5. TEMPERATURE AND HEAT

Transcription

1 5. TEMPERATURE AND HEAT You will study the concepts of temperature and heat as they apply to a sample of water and you will measure the specific heat capacity of the sample. The measurement of one property of fluids, viscosity, was the major activity of the experiment Simple Measurements. But water is more important than shampoo, since together with air, it makes possible all life on earth. Water has a greater heat capacity than does air so those of us who live near the great lakes benefit from a local climate Theory Water that is moderated by these large bodies of water. The study of water is central to the environmental sciences, but it is also important in physics and chemistry. The major objective of this experiment is the measurement of the specific heat capacity of water. The difference between hot and cold is learned in childhood. A child learns by experience that a pot of boiling water is hot to the touch, and therefore hurtful, while a glass of ice water is cold and therefore safe. Most adult non-scientists would say that a thermometer would show that a pot of boiling water is at a higher temperature than is a glass of ice water, though not being able, perhaps, to define what temperature is exactly. Most of us know that the hotness or coldness of a body causes a thermometer to respond in a visual way through expansion or contraction of something. As the temperature rises the liquid in a Hot and Cold mercury or alcohol thermometer expands and rises in a column. If the column is calibrated the temperature of the liquid can be read from a scale. And conversely, as the temperature falls the liquid contracts and the column of liquid falls. Also known to the non-scientist is the word heat and that heat has something to do with temperature. It is widely claimed, however inaccurately, that a hot body contains more heat than a cold body. In fact, heat requires some study to define it precisely in terms acceptable to a physicist. But before defining anything we prepare the way by reviewing some everyday observations. Any interested person can collect useful data about heating and cooling with an ordinary thermometer. Here are a few sample observations: 1 If a cup of hot coffee is placed in a room, the temperature of the coffee falls, soon reaching the temperature of the room. If a cup of ice coffee is placed in a room, the temperature of the coffee rises, eventually reaching the temperature of the room. Observations with Thermometers When two bodies at different temperatures are brought into contact, the temperature of the higher one always drops while that of the lower one rises until the two temperatures are equal. When two bodies at the same temperature are brought into contact, no changes in temperature take place. These examples imply that two bodies at B5-1

2 5 Temperature and Heat different temperatures in contact experience an interaction a thermal interaction. Since the bodies reach the same temperature the interaction eventually ceases. Two bodies in contact which are not interacting are said to be in a state of thermal equilibrium. We can also add the following to the above list: If the coffee is first poured into a thermos, and the thermos is placed in a room, the temperature of the coffee falls, eventually reaching the temperature of the room but only after a much longer time than without the thermos. If the ice coffee is first poured into a thermos, and the thermos is placed in a room, the temperature of the coffee rises, eventually reaching the temperature of the room, but only after a much longer time than without the thermos. The new factors of insulation and time in these examples imply that something is happening that is not described completely by thermometer readings alone. We conclude our list with the following more sophisticated observations: If two bodies of the same substance and mass but at different temperatures T 1 and T 2 are brought into contact, the combination reaches the intermediate temperature (T 1 + T 2 )/2. If two bodies of the same substance of masses and m 2 at different temperatures T 1 and T 2 are brought into contact the combination reaches the temperature T = T 1 + m 2 T 2 + m 2. [1] This last example is a more general case of the one before. Clearly, mass is also a factor in thermal interactions. The examples just described can be seen to be consistent with a thermal model of a substance. The starting fact of this model is that any substance larger than a point mass is imagined to have an internal energy, which is the sum total of the translational and vibrational kinetic energies of the molecules in the substance, the potential energies of the molecules and other forms of energy. 2 This internal energy, U, is therefore a quantity we have no hope of measuring directly with any instrument. To make comparisons possible we shall think of the internal energy contained in a unit mass of the substance, the quantity U/m. It can be shown (from other branches of physics) that the internal energy per unit mass of a substance is proportional to its temperature T: B5-2 U m = εt, Thermal Model of a Substance [2] where ε is a constant of proportionality (which in what follows we shall make no effort to determine). We are interested in establishing that the reading on the thermometer indicates the internal state of the substance; the higher the temperature the more energy is contained in each gram or kilogram of the substance. With our thermal model we can now explain what transpires during a thermal interaction between two bodies, say of masses and m 2 which are initially at different temperatures T 1 and T 2 (Figure 1). The body at the higher temperature has the greater internal energy per unit mass. During the interaction some of the internal energy of the hotter body flows to the colder body. Energy flows until both bodies reach the same temperature (Figure 2). In this state of thermal equilibrium both bodies have the same internal energy per unit mass. Thus heat is not internal energy but energy in transfer. If during the interaction no work is done the hotter body s internal energy per unit mass decreases, while the colder body s internal energy per unit mass increases. Heat is the flow of energy or energy in transition. We repeat for emphasis that this transfer of heat is not observed directly,

3 but rather is inferred from the change in thermometer readings. Heat, being energy in transfer, is measured in the same units as kinetic energy, potential energy and work, namely joules (J). To relate heat with a change in temperature we must assume an expression which has been verified from a large body of experiments. In order to increase the temperature of an object of mass m by an amount dt, an amount of heat Q must be transferred to the object given by Q = cmdt, [3] Temperature and Heat 5 where c, the constant of proportionality, is called the specific heat capacity of the object. If Q has the unit joules (J), m is in kg and T is in C then c has the units J.kg 1. C 1. The specific heat capacities for substances vary widely as can be seen from the selections in Table 1. 3 Table 1. Specific heat capacities of a few substances Substance c Aluminum 900 Iron or steel 452 Benzene 1740 Water (15 C) 4186 Heat transfer is only inferred. It cannot be observed directly. higher temperature T 1 heat higher lower is U 1 U energy 2 m 2 transferred lower temperature T 2 Figure 1. When two bodies initially at different temperatures, as indicated by different readings on thermometers, are brought into contact, heat flows from the hotter to the colder body. Temperature T Temperature T U 1 = U 2 m 2 U 1 U 2 m 2 Figure 2. After some elapsed time the two bodies of Figure 1 reach a state of thermal equilibrium indicated by the same reading on thermometers. The temperature T is given by eq[1]. The net flow of heat between the bodies is inferred to be zero. Thus we also infer that the two bodies have equal internal energies per unit mass. B5-3

4 5 Temperature and Heat Calculating the Specific Heat Capacity In this experiment an electrical resistor, immersed in the water sample, is the source of heat. An electric current is made to pass through the resistor thus heating it up. The power supplied is measured by means of digital multimeters. Since the resistor is in thermal contact with the water sample, heat is transferred to the sample. The net result is the transformation of an amount of electrical energy into an equal amount of heat, which in turn adds to the internal energy of the water sample causing its temperature to rise. The temperature is measured and logged by a detection system and a computer. We can derive an expression that will enable us to calculate the specific heat capacity of the water from the rate of the temperature rise and the value of the power input. Dividing both sides of eq[3] by an increment of time dt we have Q dt = mc dt dt. [4] As stated above the heat per unit time (the heat power) is supplied by a resistor. Neglecting loss of power to the surroundings, therefore, this heat power is equal to the electric power Q dt = P = IV = V2 R, [5] assuming the resistor obeys Ohm s Law. 4 Combining eqs[4] and [5] we have Integrating eq[6] T i dt = P mc dt. T f t f dt = P dt mc, t i so we can write more simply T = b + P mc t, [6] [7] where b is some constant (numerically equal to the temperature at a clock time of zero seconds and of little interest to us). Eq[7] represents a straight line with slope P/mc. If the datapairs (t, T) are graphed, when the mass m and power P are known, c can be found from the slope of the graph. This is the method of finding c in this experiment. One Way to Account for Heat Loss The insulated container you are issued in this experiment is fairly rudimentary (but is at least not expensive!). It is expected that some loss of heat will occur through the wall of the container and out into the surrounding air. The effect of this loss is to reduce the temperature of the water bath at some clock time from what it would otherwise have been. And over the entire run the slope of the heating line is expected to be reduced from what it would otherwise have been. We might therefore put down a revised form of eq[7] T = b + P mc + P' t, mc [8] where P represents the heat power lost. With the equipment issued in this experiment the factor P cannot be found during heating. But if we assume that the heat loss during cooling and heating are the same per second, then P can be found from a cooling run. The slope of the cooling line can then be added to the slope of the heating line before c is calculated. This method will be explained in more detail in the Experiment section. B5-4

5 The Experiment Temperature and Heat 5 Exercise 0. Preparation Orientation Many of the technical aspects of this experiment you have seen before in the experiments Linear Motion and Vibrations. LoggerPro is used here too. You will use a Serial Box Interface instead of a ULI board but the functioning is similar. Identify the apparatus from Figure 3. You have a source of electrical energy (MW regulated 12V power supply), cables connecting the supply to two resistors in the insulated container, two model 7001 digital multimeters, an electronic thermometer connected to the Serial Box Interface (SBI), and in turn to the computer. Setting up the SBI and Supply Check the following: Confirm that the power adapter for the SBI is plugged into the power bar and the light on the Interface is glowing green. Confirm that the thermometer is plugged into Port 1 of the SBI and the signal output cable from the SBI is plugged into the printer port of the computer. Confirm that the power supply is plugged into the power bar and is OFF. Connecting the Multimeters Because you have not yet completed the DC Circuits experiment you should follow these instructions closely: Select one of the DMMs and set its controls to DC 20 A. This will be your ammeter (A) in Figure 3. Connect a wire from the 20 A socket of the DMM to the red output socket of the supply. Connect one of the wires from the heating resistors coming from the insulated container to the COM socket of the DMM. Connect the other wire from the heating resistors to the black output socket of the supply. Select the other DMM and set its controls to DC 20 V. This will be your voltmeter (V) in Figure 3. Connect the COM socket to the black socket of the supply and the VΩ Hz socket to the red socket of the supply. MW 12V power supply red black A V SBI printer port resistors insulated container Figure 3. Diagram of the equipment used in this experiment. You will calculate the power delivered by the supply from the readings on the multimeters. The multimeter in ammeter function (A) is connected in series with the resistors, the multimeter in voltmeter function (V) in parallel. B5-5

6 5 Temperature and Heat Preparing the Sample The first step is to measure off 200 g of water as accurately as you can. Pour the water into your insulated container and seat the top of the cup (which holds the thermometer) firmly in place. WARNING: Make certain before inserting the top of the cup that the resistors are completely immersed in the water. The usual covering of the resistors has been removed to ensure a good thermal contact between them and the water. If the resistors are operated in the open air they may burn out. Opening LoggerPro 1 Boot the computer as you learned to do in the experiment Linear Motion. ➁ Log into your account on the college network. Remember, if you can t log in you can always save your experiment in the Student Temp Save folder on the local hard drive. You can log in or out at any time. ➂ On the local hard drive Macintosh HD locate and open folders in this order: Physics >> PHYA10/A20 >> 05. Temperature and Heat. ➃ Inside 05.Temperature and Heat double click the icon Temperature and Heat. LoggerPro and the Temperature and Heat setup will run. The Opening Screen Examine this screen (it will resemble Figure 4 without the data) and identify the following aspects: The calibration is set to display temperatures between 0 C and about 60 C. Once the program is started the calibration is set to record for 20 minutes. At the end of that time recording will stop. The calibration is set to record five temperature readings every minute, making for 50 datapoints. Notice that the current reading of temperature is displayed at the bottom center of the screen. These are the readings that get recorded when you click Collect. Unlike other experiments using this detection system there are few options you need to investigate. Let us proceed to Exercise 1. Exercise 1. Collecting Data The object of this exercise is to gather data for heating and cooling and display it on the same graph. Double Checks You must be careful to connect up your multimeters correctly and have them set to the correct function and range. Double check this before you turn on the power supply. If in doubt ask your TA to check your circuit and DMM settings. The Run Heating and Cooling When you are ready to start recording turn the power supply ON and click Collect. LoggerPro collects the data automatically. Write down the voltage and current (including the correct units) periodically. (The power, the product of V and I B5-6 should be of the order of 25 watts.) It is suggested that at the end of 10 minutes you turn the power supply OFF. Then the data you collect from this time onward will be for cooling. At the end of 20 minutes data collection will automatically stop. You screen should resemble Figure 4 without the fits. WARNING: If you have chosen to follow a different procedure than the one here make certain the power supply is OFF when the 20 minute running time expires. Otherwise the water may boil. Your TA will

7 remind you of this. You are now ready to fit functions to your data. Temperature and Heat 5 Graph produced by LoggerPro, copied to the clipboard and pasted directly into Microsoft Word. Figure 4. A typical output from LoggerPro showing the regions for heating and cooling. Shown also are linear fits to each region as obtained from Exercise 2. Exercise 2. Fitting and Calculating Fitting Since you have already used LoggerPro to fit linear and polynomial functions to data (in Linear Motion and Vibrations ) you should be an old hand at this activity. What you probably haven t realized is that you can select and fit the heating and cooling curves individually and display the results on one graph. If you have forgotten how to do a fit follow these steps: 1 Select Analyze >> Examine. This gives you the vertical cursor. ➁ With the mouse, position the cursor at the beginning of the range you wish to fit, hold down the mouse button and drag over the range desired. You may observe that the heating section of the graph has a curve at its bottom end; you may wish to avoid this region and drag over only the linear part. ➂ Select Analyze >> Linear Fit. Notice that you can position the information box anywhere on the graph area by dragging with the mouse. If you wish more information on a fit (more digits, standard deviations of the slope, etc) you can double click on the header bar of the information box and make your request in the dialog box which then appears. ➃ Do the above for the cooling region as well as the heating region. Your screen should resemble Figure 4. B5-7

8 5 Temperature and Heat Calculating the Specific Heat Calculate the specific heat capacity from the two slopes. Calculating the Uncertainty in c In physics a result is not final until its uncertainty is known. To get the uncertainty in c, c, we can first write c in terms of the slope of the linear section of a graph: c = = P m _ x _ slope, I _ x _ V m _ x _ slope. now by hand or use the Maple worksheet Quadrature Calculator. You can find the uncertainties in I and V from the specification sheets in the guidesheet for the experiment DC Circuits. The uncertainty in the slope is a standard deviation in m as returned by LoggerPro. Write your value of c in correct notation (as you learned to do in the Orientation Workshop). Questions:? Does your value of c agree with the accepted value (Table 1) to within your experimental error?? Can you speculate on the cause of the curve at the beginning of the heating graph? You must apply the quadrature formula. Do this Physics Demonstrations on LaserDisc from Chapter 32 Thermal Phenomena Demo Specific Heat Demo Boiling Water in a Paper Cup from Chapter 33 Heat Transfer Demo Two Can Radiation Demo Insulation (Dewar Flasks) Activites Using Maple Quadrature Calculator Calculating the uncertainty in a function of two or more variables from the quadrature formula is one of the more important, though stressful, activities you are called upon to do in the first year physics lab. This worksheet is intended to show you how Maple may be used to calculate the uncertainty in a function that you input. You may also input numerical values for the variables, and the uncertainty in the variables, to get an actual numerical value for the uncertainty in the function. Stuart Quick 96 EndNotes for Temperature and Heat 1 This guidesheet is strongly influenced by Arnold Arons treatment of the subject in his A Guide to Introductory Physics Teaching (Wiley & Sons, 1990). 2 You have encountered the idea of energy before in the experiment Linear Motion. The energy studied there was mechanical energy (kinetic and potential), the energy the object possessed assuming it to be a point mass. 3 Specific heat varies with temperature. The specific heat of water varies by about 1% from 0 C to 100 C. 4 We apologize here for having to introduce electrical concepts which you may not have encountered yet in the theory course. This underlines the importance of electricity in physics! In any case these concepts will be useful later in the experiments DC Circuits and The Solar Array. B5-8