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UNIVERSITY OF SURREY DEPARTMENT OF PHYSICS Level 1: Experiment 2A THERMAL EXPANSIVITY of SOLIDS and GASES 1

AIMS

1.1

Relevant Physics

These experiments investigate how the volumes of solids and gases change with temperature at a constant pressure. You will measure the length of bars of various metals (aluminium, copper and/or steel) over the temperature range between 0 and 100 °C. You will then compare the values of the linear thermal expansivity for each of the metals and you will determine whether this expansivity is a strong function of temperature. You will also do a series of experiments to measure the thermal expansivity of air in a piston and will compare your result to the values for the metals. Because gases have a high thermal expansivity, their large volume change can be used to do useful work. You will use air in a piston to build a simple heat engine and operate it in a cycle.

1.2

Skills

The particular skills you will acquire by performing these experiments are: z

Planning of a project for efficient time management.

z

Precision measurements using an analogue gauge.

z

Skills in data collection.

z

Data plotting and analysis skills.

Document last updated on 6/2/06 by JL Keddie

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INTRODUCTION

Thermal expansivity is a fundamental thermodynamic property. It is defined in terms of pressure (P), volume (V) and temperature (T). In the language of thermodynamics, these three variables are jointly referred to as state variables because they define the equilibrium state of a system. The volume thermal expansivity β is simply the fractional increase in volume per unit temperature rise. In differential form it is given as 1 ⎛ ∂V ⎞ (1) β= ⎜ ⎟ , V ⎝ ∂T ⎠ P where the ∂ designates a partial differential and the subscript P refers to constant pressure. In one dimension, a linear thermal expansivity α can be defined for a substance with a length L and under a constant force F:

α=

1 ⎛ ∂L ⎞ ⎜ ⎟ . L ⎝ ∂T ⎠ F

(2)

From comparison of the expressions for α and β, it can seen that F can be considered to be a one-dimensional analogue of P and L is the one-dimensional form of V. Note that the units for both α and β are given as reciprocal temperature (K-1), although sometimes you might see the units for α written as length/length/degree. Solids typically have values of β on the order of 10-5 K-1. If a substance is not constrained, then β ≈ 3α. Can you explain why this is the case? You will learn more about these and other thermodynamic functions in your Level Two module in Thermal Physics. The thermal expansivity has a profound effect on various engineering applications of solids. Mismatch of the thermal expansivity at the interface between two solids creates stress and can thus lead to mechanical failure. One of the major issues facing the designers of semiconductor packages is not electronic but thermal compatibility. In designing a satellite or a steel bridge, it is essential to consider the extremes of temperature to which the structure will be exposed. Ignoring thermal expansivity can lead to disaster! At the atomic level of solids, added heat increases the average amplitude of vibration between atoms in a crystal lattice and thus increases their average separation. In a single crystal, the linear expansivity can vary depending on the crystal axis along which it is measured. In a polycrystalline solid, however, in which the crystals are oriented in all different directions, the effects of crystal orientation average out, and so α is the same when measured in any direction. There is not a single, simple equation to relate P, V and T in a solid. Such an equation, called an equation of state, does exist, however, for an ideal gas. The equation of state for an ideal gas is nRT , (3)

P=

V

where n is the number of moles of the gas in the closed container, and R is the gas constant (R = 8.314 Jmol-1K-1). An ideal gas is one in which the molecules have no

3 volume and have no intermolecular attractions. In a real gas, of course, the molecules do have a finite volume and this volume must be subtracted from the volume of the container when applying the gas law. Also, in a real gas, the molecules have at least a weak attraction to each other and to the walls of the container, and this affects the pressure exerted by the gas. Nevertheless, the ideal gas law adequately describes the behaviour of air and other gases over relatively narrow ranges of pressure and temperature. Equation 3 can be used to calculate the volume expansivity β by taking its partial derivative of the right-hand side with respect to temperature while treating P as a constant. Then 1 nR nR V 1 (4) = = . β= V P V nRT T It is therefore apparent that near room temperature, β will be much greater for a gas in comparison to a solid. The fact that gasses expand so much when heated enables their use in heat engines. An expanding gas can drive a piston to do useful work. In this experiment you will operate a simple heat engine and calculate how much work is done through the application of heat. Let us define the work done by a gas on its surroundings as being a positive value. Then if a gas held at constant pressure expands from a volume of V1 to a volume of V2, the work done is given as simply: V2

W =

∫ PdV = P(V

2

− V1 )

(5)

V1

When a gas is cooled at constant pressure, and V2 < V1, W is a negative value.

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SUGGESTED WORKPLAN

This set of experiments studies the thermal expansion of both metals and gasses. In one of the weeks, you should acquaint yourself with the thermal expansion apparatus and measure the expansivity of the metals. In the other week, you should study the expansivity of air and use it in a simple heat engine. The order of the metal and gas experiments should be agreed between your group and the other group of students doing the experiment.

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OVERVIEW OF THE APPARATUS

4.1 Metal Expansion Apparatus

The apparatus consists of three metal tubes (copper, aluminium and steel) that can be mounted on a base. A dial gauge determines the position of the ends of the tube to the nearest 0.01 mm. The temperature of a particular tube is regulated by pumping either chilled or heated water through it. The temperature at the centre of the tube is recorded using a thermistor, which is embedded in a thermistor "lug". The lug is attached to

4 the tube by a thumbscrew, as illustrated in Figure 1. The thermistor's electrical resistance varies with temperature in a known way. The resistance is converted to temperature using a conversion table (on the base of the apparatus). At thermal equilibrium, the temperature is assumed to be uniform along the length of tube. Metal (Cu, Al or Steel) Tube

Thermistor lug Thumbscrew Figure 1: Schematic diagram showing how the thermistor lug is attached to the metal tube by a thumbscrew.

4.2 Gas Expansion Apparatus

The apparatus for the gas expansion experiments, shown in Figure 2, consists of a small metal chamber connected to a piston in a cylinder. The piston is made of graphite and therefore there is minimal friction between the piston and the walls of

cylinder Air chamber

graphite piston

Pressure port mating connectors

Figure 2: Gas expansion apparatus consisting of an air chamber and low-friction piston.

5 the cylinder. The cylinder has a scale marking that can is used to determine the position of the piston to the nearest mm. There are two port connectors on the base of the cylinder. One of the ports is connected to an air chamber, which is used to heat or cool the air. The other port is connected to a pressure sensor that is interfaced with a PC. The Pasco Science Workshop software can be used to determine the pressure of the gas.

5 EXPERIMENTAL PROCEDURES 5.1

Thermal Expansivity of Metals

You should follow this same procedure for each of the three tubes used. It is suggested that you study the aluminium tube first. Then, as time permits, you can study the other two in whichever order you prefer. Mount the tube on the thermal apparatus. You should make sure that the stainless steel pin on the tube fits into the slot of the mounting block and the bracket on the tube presses against the spring arm of the dial gauge. The proper positioning is illustrated in Figure 3. Then, attach the thermistor lug to the tube using a thumbscrew. The lug should be aligned along the axis of the tube, so that the two make maximum contact. Connect a digital multimeter to the thermistor by fitting banana plugs into the connections labeled "THERMISTOR" on the base of the apparatus. Reading the resistance from the multimeter will enable you to calculate the temperature of the bath. Use the conversion table on the base of the apparatus to interpolate to the nearest 0.1 °C. The temperature of the tube will be regulated by circulating water through it from a water bath. The water bath has one rubber tube from which the water is pumped out, and there is a second tube that allows water to flow back into the bath. Connect the tubes to each end of the metal bar so that water can circulate from the bath in a closed loop through the bar and back to the bath. Fill the water bath with ice, and then top it up with cold water. Set the thermostat reading on the water bath to 0 °C. Do this by rotating the black dial to adjust the setpoint on the vertical scale. Start pumping the ice-cold water through the metal tube by turning on the power to the water bath. Monitor the resistance reading on the multimeter until it stabilises. Then record this value, and use the thermistor conversion table to find the temperature. Using a metre-stick, measure the length of the metal tube L between the stainless steel pin on one end and the inner edge of the angle bracket at the other end. The position of both the pin and the bracket is shown in Figure 3. Rotate the outer casing of the dial gauge to align the zero point on the scale with the position of the large indicator needle. As the tube expands, the indicator needle will move in an anti-clockwise direction, thereby enabling you to determine the displacement.

6 Raise the thermostat setting on the bath to about 20 °C and wait for the ice to melt and the water to heat up. Circulate the 20 °C water through the tube and allow the temperature to equilibrate. This equilibration should take no longer than about 15 minutes.

L Figure 3: An illustration of how the tubes are properly positioned on the thermal apparatus. The bracket is snug against the spring arm of the dial gauge. The steel pins at the other end of the tube are fitted into the slotted brackets. The distance L is indicated. Record the new position of the dial gauge and the thermistor reading. Then increase the temperature of the water bath in increments of about 20 °C up to about 100 °C. At each temperature, allow the tube's temperature to equilibrate. Then record the gauge position and the thermistor reading once again. Important: Do not allow all of the water to evaporate from the bath. If the heating element is on when the bath is empty, it will be damaged. Prepare a plot of the length of the tube versus temperature. Do the data fall on a straight line? You can approximate α as 1 ∆L α= , (6) Lo ∆T where Lo is the original length of the tube, and ∆L is the change in length for a change in temperature of ∆T. Calculate α for each of the three metals over various temperature intervals (both large and small). How does α vary with temperature, if at all? How do your values of α compare between the three metals? Do you think there is a significant difference in the expansivity of the three metals? (If your group does not have sufficient time to analyse all three tubes, you should ask the other group simultaneously doing this experiment what value of α they obtained.) If you had a 10-metre strip of each of the three metals, and you heated each strip by 100 °C, by how much would each metal strip expand?

5.2 Thermal Expansivity of a Gas and Application to a Heat Engine One of the main goals is to determine the volume expansivity β of air. You will be able to obtain a value of β using the simple equation:

7 1 ∆V β= , (7) Vo ∆T where Vo is the initial volume of air in a closed system, ∆V is the volume change that results from a temperature change of ∆T. There are several steps in this experiment. First, you will make use of the ideal gas law to determine nR/V for your system. Then, with some additional measurements of the same kind but at different volumes, you can determine a good estimate for V. Finally, you can use the value of V and some additional measurements to find β using Equation 7. 5.2.1. Determining the Value of nR/V for your Apparatus The approach in this experiment will be to fix the volume of a closed system of gas, and then to measure the pressure at several temperatures. Using Equation 3, you will then be able to find nR/V from a plot of P versus T. To make the best use of your time, it would be a good idea to put hot water on the hot plate at the beginning of the experiment. Allow the water to heat up to about 60 °C while you set up the gas apparatus. At the base of the piston there is a rubber tube that forks into two ports (as shown in Figure 2). Open the valve to one of the ports; it is a vent for the piston. Connect a rubber tube to the other port. The opposite end of this tube should be inserted into the hole in the black rubber stopper fitted onto the air chamber, as shown in Figure 4. Make sure that the rubber stopper is fitted snugly into the air chamber, so that there are no gas leaks. On the other hand, do not press the stopper so firmly that you bend or distort the chamber. Do not re-open the chamber until the end of the experiment. Position the bottom of the piston to about 6 cm on the scale of the cylinder and make a note of the exact position. There is a screw at the top of the piston. You should tighten the screw so that the piston is held at a fixed volume. Next connect the pressure sensor to the second port on the base of the piston, making sure that the port is open. The other end of the sensor should be connected to the Pasco Interface box. A PC with the Pasco Science Workshop software should already be up and running. Now that the air chamber and piston are sealed and the volume is fixed, you are ready to make your first measurements. Click on the "monitor" panel (labelled as "MON") in the Science Workshop software to obtain a “live” reading of pressure. From a textbook or reference book, find out the standard value of atmospheric pressure in kPa (if you don't already know it!). Does your “live” value of pressure seem reasonable? You should now adjust the temperature of the air in the closed system by immersing the air chamber in water in the beaker on the hotplate. You should use the clamp to hold the chamber in place. You will need to assume that the air in the chamber is at thermal equilibrium with the water, such that they are at the same temperature. It would probably be most efficient to begin your measurements at

8 the highest temperature, which should be close to 60 °C. Record the pressure of the gas. (Note that you can heat the air chamber to higher temperatures (up to 100 °C) if you wish, but ensure that P does not exceed about 120 kPa or the stopper will shoot out!) Then adjust the temperature of the gas in the chamber by immersing it in water baths of various temperatures. You can use a beaker with room temperature water and hot water from the tap. You can immerse the air chamber in ice water to obtain the lowest T. You should then prepare a plot of pressure as a function temperature to find nR/V at this particular volume. For this part of the experiment, you can refer to the volume as Vo. You can check the quality of the data in your plot by extrapolating it to the point where P = 0. For an ideal gas, if the temperature is in units of Kelvin degrees, K, then at the point where P = 0, T should also be 0. What temperature value do you obtain? How does it compare to absolute zero: 0 K or –273.15 °C? 5.2.2. Determining the Value of V for your Apparatus The number of moles of gas in the system is unknown, so V is likewise unknown. However, if you change the volume of the system by a known amount, you can then find n for the closed system. After you have found n, then you can calculate V from measurements of P and T. Untighten the screw on the piston and move it up or down. Then re-tighten the screw and fix the piston at a new position. A decrease in height of about 2 cm should be suitable. Do not open the air chamber or you will change the amount of gas in the piston! Record the new height, h, of the piston. The diameter of the piston, D, is 32.5 mm. Hence, if you adjust the height of the piston by a carefully measured amount, dh, then the change in the volume of the system will be dV = 4π(D/2)2dh. The new volume of the system is equal to V=Vo+dV. You should repeat your measurements of P versus T and then find a value of nR/(Vo+dV). If you have decreased h, then P will be higher than atmospheric pressure at room temperature. Therefore take care in choosing temperatures for subsequent experiments so as not to build up excessive pressures. From your previous experiment, you can evaluate Vo as a function of nR. Using your second measurement, you can find the value of n. Use the ideal gas equation in a quick calculation to estimate an expected value for n as a way to check your result. 5.2.3. Measuring the Thermal Expansivity, β, of Air You are now ready to measure the expansivity of the air in the system. Keep the chamber and piston closed so that n stays constant. You should turn the piston on its side, as shown in Figure 4, so that the air will not have to lift the weight of the platform. Untighten the screw so that the piston is able to move freely. The pressure of the gas should then be the atmospheric pressure. Make sure you obtain a reasonable value.

9 You should cool the air by putting the chamber in ice (or chilled) water or in dry ice. Record P and T, and use your value of n to calculate Vo at this lower temperature. You should also make a note of the height of the piston. Then put the air chamber in hot (or boiling) water and record the temperature. Record the new height of the piston when equilibrated and use it to calculate the change in volume, ∆V. When choosing temperatures, you should ensure that the maximum and minimum V stay in the range of the piston. Use equation 7 to determine a value for β. Compare your result to literature values in the same temperature range. You should also compare your experimental value of β to the theoretical value for an ideal gas. If you have time, you should measure β over a few different temperature ranges. It has been found that the piston sometimes leaks out air during the course of the experiment. You should check to see if that has happened in your experiment. Compare V at the beginning and end of the experiment (at the same P and T) and comment on whether there has been a significant leak. Valve on the second port should connect to the pressure sensor.

Dry ice or hot water

Figure 4: The set-up for the gas expansivity experiment. The air chamber is placed into either dry ice or into water on the hot plate. 5.2.4. Construction of a Heat Pump and Doing Useful Work In this experiment, you will operate the piston in a cycle. You will change the volume by adjusting the temperature between the temperature of dry ice (-78 °C) and boiling water (~100 °C). You will change the pressure by putting a 200 g weight on top of the piston. On a plot of pressure (P) versus volume (V), the cycle will look roughly like a parallelogram. The four corner points should be: (1) (2) (3) (4)

V at the dry ice temperature; standard P. V at the dry ice temperature; P increased with a 200 g weight V at the temperature of boiling water; P increased with a 200 g weight V at the temperature of boiling water; standard P.

At each of the four points, you should record the pressure using the pressure sensor, and you should record the volume obtained from a measurement of the height of the piston (and knowing that its diameter is 32.5 mm). Design a table to

record your data in your lab diary. Use

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SI units.

Put the piston in a vertical position. To record data at point (1), you will need to cool the air in the piston. Ask a demonstrator or a technician for some dry ice. Put on protective gloves and safety glasses, then use a hammer to crush the dry ice. Cover the bottom of a large beaker with some of the crushed dry ice. Place a small beaker on this layer of dry ice, and pack additional dry ice around it. Wearing gloves, carefully remove the small beaker to leave a cylindrical hole. Press the air chamber into the hole in the dry ice. Within seconds, the height of the piston should drop. Wait until it stabilises. Determine as accurately as possible the position of the base of the piston on the linear scale on the side of the cylinder. Also determine the value of the pressure with the pressure sensor. If the piston position falls to zero, then you should restart the experiment with slightly more gas in the piston. (Then the initial height will be greater than 6 cm.) To collect data at point (2), you should place a 200 g weight on top of the piston. Then record the resulting values of P and V. To collect data at points (3) and (4), you will need to submerge the air chamber in boiling water. To make the most of your time, it would be a good idea to start heating the water while you are getting data at points (1) and (2). Fill a mediumsized beaker with water. Place the beaker on the top of a hot plate, and turn on the power to the hot plate. You should put the temperature probe for the hot plate into the beaker of water. Press the black button on the front of the hot plate, and adjust the set point to about 150 °C. The digital display on the hot plate will show the temperature of the water as measured with the probe as it heats up. When you are ready to start collecting data, submerge the air chamber in the hot water so that it is covered up to the rubber stopper, and fix it into position using the clamp on the ring stand. In the interests of safety, you should put the air chamber in the water well before it has reached the boiling point. When the water has reached its maximum temperature, record the values of P and V to determine point (3). Also, make a note of the temperature. Finally, for point (4), remove the 200 g weight from the piston, and record the values of P and V. Allow the chamber to cool down in air and then place it in the dry ice again. The piston should return to position 1. Does it? If not, can you explain why? If the values of P and V are significantly different, you should re-do the experiment, provided there is sufficient time. To analyse the output of your simple heat engine, you should use two different methods to calculate the work done. 1. One method is to calculate the work needed to raise the 200g weight from the height at position (2) to the height at position (3). If the change in the height is h, and the mass of the piston is m, what is the work done in raising the weight against gravity? 2. The second method is to calculate the work done by the gas using a P-V diagram, as shown schematically in Figure 5. According to Equation 5, the useful work done in one cycle of the heat engine will be the area inside the

11 parallelogram. You should devise a way to calculate this area. In actuality, the lines between points (1) and (2) and between (3) and (4) will not be straight, but you can make this approximation. Also, because the change in volume is needed, and not the absolute value of volume, it is not necessary to consider the volume of the air chamber, because it is roughly constant. You only need to plot the volume of the gas in the piston. Compare the values of work obtained by the two methods. Are they the same? What is the origin of the work that is done? That is, where does the energy come from?

(2)

(3)

Pressure (Pa)

(1 )

(4) Volume (m3 )

Figure 5: Schematic P-V diagram showing the four points where (P,V) data are collected. The area inside the parallelogram is the useful work done in one cycle.

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FURTHER READING

Thermal Physics by C.B.P. Finn, Chapman & Hall, London, 1998 (Available in the University Library)