# 5.6 Isobaric thermal expansion and isothermal compression (Hiroshi Matsuoka)

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1 5.6 Isobaric thermal expansion and isothermal compression Hiroshi Matsuoka he coefficient of thermal expansion as a response function to a temperature change he coefficient of thermal expansion " is basically a measure for the response of a system s volume to an increase in the system s temperature : it tells us how a material changes its volume as is varied. It is therefore important to know " when we design roads, railways, buildings, airplanes, ships, cars, etc. he larger! gets, the more sensitive the volume of a piece of material is to a change of temperature. ypical values for " in different phases are: Gases: " #10 3 K -1 Liquids: " #10 4 ~ 10 3 K -1 Solids: " #10 6 ~ 10 4 K -1 he values for " for solids are smaller than those for liquids while the values for " for liquids are smaller than those for gases. he values of " are basically determined by interatomic or inter-molecular forces on the microscopic level. Inside a solid, atoms are close together and tightly bound together so that by a small increase in atomic vibrational speed induced by an increase in the temperature does not allow the atoms to increase their inter-atomic distances very much resulting in a very small value for ". Similarly, inside a liquid, the atoms are still close together but they are a bit further away from each other compared to the atoms inside a solid so that they are less strongly bound to each other and they can increase their interatomic distances more readily in response to an increase in their speeds induced by an increase in the temperature. As we found in the last chapter, inside a gas, the atoms are far apart and hardly exert forces on each other so that they can increase its volume most readily in response to an increase in their speeds. " for low-density gases he ideal gas law or v = R leads to! = 1 # "v v " ' = 1, HW#5.6.4: show this

2 2 from which we find that at! 300 K,! " 3 #10 3 K -1 and that " decreases as is increased, which means that it gets harder to expand a gas at higher. " also remains constant when the pressure is kept constant. " for solids and liquids ypically, " for solids and liquids are positive and increases as is increased as we will find in the next subsection. Substance! 300 K K -1 Water 2.1! 10 "4 Ethanol 1.1!10 "3 Diamond 3.0!10 "6 Copper 5.0!10 "5 NaCl 1.2! 10 "4 How do we measure " of a solid? We can measure the molar volume of a solid directly by using the x-ray diffraction technique, with which we can find a crystal lattice structure of atoms inside the solid and its lattice constant, which is directly related to the average inter-atomic distance, from which we can estimate the molar volume as a function of temperature and pressure and finally the coefficient of thermal expansion ". As mentioned above, it is quite time-consuming to directly obtain the molar volume and " this way. We can also measure " indirectly by measuring the linear coefficient of thermal expansion defined by! l " 1 #l ' l # v = l 3. " and! l are then related by! = 3! l. HW#5.6.5: show this. Hint: substitute v = l 3 into the definition of!

3 o measure " l, we must amplify a small length change by using 3 Interference fringes of light Variation of capacitance Variation of light intensity For more detail on these specific techniques, see, for example, Heat and hermodynamics sixth edition by Zemansky and Dittman McGraw-Hill. Using " l " l is also a useful quantity in engineering. For example, we can answer the following question by using " l. How much does the length of a steel arch bridge, whose length at 20 C is 500 m, change when its temperature is increased to 40 C? Assume that the value of " l for steel between 20 C and 40 C is roughly constant at " l =1#10 5 K 1. According to the definition of " l, the length increase "l can be estimated by "l = #l ' # = 500 m 40 C " = l* l " = 500 m 1+10,5 K,1 60 K 1+10,5 K,1 = 0.3 m {,,20 C} his length change is sizable enough that when constructing such a bridge a designer must take into account this change henomenology of " for solids at =1 atm Under the atmospheric pressure, the coefficient of thermal expansion " of a solid becomes a function of its temperature only: " = ", =1 atm. For solids, ", =1 atm as a function of has the following two common features see the figures on the next page: ", =1 atm approaches zero when is decreased toward absolute zero. ", =1 atm increases as is increased.

4 With further examination of data for ", =1 atm of various solids, we can identify two universality classes according to low-temperature behaviors of ", =1 atm. 4 Insulators as a universality class Solid insulators such as sodium chloride NaCl share the same temperature dependence of ", =1 atm at low temperatures see the top figure on the next page. More specifically, at low temperatures, we find " # 3 low, where the proportionality constant between " and 3 varies from one solid to another. As you can see in the figure below, because of this 3 dependence, the slope of " as a function of is zero at = 0 and " increases very slowly as is increased near = NaCl! K K Simple metals as a universality class ", =1 atm for simple metals such as alkali metals e.g., sodium, etc. and noble metals e.g., copper behaves, at low temperatures, as " = A + B 3 low, where the constants A and B vary from one metal to another.

5 Cu ! K K As you can see in the figure above, because of the linear temperature term, the slope of " as a function of is not zero at = 0 and " increases relatively sharply as is increased near = 0. Expansion of a crystal lattice and of a free electron gas For insulators, thermal expansion comes from an expansion of a crystal lattice of atoms or ions e.g., inside a NaCl crystal, positive Na ions and negative Cl ions are placed at corners of cubes and each positive Na ion is surrounded by 6 negative Cl ions. he crystal lattice expands because the amplitudes of thermal vibrations of atoms or ions become larger at higher temperatures so that each atom or ion will claim a larger volume to itself. At low temperatures, inter-atomic or inter-ionic forces strongly bind atoms or ions together and generate collective vibration of atoms or ions or lattice waves. he cubic term in at low temperatures reflect how these lattice waves affect thermal expansion. At high temperatures, individual atoms or ions vibrate almost independently so that the temperature dependence of! becomes different from that at low temperatures. For metals, besides a crystal lattice of positive ions, we must also take into account the presence of conduction electrons that can almost freely move through the crystal lattice. It turns out that for a class of simple metals such as sodium and copper, we can regard these electrons as a gas of non-interacting particles i.e., a free electron gas. For temperatures much lower than roughly 10,000 K, this free electron gas contributes a linear term in to!. In other

6 words, temperatures much lower than 10,000 K for example, the room temperature is still low temperatures for the free electrons inside a simple metal more on this below. 6 he Debye temperature! separates low from high. he characteristic temperature that separates high temperatures from low temperatures for lattice waves in a solid is called the Debye temperature!. emperatures that are much lower than! are low temperatures, while those that are much higher than! are high temperatures. he Debye temperatures for various solids range roughly between 100 K and 1000 K. For example,! = 321 K for NaCl while! = 343 K for copper. Where does! come from? he Debye temperature! comes from the maximum energy that lattice waves can have inside a solid. According to quantum mechanics, the lattice waves behave also as a type of quantum particles called phonons with their energy given by! = hf, where f is the frequency of atomic vibrations accompanying the lattice waves. he maximum energy called the Debye energy! D that the lattice waves can have in a solid is then controlled by the maximum frequency called the Debye frequency for these waves. For any wave, we can use the following general formula: f! = w, where! is in our case the wavelength of a lattice wave and w should be the speed of sound. he maximum frequency for the lattice waves then corresponds to their minimum wavelength called the Debye wavelength, which is on the order of the so that average distance among the atoms or ions inside the solid or! D ~ O 1 A = O 10 "10 m f D = w # ~ O 103 m s ~ O Hz! D 10 "10 m ' High temperatures for the lattice waves are the temperatures that satisfy k B >> " D = hf D or >>! " # D k B = hf D k B, where k B is the Boltzmann constant, which is related with the universal gas constant R by

7 k B = R N Avogadro = 1.38!10 "23 J K, 7 and k B gives the order of magnitude for the average energy per atom. k B >> " D then means that at high temperatures the typical energy per atom is much higher than the maximum energy for the lattice waves so that all the lattice waves are excited in the solid while at low temperatures only the lattice waves with low energy are excited. We can estimate the typical order of magnitude for the Debye temperature as s "! ~ O hf * D # k ' B ~ O, J s +, J K - /. / ~ O 100 K he Debye model as a minimal model for quantum lattice waves or phonons he minimal model for quantum lattice waves or phonons is called the Debye model and treats phonons as a type of quantum particles called bosons and neglects forces or interactions among them. Although the Debye model as it is cannot explain thermal expansion of the crystal lattice, we can modify it to show the 3 -dependence of " at low temperatures. his modified version of the Debye model is sometimes called the Gruneisen model. Free electrons in a simple metal are always in the low temperature regime he characteristic temperature that separates high temperatures from low temperatures for free electrons in a simple metal is called the Fermi temperature F, which is directly related with the maximum energy called the Fermi energy for the free electrons in a simple metal at = 0. According to quantum mechanics, an electron behaves as a wave whose wavelength! controls its energy! through! = h 2 2" ' 2m # 2, where m is the electron mass and h is lanck s constant h divided by 2!. Inside a simple metal, the minimum wavelength called the Fermi wavelength that corresponds to the maximum energy is on the order of the average distance among the free electrons, which is comparable to the average distance among positive ions inside the solid so that

8 8! F ~ O 1 A = O 10 "10 m so that the Fermi energy is on the order of! F = h2 2" ' 2m # F 2 ~ O J *s kg m 2 ' ~ O J ~ O 10 ev, where we have used 1 ev ~ 10 "19 J. he Fermi temperature for the free electrons is then on the order of F! " # F ~ O J k B ~ O 10 5 K J K' so that the Fermi temperature is much higher than the melting temperature of the metal: F >> m ~ O 100 K!1000 K. herefore, for the free electrons in a simple metal, the temperature range for the solid phase is always in their low temperature regime. he free electron gas model as a minimal model for conduction electrons in simple metals he minimal model for conduction electrons in simple metals is called the free electron gas model and treats these electrons as a type of quantum particles called fermions and neglects forces or interactions among them. We will show later that the free electron gas model combined with the modified Debye model or Gruneisen model can explain the linear temperature term in " found for simple metals he molar volume v for solids at =1 atm As mentioned above, if we know ", and ",, we can calculate v,. When the pressure is kept constant at 1 atm, we then find v, =1 atm = v 0, =1 atm exp' " #, d #* ' *. 0

9 As 9 " #, d # ~ O " 0, 0 [ 0 ] <<1, we can use e x "1+ x for x <<1 to get v, =1 atm = v 0, =1 atm ' 1+ " # ' 0,d #* *. If " > 0, the molar volume then increases as is increased. Insulators e.g., NaCl At low temperatures, we have found! " A 3, where A is a positive constant specific to a particular insulator, so that v, =1 atm " v = 0 K, =1 atm 1+ A 4 4 #. HW#5.6.6: show this ' As shown on the figure on the next page, because of the 4 term, the slope of the molar volume v as a function of is zero at = 0 and v increases very slowly as is increased near = 0. Simple metals e.g., copper At low temperatures, we have found " # A + B 3, where A and B are positive constants specific to a particular simple metal, so that v, =1 atm " v = 0 K, =1 atm 1+ A B 4 4 #. HW#5.6.7: show this ' As shown on the figure on the next page, because of the 2 term, the slope of the molar volume v as a function of is zero at = 0 and v increases relatively sharply compared with v for NaCl as is increased near = 0.

10 NaCl v m 3 /mol K Cu v m 3 /mol K

11 SUMMARY OF SEC HROUGH SEC ypical orders of magnitude for " are: Gases: " #10 3 K -1 Liquids: " #10 4 ~ 10 3 K -1 Solids: " #10 6 ~ 10 4 K For low-density gases: " = 1 #v ' v # = We measure the linear coefficient of thermal expansion defined by " l # 1 l ' * l v = l 3 and calculate " using " = 3" l. 4. he Debye temperature of a solid separates the low temperature regime from the high temperature regime for lattice waves in the solid and ranges between 100 K and 1000 K. 5. he Fermi temperature of a simple metal separates the low temperature regime from the high temperature regime for free electrons in the metal and is on the order of 10 5 K. 6. For solid insulators such as NaCl, " # A 3 at low temperatures so that their molar volumes at low temperatures behave as v, = 1 atm = v = 0 K, =1 atm 1+ A " For simple metals such as Cu, " # A + B 3 at low temperatures so that their molar volumes at low temperatures behave as v, = 1 atm " v = 0 K, =1 atm 1+ A # 2 + B # 4.

12 Answers for the homework questions in Sec and Sec HW#5.6.4 " = 1 #v ' v # = 1 * # R '-, / v + #. = R v = R R = 1 HW#5.6.5 " = 1 #v ' v # = 1 #l 3 ' l 3 # = 3l2 #l ' l 3 # = 3 1 #l ' l # = 3" l HW#5.6.6 HW#5.6.7 v, =1 atm = v = 0 K, =1 atm ' 1+ " # + v = 0 K, =1 atm ' 1+ A #, = v = 0 K, =1 atm A ,d #* 3 d #* v, =1 atm = v = 0 K, =1 atm ' 1+ " # + v = 0 K, =1 atm ' 0 / 1 0,d #* d #* 1+ A + B 3, = v = 0 K, =1 atm A B / 1 0

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