Chapter 1: Thermometry and Thermal Expansion PDF

Chapter 1 Thermometry and Thermal Expansion 1.1 Temperature and the Zeroth Law of Thermodynamics We often associate the concept of temperature with how hot or cold an object feels when we touch it. Thus, our senses provide us with a qualitative indication of temperature. However, our sense...

Chapter 1 Thermometry and Thermal Expansion 1.1 Temperature and the Zeroth Law of Thermodynamics We often associate the concept of temperature with how hot or cold an object feels when we touch it. Thus, our senses provide us with a qualitative indication of temperature. However, our senses are unreliable and often mislead us. For example, if we remove a metal ice tray and a cardboard box of frozen vegetables from the freezer, the ice tray feels colder than the box even though both are at the same temperature. The two objects feel different because metal transfers energy by heat at a higher rate than cardboard does. What we need is a reliable and reproducible method for measuring the relative hotness or coldness of objects rather than the rate of energy transfer. Scientists have developed a variety of thermometers for making such quantitative measurements. We are all familiar with the fact that two objects at different initial temperatures eventually reach some intermediate temperature when placed in contact with each other. For example, when hot water and cold water are mixed in a bathtub, the final temperature of the mixture is somewhere between the initial hot and cold temperatures. Likewise, when an ice cube is dropped into a cup of hot coffee, it melts and the coffee’s temperature decreases. To understand the concept of temperature, it is useful to define two often-used phrases: thermal contact and thermal equilibrium. To grasp the meaning of thermal contact, imagine that two objects are placed in an insulated container such that they interact with each other but not with the environment. If the objects are at different 5 temperatures, energy is exchanged between them, even if they are initially not in physical contact with each other. The energy transfer mechanisms are heat and electromagnetic radiation. For purposes of the current discussion, we assume that two objects are in thermal contact with each other if energy can be exchanged between them by these processes due to a temperature difference. Thermal equilibrium is a situation in which two objects would not exchange energy by heat or electromagnetic radiation if they were placed in thermal contact. Fig. 1.1 The zeroth law of thermodynamics. (a) and (b) If the temperatures of A and B are measured to be the same by placing them in thermal contact with a thermometer (object C), no energy will be exchanged Let us consider two objects A and B, which are not in thermal contact, and a third object C, which is our thermometer. We wish to determine whether A and B are in thermal equilibrium with each other. The thermometer (object C) is first placed in thermal contact with object A until thermal equilibrium is reached, as shown in Figure 1.1a. From that moment on, the thermometer’s reading remains constant, and we record this reading. The thermometer is then removed from object A and placed in thermal contact with object B, as shown in Figure 1.1b. The reading is again recorded after thermal equilibrium is reached. If the two readings are the same, then object A and object B are in thermal equilibrium with each other. If they are placed in contact with each other as in Figure 1.1c, there is no exchange of energy between them. 6  We can summarize these results in a statement known as the zeroth law of thermodynamics (the law of equilibrium): If objects A and B are separately in thermal equilibrium with a third object C, then A and B are in thermal equilibrium with each other. This statement can easily be proved experimentally and is very important because it enables us to define temperature. We can think of temperature as the property that determines whether an object is in thermal equilibrium with other objects. Two objects in thermal equilibrium with each other are at the same temperature. Conversely, if two objects have different temperatures, then they are not in thermal equilibrium with each other. Quick Quiz 1.1 Two objects, with different sizes, masses, and temperatures, are placed in thermal contact. Energy travels (a) from the larger object to the smaller object (b) from the object with more mass to the one with less (c) from the object at higher temperature to the object at lower temperature. 1.2 Thermometers and the Temperature Scales Thermometers are devices that are used to measure the temperature of a system. All thermometers are based on the principle that some physical property of a system changes as the system’s temperature changes. Some physical properties that change with temperature are (1) the volume of a liquid, (2) the dimensions of a solid, (3) the pressure of a gas at constant volume, (4) the volume of a gas at constant pressure, (5) the electric resistance of a conductor, and 7 (6) the color of an object. A temperature scale can be established on the basis of any one of these physical properties. Fig. 1.2 As a result of thermal expansion, the level of the mercury in the thermometer rises as the mercury is heated by water in the test tube. A common thermometer in everyday use consists of a mass of liquid—usually mercury or alcohol—that expands into a glass capillary tube when heated (Fig. 1.2). In this case the physical property that changes is the volume of a liquid. Any temperature change in the range of the thermometer can be defined as being proportional to the change in length of the liquid column. The thermometer can be calibrated by placing it in thermal contact with some natural systems that remain at constant temperature. One such system is a mixture of water and ice in thermal equilibrium at atmospheric pressure. On the Celsius temperature scale, this mixture is defined to have a temperature of zero degrees Celsius, which is written as 0°C; this temperature is called the ice point of water. Another commonly used system is a mixture of water and steam in thermal equilibrium at atmospheric pressure; its temperature is 100°C, which is the steam point of water. Once the liquid levels in the thermometer have been established at these two points, the length of the liquid column 8 between the two points is divided into 100 equal segments to create the Celsius scale. Thus, each segment denotes a change in temperature of one Celsius degree. Thermometers calibrated in this way present problems when extremely accurate readings are needed. For instance, the readings given by an alcohol thermometer calibrated at the ice and steam points of water might agree with those given by a mercury thermometer only at the calibration points. Because mercury and alcohol have different thermal expansion properties, when one thermometer reads a temperature of, for example, 50°C, the other may indicate a slightly different value. The discrepancies between thermometers are especially large when the temperatures to be measured are far from the calibration points. An additional practical problem of any thermometer is the limited range of temperatures over which it can be used. A mercury thermometer, for example, cannot be used below the freezing point of mercury, which is 39°C, and an alcohol thermometer is not useful for measuring temperatures above 85°C, the boiling point of alcohol. To surmount this problem, we need a universal thermometer whose readings are independent of the substance used in it. The gas thermometer, discussed in the next section, approaches this requirement. 9 Fig. 1.3 A constant-volume gas thermometer measures the pressure of the gas contained in the flask immersed in the bath. The volume of gas in the flask is kept constant by raising or lowering reservoir B to keep the mercury level in column A constant. Figure 1.4 A typical graph of pressure versus temperature taken with a constant-volume gas thermometer. The two dots represent known reference temperatures (the ice and steam points of water). 1.3 The Constant-Volume Gas Thermometer and the Absolute Temperature Scale One version of a gas thermometer is the constant-volume apparatus shown in Figure 1.3. The physical change exploited in this device is the variation of pressure of a fixed volume of gas with temperature. When the constant-volume gas thermometer was developed, it was calibrated by using the ice and steam points of water as follows. (A different calibration procedure, which we shall discuss shortly, is now used.) The flask was immersed in an ice-water bath, and mercury reservoir B was raised or lowered until the top of the mercury in column A was at the zero point on the scale. The height h, the difference between the mercury levels in reservoir B and column A, indicated the pressure in the flask at 0°C. 10  The flask was then immersed in water at the steam point, and reservoir B was readjusted until the top of the mercury in column A was again at zero on the scale; this ensured that the gas’s volume was the same as it was when the flask was in the ice bath (hence, the designation “constant volume”). This adjustment of reservoir B gave a value for the gas pressure at 100°C. These two pressure and temperature values were then plotted, as shown in Figure 1.4. The line connecting the two points serves as a calibration curve for unknown temperatures. (Other experiments show that a linear relationship between pressure and temperature is a very good assumption.) If we wanted to measure the temperature of a substance, we would place the gas flask in thermal contact with the substance and adjust the height of reservoir B until the top of the mercury column in A is at zero on the scale. The height of the mercury column indicates the pressure of the gas; knowing the pressure, we could find the temperature of the substance using the graph in Figure 1.4. Now let us suppose that temperatures are measured with gas thermometers containing different gases at different initial pressures. Experiments show that the thermometer readings are nearly independent of the type of gas used, as long as the gas pressure is low and the temperature is well above the point at which the gas liquefies (Fig. 1.5). The agreement among thermometers using various gases improves as the pressure is reduced. If we extend the straight lines in Figure 1.5 toward negative temperatures, we find a remarkable result—in every case, the pressure is zero when the temperature is 273.15°C! This suggests some special role that this particular temperature must play. It is used as the basis for the absolute temperature scale, which sets 273.15°C as its zero point. This temperature is often referred to as absolute zero. The size of a degree on the absolute temperature scale is chosen to be identical to the size of a degree on the Celsius scale. Thus, the conversion between these temperatures is TC  T  273.15 (1.1) 11  where TC is the Celsius temperature and T is the absolute temperature. Figure 1.5 Pressure versus temperature for experimental trials in which gases have different pressures in a constant-volume gas thermometer. Note that, for all three trials, the pressure extrapolates to zero at the temperature 273.15°C. Fig. 1.6 Absolute temperatures at which various physical processes occur. Note that the scale is logarithmic. Because the ice and steam points are experimentally difficult to duplicate, an absolute temperature scale based on two new fixed 12 points was adopted in 1954 by the International Committee on Weights and Measures. The first point is absolute zero. The second reference temperature for this new scale was chosen as the triple point of water, which is the single combination of temperature and pressure at which liquid water, gaseous water, and ice (solid water) coexist in equilibrium. This triple point occurs at a temperature of 0.01°C and a pressure of 4.58 mm of mercury. On the new scale, which uses the unit kelvin, the temperature of water at the triple point was set at 273.16 kelvins, abbreviated 273.16 K. This choice was made so that the old absolute temperature scale based on the ice and steam points would agree closely with the new scale based on the triple point. This new absolute temperature scale (also called the Kelvin scale) employs the SI unit of absolute temperature, the kelvin, which is defined to be 1/273.16 of the difference between absolute zero and the temperature of the triple point of water. Figure 1.6 shows the absolute temperature for various physical processes and structures. The temperature of absolute zero (0 K) cannot be achieved, although laboratory experiments incorporating the laser cooling of atoms have come very close. What would happen to a gas if its temperature could reach 0 K (and it did not liquefy or solidify)? As Figure 1.5 indicates, the pressure it exerts on the walls of its container would be zero. In a next Chapter we shall show that the pressure of a gas is proportional to the average kinetic energy of its molecules. Thus, according to classical physics, the kinetic energy of the gas molecules would become zero at absolute zero, and molecular motion would cease; hence, the molecules would settle out on the bottom of the container. Quantum theory modifies this prediction and shows that some residual energy, called the zero-point energy, would remain at this low temperature. The Celsius, Fahrenheit, and Kelvin Temperature Scales Equation 1.1 shows that the Celsius temperature TC is shifted from the absolute (Kelvin) temperature T by 273.15°. Because the size 13 of a degree is the same on the two scales, a temperature difference of 5°C is equal to a temperature difference of 5 K. The two scales differ only in the choice of the zero point. Thus, the ice-point temperature on the Kelvin scale, 273.15 K, corresponds to 0.00°C, and the Kelvin- scale steam point, 373.15 K, is equivalent to 100.00°C. A common temperature scale in everyday use in the United States is the Fahrenheit scale. This scale sets the temperature of the ice point at 32°F and the temperature of the steam point at 212°F. The relationship between the Celsius and Fahrenheit temperature scales is 9 TF  TC  32 F (1.2) 5 We can use Equations 1.1 and 1.2 to find a relationship between changes in temperature on the Celsius, Kelvin, and Fahrenheit scales: 5 (1.3) TC  T  TF 9 Of the three temperature scales that we have discussed, only the Kelvin scale is based on a true zero value of temperature. The Celsius and Fahrenheit scales are based on an arbitrary zero associated with one particular substance—water—on one particular planet—Earth. Thus, if you encounter an equation that calls for a temperature T or involves a ratio of temperatures, you must convert all temperatures to kelvins. If the equation contains a change in temperature T, using Celsius temperatures will give you the correct answer, in light of Equation 1.3, but it is always safest to convert temperatures to the Kelvin scale. Applications and problems Relation between the different scales Consider five identical thermometers marked in Celsius, Fahrenheit, Kelvin, Reaumur and Rankin scales. Place the five 14 thermometers in a bath at a certain fixed temperature. Mercury in each thermometer stands up to the same level. In the next Fig the temperatures of the absolute zero , the melting point of ice and the boiling point of water as measured on the Celsius (centigrade ), Kelvin (absolute ) Fahrenheit and Rankin scales are shown. Celsius and Fahrenheit scales show the same reading at -40 i.e., -40°C =-40 F The temperatures represented in the Fig have been rounded off to the nearest degree. Relation between different scales: C0 F  32 k  273 Rank  492 R  0     100  0 212  32 373  273 672  492 80  0 Example 1: The temperature of the surface of the sun is about 6500C what is this temperature : i) on the Rankin scale and ii) on the Kelvin scale? 15  C0 k  273 Rank  492   100  0 100 180 C= 6500 6500 K  273 Rank  492 Here   100 100 180 K= 6500 + 273= 6773 The temperature of the surface of the sun corresponding to 6500C is i) 6773K and ii) 12192 Rankin Example 2: The normal boiling point of liquid oxygen is -183C what is this temperature on : i) Kelvin ii) Rankin scale. Solution: C0 K  273 Rank  492   100 100 180 Here C= - 183C  183 K  273 Rank  492    100 100 180 K= 90 16 Rank= 162 The boiling point of liquid oxygen corresponding to-183C is i) 90 Kelvin iii) = 162 Rankin scale Example 3: At what temperature do the Kelvin and Fahrenheit scales coincide. Solution: K  273 F  32  100 180 X  273 X  32  100 180 X= 574.25 574.25K= 574.25F Example 4 At what temperatures do Celsius and the Fahrenheit scales coincide? Solution: C  0 F  32  100 180 X  0 X  32  100 180 17 X= -40  - 40 Celsius= - 40F Quick Quiz 1.2 Consider the following pairs of materials. Which pair represents two materials, one of which is twice as hot as the other? (a) boiling water at 100°C, a glass of water at 50°C (b) boiling water at 100°C, frozen methane at 50°C (c) an ice cube at 20°C, flames from a circus fire-eater at 233K (d) No pair represents materials one of which is twice as hot as the other Example 1.1 On a day when the temperature reaches 50°F, what is the temperature in degrees Celsius and in kelvins? Solution Substituting into Equation 1.2, we obtain TC  5 TF  32  5 50  32  10 C 9 9 From Equation 1.1, we find that T  TC  273.15  10 C  273.15  283 K A convenient set of weather-related temperature equivalents to keep in mind is that 0°C is (literally) freezing at 32°F, 10°C is cool at 18 50°F, 20°C is room temperature, 30°C is warm at 86°F, and 40°C is a hot day at 104°F. Example 1.2 A pan of water is heated from 25°C to 80°C. What is the change in its temperature on the Kelvin scale and on the Fahrenheit scale? Solution From Equation 1.3, we see that the change in temperature on the Celsius scale equals the change on the Kelvin scale. Therefore, T  TC  80 C  25 C  55 K From Equation 1.3, we also find that TF  9 5 9   TC  55 C  99  F 5 Platinum resistance thermometer It is based on the principle of change of resistance with change of temperature. It was first designed by Siemen in 1871 and later on improved by Calendar and Griffiths. A platinum resistance thermometer consists of a pure platinum wire wound in a double spiral to avoid inductive effects. The wire is wound on a mica. The resistance of a wire at t °C = Rt and at 0 °C = R0. These resistance are connected by the relation: Rt =R0 (1+t + t2) (1) Here  and  are constants, which can be determined. Let the resistance at the boiling point of pure water is R100 19 Neglecting t2 in eq.(l) (because  is very small) Rt = R0 (1+ t) (2) R100 = R0 (l +  100) (3) Rt-R0=R0 t (4) R100-R0 = R0. 100 (5) divided eq.(4) by eq.(5). Rt  R0 t  R 100  R 0 100 or  R  R0  t   t   100  R 100  R 0  knowing the values of R0 , R100 and Rt , t can be calculated. The resistance of the platinum wire is found accurately using Calendar and Griffith's bridge. Calendar and Griffiths Bridge: This is a modified form of Whetstone's bridge, used to measure the change in resistance with temperature of the platinum wire used in a platinum resistance thermometer. R1 and R2 are two resistances of equal value and r is a standard fractional resistance box. The platinum wire whose resistance is to be calculated is connected in one of the arms of the Whetstone's bridge as shown in the Fig. 20 The bridge wire of length L. Its total resistance is R and S its resistance per unit length. S = R/L Ohm/m To determine the resistance of the Platinum wire, it can be connected in Calendar and Griffith bridge as the fourth arm. In the third arm a standard resistance r is connected. At the balance point R1 /R2 = R3/ R4 (7) R1 and R2 can be adjusted at known values. R3 = r + S L1 (8) Where L1is the length of the bridge wire on the left hand side from the balance point, while R4 = P + S L2 (9) Where L2 is the length of the bridge wire on the right hand side from the balance point, and P is the resistance of the platinum wire at the temperature to be measured. Using the eqs. (7), (8) and (9) we get 21 R1 r  SL1  R2 P  SL2 If we choose R1= R2.we get P + S L2 = r + S L 1 P=r+ S (L1-L2) By this way the resistance of the platinum wire can be calculated at a given temperature. Thus using a Calendar and Griffiths bridge the resistance Rt of the platinum wire can be determined at temperature t A scale can be fixed along the bridge wire to read platinum scale temperatures directly. Advantages :The platinum resistance thermometer can be used with great accuracy to measure temperatures ranging from - 200°C to 1200°C, Its accuracy is 0.l°C. Once the platinum resistant thermometer has been standardized with a constant volume hydrogen thermometer. It can be used as a standard thermometer 1.4 Thermal Expansion of Solids and Liquids Our discussion of the liquid thermometer makes use of one of the best-known changes in a substance: as its temperature increases, its volume increases. This phenomenon, known as thermal expansion, has an important role in numerous engineering applications. For example, thermal-expansion joints, such as those shown in Figure 1.7, must be included in buildings, concrete highways, railroad tracks, brick walls, and bridges to compensate for dimensional changes that occur as the temperature changes. Thermal expansion is a consequence of the change in the average separation between the atoms in an object. At ordinary temperatures, the atoms in a solid oscillate about their equilibrium 22 positions with an amplitude of approximately 1011 m and a frequency of approximately 1013 Hz. The average spacing between the atoms is about 1010 m. As the temperature of the solid increases, the atoms oscillate with greater amplitudes; as a result, the average separation between them increases. Consequently, the object expands. Figure 1.7 (a) Thermal-expansion joints are used to separate sections of roadways on bridges. Without these joints, the surfaces would buckle due to thermal expansion on very hot days or crack due to contraction on very cold days. (b) The long, vertical joint is filled with a soft material that allows the wall to expand and contract as the temperature of the bricks changes 23 Figure 1.8 Thermal expansion of a homogeneous metal washer. As the washer is heated, all dimensions increase. (The expansion is exaggerated in this figure.) If thermal expansion is sufficiently small relative to an object’s initial dimensions, the change in any dimension is, to a good approximation, proportional to the first power of the temperature change. Suppose that an object has an initial length Li along some direction at some temperature and that the length increases by an amount L for a change in temperature T. Because it is convenient to consider the fractional change in length per degree of temperature change, we define the average coefficient of linear expansion as L L i  T Experiments show that  is constant for small changes in temperature. For purposes of calculation, this equation is usually rewritten as L  Li T (1.4) 24  or as Lf  Li  Li Tf  Ti  (1.5) where Lf is the final length, Ti and Tf are the initial and final temperatures, and the proportionality constant  is the average coefficient of linear expansion for a given material and has units of (°C)1. It may be helpful to think of thermal expansion as an effective magnification or as a photographic enlargement of an object. For example, as a metal washer is heated (Fig. 1.8), all dimensions, including the radius of the hole, increase according to Equation 1.4. Notice that this is equivalent to saying that a cavity in a piece of material expands in the same way as if the cavity were filled with the material. Table (1.1) Average Expansion Coefficient for Some Materials Near Room Temperature Average Volume Average Linear Expansion Material Expansion Material Coefficient() Coefficient () (C)-1 (C)-1 Aluminum 24 10-6 Alcohol ethyl 1.12 10-4 Brass and bronze 19 10 -6 Benzene 1.24 10-4 Copper 17 10-6 Acetone 1.5 10-4 Glass (ordinary) 9 10 -6 Glycerin 4.85 10-4 Lead 29 10-6 Mercury 4.82 10-4 Steel 11 10 -6 Turpentine 9.0 10-4 Glass (Pyrex) 3.2 10-6 Gasoline 9.6 10-4 Invar (Ni-Fe Alloy) 0.9 10 -6 a Air at 0C 3.76 10-5 Concrete 12 10-6 Helium 3.665 10-5 Gases don not have a specific for the volume expansion coefficient because the surmount of expansion depends on the type 25 of process through which the gas is taken,. The value given here assure that the gas undergoes expansion at constant pressure. Table 1.1 lists the average coefficient of linear expansion for various materials. Note that for these materials  is positive, indicating an increase in length with increasing temperature. This is not always the case. Some substances—calcite (CaCO3) is one example—expand along one dimension (positive ) and contract along another (negative ) as their temperatures are increased. Because the linear dimensions of an object change with temperature, it follows that surface area and volume change as well. The change in volume is proportional to the initial volume Vi and to the change in temperature according to the relationship V  Vi T (1.6) where  is the average coefficient of volume expansion. For a solid, the average coefficient of volume expansion is three times the average linear expansion coefficient:  = 3. (This assumes that the average coefficient of linear expansion of the solid is the same in all directions—that is, the material is isotropic.) To see that  = 3 for a solid, consider a solid box of dimensions l, w, and h. Its volume at some temperature Ti is Vi= l w h. If the temperature changes to Ti + T, its volume changes to Vi + V, where each dimension changes according to Equation 19.4. Therefore, Vi  V  l  l w  w h  h   l  lT w  wT h  hT   lwh1  T  3   Vi 1  3T  3T   T  2 3  If we now divide both sides by Vi and isolate the term V/Vi , we obtain the fractional change in volume: 26  V  3T  3T   T  2 3 Vi Because T