HEAT AND THERMODYNAMICS

Transcription

1 HEAT AND THERMODYNAMICS 1. THE ABSOLUTE TEMPERATURE SCALE In the textbook you have been introduced to the concept of temperature, and to the fact that there is a natural zero of temperature, the temperature at which the motion of molecules ceases. This temperature is called absolute zero. With a simple experiment we can determine the location of absolute zero on the Celsius scale, by making use of the ideal gas law, pv = nrt, where p is the pressure in N/m 2 K, n is the number of moles, R is the universal gas constant (R = 8.31 Joule/mol-K), and T K is the absolute temperature in Kelvins. For our purposes the formula can be re-written nr nr p = TK = ( TC + TK 0 ) (1) V V Where T C is the temperature measured in Celsius and T K0 is the absolute temperature of the gas (measured in Kelvins) when it is at 0 degrees Celsius. (We will be measuring the constant T K0 ). This formula tells us that if we plot the pressure of a fixed, sealed volume of gas, then the pressure should be proportional to the absolute temperature. Thus, we should be able to determine the value of absolute zero (expressed in degrees Celsius) by taking our fixed, sealed volume of gas and measuring the pressure at two known temperatures; e.g. 0 degrees Celsius and 100 degrees Celsius, corresponding to ice water and boiling water. Experiment: You will find several stainless-steel bulbs, containing air at various pressures. Take each bulb and immerse it sequentially in the given bath of ice water and the bath of boiling water, recording the respective pressures. Plot the pressure versus temperature for each bulb, as shown in Figure 1, and graphically extrapolate to zero pressure. Do the two lines intersect the p = 0 axis at the same temperature? What is that temperature? Given your measurements of the absolute pressure at 0 C and 100 C, how would you mathematically extrapolate back to zero pressure? (Hint: consider the ratios of sides of similar triangles). What temperature do you get if you do so, for each of the starting pressures that you used?

2 30 25 Absolute Pressure (psi) Temperature (C) Figure 1 The pressure of an ideal gas in a sealed bulb as a function of temperature in Celsius. The three curves depict the same volume but with different numbers of atoms in the volume, and hence different pressures at a given temperature. 2. THE MECHANICAL EQUIVALENT OF HEAT When you rub your hands together you do mechanical work against the frictional force exerted by one hand against the other. Try rubbing your hands together. Your hands get warm as the mechanical work (force times distance) is converted into thermal energy. As we saw in the previous demonstration, thermal energy is just the energy associated with the random jiggling of atoms. The frictional force pushes and pulls on the individual atoms, giving them increased kinetic energy of jiggling. You may also warm your hands by putting them close to a fire, that is, by heating them. Whether you rub your hands or put them close to a fire, the end result increased jiggling of the atoms and increased temperature is the same.

3 Prior to the middle of the nineteenth century it was not realized that heat was a form of energy. Heat was thought to have the properties of a kind of invisible material fluid, such that when it flowed into a body, that body got hot. We now realize that heat is merely another form of energy, and that the temperature of a body can be increased either by rubbing it or by holding it over a source of heat. The amount of rubbing work needed to raise a gram of water by one degree Celsius is called the mechanical equivalent of heat. The apparatus, shown in Figure 2, consists of an aluminum spool of known mass m and diameter d, which may be heated by rubbing it with a known frictional force. As a result of this rubbing, the temperature of the spool will rise. To measure the temperature of the spool, there is, embedded in the spool, a device called a thermistor a temperaturesensitive resistor whose resistance is a unique function of its temperature. If this functional relationship is provided, the temperature rise of the aluminum spool may be determined by monitoring the resistance of the thermistor with an ohmmeter. The calibration of our thermistor is given in Table 1. Figure 2 The mechanical equivalent of heat apparatus. The known frictional force can be exerted by a length of cord that is wrapped a few turns around the spool. The upper end of the cord is attached to a post, while the lower end of the cord is attached to a known mass M, typically several kilograms. If the crank is turned quickly enough to make the cord tension vanish where it is attached to the post, but not so quickly that the weight accelerates upwards, the frictional force will just equal Mg, the weight of the suspended mass. The work done by this force for each revolution of the cylinder will be given by work per turn = force times distance = ( Mg)( π d) (2)

4 If the crank is turned N revolutions, the total work done will be NMgπ d. This work, which will be measured in Joules if M is measured in kilograms and d is measured in meters, 1 will be completely transformed into heat. Almost all of this heat will be absorbed by the spool, since the cord is a good insulator. TEMPERATURE (C) RESISTANCE (Ω) , , , , , , , , , , ,422 Table 1. Resistance vs. temperature for the PASCO thermistor. The total heat absorbed by the cylinder, which we call Q and which is measured in calories, is equal to the heat capacity of the cylinder times the temperature rise ΔT. Q = CmΔT (3) Here C is the specific heat of the pure aluminum cylinder (0.214 calories/gram-c), m is the mass of the cylinder in grams 2 (not to be confused with M, the much larger mass of the suspended weight), and ΔT is the temperature rise in degrees Celsius. Experiment: With the mass suspended from the cord, and with the cord wrapped 3-4 turns around the cylinder, turn the crank about 100 turns. After you stop, wait about a minute (but not much more) for the temperature in the cylinder to come to equilibrium throughout itself. 3 Determine the total work done by the frictional force in Joules, and determine the amount of heat absorbed by the cylinder in calories. The number of Joules per calorie is called the mechanical equivalent of heat, and has been determined, by experiments similar to the one that you have just done, to be approximately Joules per calorie. How well does your measured value agree with this? 1 For this apparatus, d = m. 2 For this apparatus, m = kg. 3 If you wait much longer than a minute, the spool will begin to noticeably cool overall.

5 PRE-LABORATORY QUESTIONS 1. Measurement of absolute zero: Suppose the pressure of the bulb is 20 psi (absolute) at 20 C. What would you predict the pressure to be at 0 C? What would you predict the pressure to be at 100 C? 2. The Mechanical equivalent of heat: Using the table provided, what is the change in the thermistor resistance if its temperature rises from 21 C to 30 C? Does it matter how many turns of cord surround the cylinder? Why? Does it matter how fast you turn the crank? Why? Suppose the cord, where it is attached to the post, is under tension during the experiment. Would this lead to an error in your measured value of the mechanical equivalent of heat? If so, would your value be too large, or too small?