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Transcript
1 Temperature 1.1 Introduction Many of the concepts of thermodynamics are familiar from everyday experience but are difficult to define in a scientifically rigorous way. Foremost of these is temperature. Everyone uses the concepts of hotness and coldness and understands the meaning of the air temperature given on the radio; however, few people can give a definition of exactly what is meant by “the temperature of the water was 68°F.” Introductory physics text usually explain the concept of temperature of an object in terms of the kinetic energy of the molecules which make up the object. Although this approach is useful in gaining an intuitive understanding, it does not meet our needs for two reasons. First, it is a microscopic definition and so does not fit with the aims of thermodynamics. Second, it does not provide any method to experimentally measure temperature. Thus it is the goal of this chapter to present a rigorous, macroscopic definition of temperature, as well as to demonstrate ways to measure it. 2 TEMPERATURE 1.2 Thermal Equilibrium and the Concept of Temperature The everyday concept of temperature is based on the sensory perception of hotness and coldness. One can compare the temperature of two objects by touch and thus determine which has a higher temperature. From this experience, we expect to be able to rank objects by temperature. Therefore, temperature must be a scalar quantity and can be specified by a single value. A second observation based on experience is that if two objects with different temperatures, as determined by touch, are brought into contact, they will both change temperature. Eventually the objects will reach a common temperature, which is between the original temperatures. Similarly, if two objects with the same temperature are brought into contact, neither will change temperature. These two observations will serve as the basis of our scientific definition of temperature, but first we need several definitions. thermal contact diathermal wall thermal equilibrium We say two objects are in thermal contact if they are separated only by a wall which is a good conductor of heat, such as a thin sheet of copper. Such a wall, called a diathermal wall in thermodynamics, prevents particles from passing through but allows some energy to pass. When two objects are first brought into thermal contact, they may exchange energy and so change temperature. Eventually the objects will stop changing. We can then say they are in thermal equilibrium. Thus, we define thermal equilibrium as follows: Two objects are in thermal equilibrium if they do not change when separated by a diathermal wall. temperature The concept of thermal equilibrium is used to define the concept of temperature as follows: The temperature is that property of an object that determines whether it is in thermal equilibrium with other objects. THERMAL E QUILIBRIUM AND THE C ONCEPT OF T EMPERATURE For temperature to be a useful concept, we need the following postulate: Two objects in thermal equilibrium with a third object will be in thermal equilibrium with each other. The need for this postulate was recognized by R. H. Fowler in 1939 and we shall follow him in calling it the zeroth law of thermodynamics. The first and second laws were already named by this time. 3 This postulate seems selfevident and was taken for granted by physicists throughout the development of thermodynamics. We cannot expect, however, that relationships are always transitive. 1.3 Thermometry The above definition of temperature meets our need for a rigorous physical definition but does not provide a way to measure and quantify temperature. We need a way to measure temperature which is based on an experimental definition which is reproducible. Such a definition is called an empirical temperature scale. To define an empirical temperature scale, one must choose a system to act as a thermometer and at least two reference temperatures. Some measurable property of the thermometer, called the thermometric property, must change with changing temperature. All other properties of the system which affect this thermometric property must be held constant. Several examples of thermometers are considered below. •Common Thermometers The thermometers with which the student is most familiar are probably liquid expansion thermometers. The basic design of this type of thermometer, shown in figure 1.1, dates back to 1650, when it was known as a Florentine thermometer. An arbitrary scale was engraved on the sealed tube. Later, two reference temperatures were chosen and the separation between the two was divided into equal parts or degrees. As we will see later, many different choices for reference temperatures and number of degrees were used. empirical temperature scale thermometric property 4 TEMPERATURE Figure 1.1: Florentine Thermometer. sealed tube with arbitrary scale glass bulb filled with liquid The bimetallic strip thermometer is a second common type of thermometer with which the student is probably familiar. This thermometer, as shown in figure 1.2, consists of a coiled strip consisting of two metals. The metals used have very different coefficients of thermal expansion. As a result, a change in temperature will cause the coil to expand or contract, moving a needle on a calibrated dial. Figure 1.2: Bimetallic Thermometer. THERMOMETRY 5 •Precision Thermometers Some of the systems and thermometric properties commonly used to make precision measurements of temperature are summarized in table 1.1. The most important of these, the gas thermometer, is discussed in section 1.4. The physical theory needed for the optical pyrometer and paramagnetic thermometer will be discussed in later chapters. Two of the remaining thermometers merit special discussion. First is the thermocouple, which consists of two wires made of dissimilar metals in contact with each other. A small thermal electric potential is produced at the junction. This emf is temperature-dependent and, therefore, can be used as a thermometric property. For a more accurate temperature measurement, the difference in emf from a thermocouple at the test temperature and a second at a reference temperature, often 0°C, can be measured by a microvoltmeter or a potentiometer. This difference in emf, φ, is generally between 5 and 50µV/K and does not vary linearly with the temperature. The thermocouple must be calibrated at a number of known temperatures. The variation of φ as a function of the Celsius temperature can usually be fit to a cubic equation, φ = a + bt + ct 2 + dt 3 , (1.1) to within the desired accuracy over a large range of temperatures. As an example, a commonly used thermocouple is constructed from platinum and 90% platinum, 10% rhodium alloy. It can be fit to a cubic function to an accuracy of ±0.1K over a range of 400K. Thermometer Thermometric Property Constant Volume Gas pressure Acoustic speed of sound Range of Use 2K < T < 30K Thermal Noise thermal noise voltage 0.01K < T < 1K Paramagnetic magnetic susceptibility 0.001K < T < 1K Thermocouple thermal emf 150K < T < 2000K Electrical Resistance electrical resistance 10K < T < 1000K Optical Pyrometer radiant emittance T > 1000K Table 1.1 : Some thermometers, thermometric properties, and typical ranges of application. This phenomenon was discovered by the German physicist Thomas Johann Seebeck (1770-1831) in 1821 and is called the Seebeck effect. 6 TEMPERATURE Thermocouple thermometers have a number of advantages which makes their use widespread. The junctions and wires can be made very small so that the heat capacity of the thermometer does not significantly affect the experiment and the thermocouple can reach the temperature of the sample very quickly. With modern semiconductor technology, it is inexpensive to produce a combination microvoltmeter and dedicated microprocessor which can measure φ, convert it to a temperature using a cubic or quadratic equation of fit, and display the result. This type of thermometer is now common not only in physics labs but also in homes. The second type of precision thermometer we need to consider is the electric resistance thermometer. The most important of these uses the resistance of platinum as the thermometric property. This resistance increases with increasing temperature and can be very accurately measured. The most common experimental approach is shown in figure 1.3. The current through the circuit is held constant by varying a rheostat (variable resistor) while the voltage drop across the resistor is measured using a potentiometer. Since the variation of resistance with temperature is not linear, the thermometer is calibrated at several known temperatures. Thermometers using platinum or other metals can be very accurate but have several inherent problems. First, since a current is running through the resistor, there is heat being dissipated into the sample. This can be significant problem if the sample is very small. Second, the resistor requires very careful handling since small physical changes in the resistor can affect the reproducibility of the measurement. resistor in sample battery to potentiometer µA rheostat Figure 1.3: Basic circuit for electric resistance thermometry. THERMOMETRY 7 Semiconductors may also be used in resistance thermometry. For these materials, the resistance decreases approximately exponentially with increasing temperatures. Semiconductor thermometers are commonly used for precision measurements at very low temperatures. They are also used at higher temperatures in inexpensive digital thermometers and as thermostats. •Temperature Scales In addition to specifying the system and thermometric property used as a thermometer, it is necessary to specify either two reference temperatures or one reference temperature and the size of the degree to completely define an empirical temperature scale. Many scales have been used historically, but the most familiar are the Fahrenheit and Celsius scales. The Fahrenheit scale has a long history beginning with Isaac Newton, who suggested using the melting point of ice as the lower reference temperature, 0, and the temperature of a healthy male as the higher reference temperature. He suggested dividing this range into 12 parts. Fahrenheit made two improvements. First, he was able to produce a lower temperature for his zero point by using a mixture of salt and ice. Second, because of his skill in making mercury-in-glass thermometers, he was able to measure much smaller differences in temperatures. Following the usual practice of the times, he divided his degrees into eighths. As part of his investigations, he measured the melting point of ice as 32°F and the boiling point of water as 212°F. Soon after his death, these two temperatures were adopted as the fixed reference temperatures. The widespread use of this scale resulted primarily from the high quality thermometers produced by Fahrenheit and his successors. Dividing the temperature interval between the freezing point and boiling point of water into 100 degrees was proposed by several scientists prior to 1740. The resulting scale, which also depends on the choice of thermometric property, is called a centigrade scale. The most commonly used temperature scale today is the Celsius temperature scale, in which the freezing point of water is 0°C and the boiling point of water is (approximately) 100°C. Gabriel Daniel Fahrenheit (1686-1739), German craftsman and physicist. The system was named in honor of Anders Celsius (1701-1744), a Swedish astronomer who suggested the use of a centigrade scale. In his scale, the boiling point was 0 and the freezing point was 100. 8 This scale was suggested by René Antoine Ferchault de Réaumur (1683-1757), a French physicist, and found common acceptance in pre-revolutionary France. TEMPERATURE A third system still in use in parts of Europe is the Réaumur scale. This scale uses the same fixed reference temperatures but divides the interval into 80 degrees, with the freezing point being 0°Ré and boiling point being 80°Ré. 1.4 Gas Thermometry and Absolute Temperature In an attempt to extend the range of temperature and to avoid some of the arbitrariness of the choice of mercury as a standard, physicists devised several thermometers which use a gas as the thermometric substance. The most important of these is the constant volume gas thermometer. •The Constant Volume Gas Thermometer Figure 1.4 is a diagram of a simple constant volume gas thermometer. The thermometric gas is contained in a bulb made of metal, which is connected to a mercury manometer. The mercury reservoir is raised or lowered to keep the volume of the gas constant. The pressure of the gas, which is equal to atmospheric pressure plus ρgh, is used as the thermometric property. The thermometer shown in the figure, while sufficient for lowprecision measurements at moderate temperatures, suffers from a number of technical problems which must be overcome in designing a thermometer for precision work. These include the following: 1. The gas in the tube and above the mercury is at a temperature different from the gas in the bulb and the sample. The gas does not have uniform temperature. This problem may be overcome by using a second gas, separated from the thermometric gas by a diaphragm, as the manometric gas. 2. The bulb and tube change volume (slightly) with changes in temperature and pressure. The pressure effects can be largely eliminated by surrounding the bulb and THERMOMETRY 9 tubing with a gas, often the manometric gas, whose pressure is varied to equal that of the thermometric gas. 3. The density of the mercury varies with the pressure and temperature. The temperature dependence can be eliminated by maintaining the mercury at a standard temperature. This heats the manometric gas near the mercury, but the conductivity of gases is low enough that little heat flows into the bulb. The pressure variation must be measured and calibrated, and a correction applied to the final measurement. 4. At very low temperatures, the gas may adhere to or be absorbed by the walls of the bulb or tube. This effect must be measured and a correction applied. h gas bulb to be placed in sample mercury manometer flexible rubber tubing Figure 1.4: Simple constant volume gas thermometer in which the mercury reservoir is raised or lowered so that the left meniscus remains at the same height. The difference in column heights, h, gives the gauge pressure. 10 TEMPERATURE A temperature scale using the pressure of a constant volume gas thermometer as the thermometric property still suffers from one difficulty— different gases give slightly different temperatures. To eliminate this problem, we use the fact that all gases act like an ideal gas at low enough pressure. •The Ideal Gas Temperature Scale The concept of an ideal gas is discussed in chapter 2. absolute zero absolute temperature The ideal gas temperature scale is defined using a constant volume gas thermometer. The thermometer is used to measure the pressure, Pi, at the freezing point of water and the pressure, P, at the temperature of the sample. The measurements are repeated with a lower density of gas in the bulb, which will reduce both Pi and P. When extrapolated to Pi=0, the ratio of pressures is found to be independent of the gas used. This limit is used as the thermometric property. Thus T = lim P . T i Pi→ 0 Pi This equation also specifies one of the reference temperatures as T=0 when P=0. This temperature is called absolute zero and a temperature scale with zero at absolute zero is called an absolute temperature scale. No gas can be used to actually measure temperatures at very low temperatures, so absolute zero on the ideal gas scale exists only as an extrapolation. To complete the specification of the temperature scale, we must specify a second fixed temperature or the size of the degree. Originally the size of the degree was chosen to be equal to a Celsius degree. This requires Ts - Ti = 100 . (1.2) We get a second equation by measuring Ts, the boiling point of water. Ts P = lim s . i Ti → 0 Pi THERMOMETRY 11 This measurement has been made by many experimenters using many different gases. Typical data for such experiments are graphed in figure 1.5. The accepted value is Ts = 1.36609 ± 0.00004 . (1.3) Ti Solving equations (1.2) and (1.3) yields Ti = 273.16 and Ts = 373.16 . As thermometers became more precise and the temperatures measured became colder, the use of Ti as a standard became a problem. The ice point, defined as the temperature of a mixture of ice and water saturated by air at a pressure of 1 atmosphere, is reproducible only to an accuracy of ±0.01°C. The major source of uncertainty in this measurement is the amount of air in the water. To eliminate this uncertainty, the use of the ice and steam points in the definition was replaced with a single fixed reference point whose value was chosen to reproduce approximately the earlier scale. The General Conference of Weights and Measures in 1954 chose the triple point of water, 273.16, as the fixed point. The 373.8 O2 373.6 T (K) 373.4 N2 373.2 H2 He 373 0 20 40 60 PTP (kPa) 80 100 120 Figure 1.5: Graph of measurements made in determining the temperature of a sample using four different gases. For this case, the sample is at Ts. 12 TEMPERATURE triple point of water is the temperature at which ice, water, and water vapor can coexist. This temperature is easily reproducible to within ±0.0001°C. Figure 1.6 shows a simple triple point cell used to maintain the thermometer bulb at TTP. The cell is filled with pure water and sealed. The central well is filled with a mixture of salt and ice until a fraction of the water in the cell has frozen. The mixture is then replaced with the thermometer bulb. Making a single temperature measurement requires the following procedure: 1. Fill the bulb with the chosen thermometric gas, generally hydrogen. 2. Allow the gas and bulb to reach equilibrium with a triple point cell. Take repeated measurements of the pressure. 3. Move the cell or thermometer so that the bulb can be placed in the sample. 4. Allow the gas and bulb to reach equilibrium with the sample. Take repeated measurements of the pressure. 5. Decrease the amount of gas in the bulb and repeat steps 2-4. vapor ice water Figure 1.6: Simple triple point cell. The cell is symmetric about a vertical axis, thus forming a central well. THERMOMETRY 13 6. Extrapolate P/PTP to PTP = 0. The temperature of the system is then given by T = 273.16K lim P . PTP → 0 PTP See example 1.1 for one way in which the extrapolation can be done. A single high-precision temperature measurement with an ideal gas thermometer typically takes many months. Using this new definition to measure the ice point and steam point experimentally, one finds Ti=273.15 and Ts=373.125 on the ideal gas temperature scale. 1.5 The International Practical Temperature Scale (IPTS-68) It has been our ultimate goal to define a temperature scale which is not dependent on an arbitrary choice of thermometric property. We will see, in Chapter 5, that such a scale can be defined theoretically and is called the Absolute Thermodynamic Temperature Scale or, more simply, the Kelvin scale. To make such a theoretical temperature scale useful, one must find empirical temperature scales which coincide with it over the desired range of temperatures. We will show, in Chapter 5, that the ideal gas temperature scale coincides exactly with the Kelvin scale over the range of temperatures for which the ideal gas scale is defined. We do not use the ideal gas thermometer as our empirical temperature scale, however, because a single precision measurement with a gas thermometer can take many months. To meet the need for a practical way to measure temperatures with high precision, the Seventh General Conference of Weights and Measures adopted an international practical temperature scale. This scale has been revised several times to take advantage of improving technology and better data. The currently accepted scale is called the International Practical Temperature Scale of 1968 (IPTS-68). This scale was named in honor of William Thomson, Lord Kelvin (1824-1907) the British physicist who suggested the use of an ideal gas thermometer and the triple point of water to define a temperature scale. 14 TEMPERATURE Example 1.1 A student uses a low-precision constant volume gas thermometer to measure the temperature of a water bath. Use a least square fit to a line to calculate the temperature and uncertainty in temperature from the following data (in kPa): PTP 40.0 60.0 80.0 101.0 120.0 P 51.2 76.5 102.2 128.8 152.6 Solution: To use the definition of ideal gas temperature P , PTP we plot P/PTP versus PTP. We then fit a line to the data using a least square fit. That line will have the equation P =m P + b , TP PTP where m is the slope and b is the intercept. The limit as PTP → 0 is then b. First calculate P/PTP. T = 273.16K lim PTP → 0 PTP 40.0 P/PTP 1.2800 60.0 1.2750 80.0 1.2775 101.0 1.2752 120.0 1.2717 A computer program was used to calculate the slope and intercept and their uncertainties. The slope was (-8.1356±.0003) kPa-1 and the intercept was 1.2824±.002. Therefore, T = 273.16K (1.2824±.002) = 350.3K ± 0.5K 1.285 1.280 P/ PTP 1.275 20 40 PTP 60 (kPa) 80 100 120 THE INTERNATIONAL PRACTICAL TEMPERATURE SCALE Table 1.2: Temperatures of primary fixed points of the International Practical Temperature Scale of 1968. Fixed Points T68 (K) t68 (°C) triple point of hydrogen 13.81 -259.34 boiling point of hydrogen at 25/76 atm 17.042 -256.108 boiling point of hydrogen at 1 atm 20.28 -252.87 boiling point of neon at 1 atm 27.102 -246.048 triple point of oxygen 54.361 -218.789 triple point of argon 83.798 -189.352 boiling point of oxygen at 1 atm 90.188 -182.962 boiling point of water at 1 atm 373.15 99.975 melting point of zinc at 1 atm 692.664 419.514 melting point of silver at 1 atm 1235.08 961.93 melting point of gold at 1 atm 1337.58 1064.43 IPTS-68 consists of a table of primary and secondary fixed point temperatures, a thermometer and thermometric property to be used for different ranges of temperature, and a method for interpolating between the fixed point temperatures. These are summarized in tables 1.2 and 1.3. Temperature Range Thermometer Interpolation Method 13.81K to 273.15K platinum resistance thermometer polynomial fit to tabulated function 273.15K to 903.89K platinum resistance thermometer quadratic fit to triple point of water and primary fixed points 903.89K to 1337.58K platinum and 90% Pt-10%Rh thermocouple quadratic fit to 903.89K and melting points of silver and gold above 1337.58K optical pyrometer Planck radiation law Table 1.3: Thermometers, thermometric properties, and methods of interpolation for the International Practical Temperature Scale. 15 16 TEMPERATURE The Kelvin scale and thus IPTS-68 are now used as the basis of the definition of the Celsius and Fahrenheit temperature scales. These definitions are summarized as tc = T - 273.15K 1°C K 9°F and t f = tc + 32° F . 5°C For the remainder of this book, we will use the Kelvin scale and give temperatures in °C. Remember, however, that this usage assumes an identification, which we have not yet shown, between the empirical temperature scales of IPTS-68 and the thermodynamic temperature scale, which we haven’t yet defined. PROBLEMS Problems 1.1 The normal boiling point for helium is the lowest of any substance at -268.93°C. Tungsten has the highest melting point of any metal at 3410°C. What are the corresponding temperatures on the Kelvin, Fahrenheit, and Réaumur scales? 1.2 The Rankine temperature scale is an absolute temperature scale with the degree size chosen to be equal to 1°F. The defining equations are equation (1.3) and T s - Ti = 180 . Find Ts and Ti on the Rankine scale. 1.3 At what temperature do Celsius and Fahrenheit temperatures have the same numerical values? At what temperature do Kelvin and Fahrenheit temperatures have the same numerical values? At what temperature do Kelvin and Rankine (see problem 1.2) temperatures have the same numerical values? 1.4 The data in the following table were taken from a constant volume gas thermometer. Determine the temperature of the sample and its uncertainty by plotting Ps/PTP versus PTP and using a least square fit to extrapolate to zero pressure. PTP in torr 100.0 200.0 300.0 400.0 500.0 Ps in torr 233.9 471.7 714.7 962.7 1215.4 1.5 The resistance of a platinum resistance thermometer is found to be 7.26Ω at the ice point, 9.14Ω at the steam point, and 16.17Ω at the normal melting point of zinc. Use these data to determine the coefficients of a quadratic fit. Plot the resistance versus temperature for t=0°C to 450°C. The resistor is placed in a sample and allowed to come to equilibrium. The resistance is measured as 12.24Ω. What is the temperature of the sample? 17 18 TEMPERATURE 1.6 The electromotive force produced by a thermocouple can be described by the equation in t (Celsius temperature) φ = a + bt + c t2 . a) Find the expression for φ as a function of T (Kelvin temperature). b) Define a centigrade temperature based on φ. That is, the scale should be linear in φ and tφ=0 at t=0°C and tφ=100 at t=100°C. c) Define an absolute temperature based on φ, φ T φ ≡ 273.16 . φ TP Give Tφ as a function of t and the parameters a, b, and c. 1.7 What is the ratio of the pressure of a gas at the normal melting point of zinc to that at the triple point of water? What must be the maximum uncertainty in your pressure measurements if you are to get an answer with as many significant figures as given in Table 1.2? 1.8 A thermocouple has been calibrated and the following voltages have been measured: 50°C 1.43 mV 100°C 2.94 mV 200°C 5.93 mV 300°C 8.94 mV When immersed in a sample, the thermocouple produces 2.00 mV. Determine the temperature of the sample by fitting the data exactly to a cubic equaiton. Get a second value for the temperature by determining the quadratic equation which best fits the data. Which estimate of the temperature is better? PROBLEMS 1.9 An optical pyrometer is used to measure temperatures above the normal melting point of gold, called the gold point. The sample and a black body at the gold point temperature are each sighted with the same optical pyrometer. When viewed through a filter (λ=6000Å), the sample appears as bright as the blackbody only when crossed polarizers are used to dim the sample by 50%. Use equation (3.4) to determine the temperature of the sample. 19 20