a wide range of temperatures at low pressure, gases don t liquify or freeze until the temperature is very low In addition note something interesting since negative pressure doesn t make sense, the point where the dashed line crosses 0 pressure represents the lowest temperature we can possibly measure And this point is at the same temperature no matter what gas or pressure we use! This absolute zero is the lowest temperature any system can have it represents a system with no energy (e.g. all atoms at rest)

4 Absolute Zero and Temperature Scales We find that absolute zero occurs at o C This is defined as the 0 temperature point of the Kelvin scale We use the Kelvin scale in all equations where temperature appears The translation between the Kelvin and Celsius scale is simple: The Fahrenheit scale is commonly used in the U.S. Conversion rules are: T = T C T C = 5 ( 9 T! F 32o F) T = 5 ( 9 T F! 32 o F) K

5 Thermal Expansion The liquid thermometer we discussed in Lecture 1 was based on the fact that mercury and alcohol increase in volume as their temperature rises This is a property of most solids and liquids For small changes in temperature, the change in length is proportional to the change in temperature:!l = "L i!t Coefficient of linear expansion Initial length Remember that ΔΤ must be expressed in Kelvin or Celsius The value of α depends on the material

6 Engineering considerations Thermal expansion is an important factor in many engineering projects if not properly accounted for (with expansion gaps or restraints) the design can fail:

7 Although we ll typically assume that materials expand as the temperature rises, there s one important example of an exception: water! At high temperatures, water contracts with decreasing temperature, as we d expect But as one nears the freezing point (below 4 o C) this turns around the water expands as the temperature decreases from 4 o to 0 o C! This odd fact of water s chemistry is the reason that lakes freeze from the top down rather than the bottom up a very convenient fact for the fish that live there!

8 Expanding a doughnut What happens when an object with the following shape is heated? Does the hole get bigger or smaller? Answer: bigger! The entire object expands

9 Area and volume changes If we have a sheet of material with length l and width w, the change in area is given by: A +!A = l +!l ( )( w +!w) = lw + w!l + l!w +!l!w " A + w#l!t + l#w!t = A + 2#A!T This is small enough to ignore So that!a = 2"A!T Similarly, the change in volume is:!v = 3"V!T Note that these area and volume formulas work only for isotropic materials that expand the same way in each direction there are some that don t behave this way!

10 Ideal Gases For a gas, there is no intrinsic volume the gas fills whatever container it s placed into An ideal gas is one for which the molecules don t influence each other much i.e., one at low pressure We can find rules that relate the quantity of gas (in terms of its mass m) and its pressure, volume and temperature Actually, it s often more convenient to express the quantity of gas in moles, defined as: n = m M Molar mass (atomic weight expressed in grams) Turns out there are x molecules in a mole (Avagadro s number)

11 Equation of State For any gas that is near ideal the following equation of state holds: PV = nrt Ideal Gas Law R is a number that is constant, no matter what type of gas is used the value of this universal gas constant is J/mol K We can also write the equation of state in terms of the number of molecules rather than the number of moles: PV = Nk B T k B is Boltzmann s constant (1.38 x J/K)