In these laboratory exercises we wish to study the relationships among heat and heat flow, temperature, specific heat, energy, power, and efficiency. We will also study the ideal gas law and measure the universal gas constant, $R$.

In order to illustrate these concepts, we will divide the work into three parts. They can be performed in any order, so start on any part as the apparatus becomes available.

In this lab, we will do the following:

• measure the mechanical equivalent of heat;
• test the ideal gas law; and
• study heat pumps and heat engines.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

In this experiment you will…

• heat metal samples using the friction between the sample and a taut string sliding over the sample; and
• compare the amount of heat added to the sample (as measured by using the sample's specific heat and measured temperature rise) to the amount of work put in.

If we put energy $Q$ into an object having mass $m$, we will raise its temperature by an amount $\Delta T$ as

where $c$ is the specific heat of the object.

In this experiment we will displace a string by a distance $x$ over the sample by applying a force $F$ to the string. This process will require expenditure of energy $Q$ which we assume goes into the sample as

This equation is equivalent to

In our experiment the sliding string always adds energy to the sample, independent of the direction in which the string slides over it. Also, the force and velocity of the string will always act along the same line. Therefore, in our experiment, Eq. (3) may be re-written as

The apparatus we will use is shown in Fig. 1.

Figure 1: Mechanical equivalent of heat apparatus

A thick string is used to lift a mass, $M = 200$ g. The string is placed on the pulley of a rotary encoder, the rotation of which enables us to measure the displacement of the string. The string will also slide around the metal sample in its spiral groove. We use a force sensor to pull the string and lift the mass. The friction between the string and the sample will heat the sample. Embedded in the sample is a thermistor, a temperature-sensitive resistor that will enable us to measure the temperature of the sample.

The rotary encoder should be plugged into Dig 1 of the LabQuest Mini (LQM) interface. The force sensor should be switched to the 50 N range, and plugged into the Ch 1 input of the LQM. The thermistor should be connected to Ch 2, using the special cable assembly, to provide temperature measurement. The LQM digitizes the data and sends it to a computer running the Logger Pro software.

How does the thermistor work?

The temperature will be determined by measuring the resistance of the thermistor embedded in each sample. The relation between resistance and temperature is given on the base holding the samples and is plotted in Fig. 2

Figure 2: Plot of temperature, $T$, as a function of resistance, $R$, for thermistor

The solid line in Fig. 2 is a least-squares fit of a second-order polynomial to the $T$ vs. $R$ data. The resulting formula is given by

$T = 61.03 ~{}^{\circ}\textrm{C} - (4.771 \times 10^{-3} ~{}^{\circ}\textrm{C}/\Omega)R + (1.176 \times 10^{-7} ~{}^{\circ}\textrm{C}/\Omega^2)R^2$.

(5)

This relation has already been entered into the Friction software to give the temperature readout.

The Logger Pro configuration file named Friction, displays the following four graphs:

• force vs. time (from the force sensor);
• height of mass above floor vs. time (from the rotary encoder);
• temperature vs. time (from a measured thermistor voltage, using Eq. (5)); and
• (force x velocity) vs. time.

Go to a lab station with the apparatus shown in Fig. 1. At each such station there will be ONE sample – either a block of aluminum (silvery in color) or a block of brass (golden in color) – mounted on a black base. Do not remove the samples from their bases, since the wires to the thermistor are easily broken. We have measured the mass of each sample for you as follows:

• Aluminum sample mass, $m_{Al} = 0.0218 \pm 0.0001$ kg
• Brass sample mass, $m_{Br} = 0.0660 \pm 0.0001$ kg

The aluminum sample mass	The brass sample mass

The force sensor is a strain gauge which provides a voltage proportional to force applied to the hook. It is necessary to provide information to the software, correlating the voltage to a known force. After calibration, the force may be read from the computer.

Calibrate as follows:

• Connect the rotary encoder to Dig 1 of the LQM. Connect the force sensor to Ch 1 input of the LQM.
• Determine the mass of the object to be hung over the rotary encoder pulley. Calculate this object’s weight.
• Attach one end of the string to the hanging mass (leave the mass resting on the floor for now). Pass the string over the encoder pulley and connect the free end of the string to the hook on the force sensor.
• Hold the force sensor horizontal, label side up, with no tension on the hook. We will define this condition to be zero force. To do so, pull down the Experiment menu and select Calibrate LabQuest Mini: 1 Ch1: Dual Range Force. Click on Calibrate now and type 0 into the Reading 1 box. Click on Keep.
• Next, use the force sensor to raise the mass off the floor (with the string passing over the pulley), still holding the sensor horizontal. Enter the weight of the hanging mass in the force box. Click on the Reading 2 button.

The force sensor should now be calibrated.

Arrange the apparatus as shown in Fig. 1, wrapping the string in the sample’s spiral groove. Be sure that the string will not touch the thermistor wires.

Start with the hanging mass on the floor. Click on the Collect button or press Return to start taking data. Smoothly, but quickly move the force sensor about 1/2 meter to raise and lower the mass repeatedly for about 25 seconds. You should see data appear on the graphs as you move the force sensor. After you stop moving the force sensor, lower the mass to the floor. Cooling of the sample will begin soon after you stop adding energy.

Questions/Tasks:

• Eq. (4) suggests that integrating the (force x velocity) vs. time plot will give the total energy (work) you have put into the system; we will call this quantity $W$. Determine this value by selecting the ($F \times v$) vs. $t$ plot and clicking on the Integrate icon at the top of the screen; the value of the integral appears in a box on the plot. (No need to estimate uncertainties here.)
• From Eq. (1), determine how much heat, $Q$, was added to the sample by using the measured rise in temperature and the sample's known specific heat. (No need to estimate uncertainties here.) The specific heats of the two metals are as follows:
  • $c$ (aluminum) = 900 J/kg $^{\circ}$C
  • $c$ (brass) = 380 J/kg $^{\circ}$C.
• Did you find that all that work done, $W$, was turned into heat inside the sample, $Q$? If not, where might there have been energy losses?

The goals of this exercise are as follows:

• to study the relationship among volume, pressure and temperature of a sample of gas, and
• to determine a value of the universal gas constant, $R$.

An ideal gas is one for which the gas molecules may be considered to be point-like particles which move randomly and do not interact with each other. We shall see if air, at atmospheric pressure and room temperature, acts as an ideal gas.

Assume that the number of gas molecules is held constant in a closed volume. Then the relation among pressure, volume and temperature of the ideal gas is given by the ideal gas law,

where the symbols (and their SI units) are as follows:

• $P$ is pressure (N/m$^2$ = Pa),
• $V$ is volume (m$^3$),
• $n$ is the number of moles of gas (mol),
• $R$ is the universal gas constant (J/mol K), and
• $T$ is the absolute temperature (K).

We will test the ideal gas law, Eq. (6), and determine the value of $R$.

You will need to know how many moles of gas will be contained in the apparatus. To do so, use the relation

where $\rho$ is the gas density and $M$ is the molecular mass.

At standard temperature and pressure (20 $^\circ$C and 1 atm), $\rho_{air} = 1.20$ kg/m$^3$. To obtain the molar mass of air, $M_{air}$, we note that air is approximately 80% nitrogen and 20% oxygen. The molecular mass of $\textrm{N}_2$ is $M_{N2}$ = 0.028 kg/mol and the molecular mass of $\textrm{O}_2$ is $M_{O2}$ = 0.032 kg/mol. Therefore,

The apparatus consists of a cylinder into which we may introduce gas. The volume of the cylinder may be varied by moving a piston up or down inside the cylinder.

The parts of the apparatus shown in Fig. 3 are as follows:

• a. piston to compress the gas
• b. cylinder to contain the gas
• c. valves to admit the gas into the cylinder
• d. linear potential divider to measure the position of the piston
• e. sliding electrical contact on the linear potential divider indicates piston position
• f. sealed base containing piezo-electric pressure sensor
• g. fine nickel wire temperature sensor
• h. transparent millimeter scale
• i. motion limiting pin (not used here)

Figure 3: Ideal gas law apparatus

There are three sensors in the apparatus to measure the pressure, volume and temperature, as described below:

• Pressure is measured by a piezo-electric crystal, mounted under the base of the gas cylinder. When stressed by gas pressure, the crystal produces a voltage $V_P$ (which is labeled “potential 3” in the software).
• Volume is measured by a linear potential divider resistor attached to the piston. A constant voltage is applied to the resistor. As the piston is raised or lowered, the resistance changes. Thus, the voltage $V_V$ (“potential 2”) changes with piston height.
• Temperature is measured by a very fine wire near the bottom of the cylinder. The resistance of this wire depends on its temperature. The wire is part of a circuit that generates a voltage $V_T$ (“potential 1”).

The formulae relating $P$, $V$ and $T$ to their respective voltages are posted on the side of each apparatus. You will have to check these formulae in software and modify them to match the posted formulae.

Check that the following connections are properly made:

where Ch 1, Ch 2 and Ch 3 are inputs on the LabQuest Mini.

Open the Ideal Gas Law file and check that the equations relating the sensor voltages are consistent with those posted on the side of your apparatus. To do so, double-click on the appropriate column heading (for example, Temperature). Check that the equation entered is the same as that posted on the apparatus. Repeat this process for Pressure and Volume. The powers of 10 have been chosen to provide sufficient precision when plotted.

Clamp the apparatus to the lab bench with the C-clamp. Open one of the gas valves and SLOWLY raise the piston to the top of its travel. Doing so draws in air at today’s room temperature and atmospheric pressure. Close the valve and hold the piston at its highest position.

When ready to take data, start the data collection and steadily move the piston all the way to the bottom of its travel over a time of 1 to 2 seconds. Continue taking data for a few seconds while holding the piston at its lowest position. Your software should display plots of temperature vs. time, volume vs. time, and pressure vs. time. In addition, there should be a plot of $PV$ vs. $T$.

Questions:

What would you expect to be the shape of the $PV$ vs. $T$ plot, using Eq. (6) as your model?

Take a screenshot of the $PV$ vs. $T$ plot. This plot likely doesn't quite look like you expect… can you explain a possible mechanism for any unanticipated behavior?

In order to avoid this effect, try taking data again but stopping the data collection just before the piston reaches the bottom.

Questions/Tasks:

Use Eq. (7) to calculate the number of moles of air trapped in the cylinder. The inside diameter of the cylinder is 4.4 cm. We will approximate today’s temperature and atmospheric pressure to be standard (i.e. $20^{\circ}$, 1 atm). Therefore, you may use air density given after Eq. (7).

According to Eq. (6), what is the physical significance of the slope of the $PV$ vs. $T$ plot?

From your data plot of $PV$ vs. $T$, obtain a value of $R$, the universal gas constant.

Compare your value of $R$ with the literature, $R$ = 8.314 J/mol K. (You don't need to compute an uncertainty on your experimental value or do a $t^{\prime}$ test. Instead, just comment on whether your value is the right order of magnitude and consider what might be the cause of any discrepancies.)

Does air approximate an ideal gas? Justify your answer.

Here we shall use the terms heat, internal energy, and work. It should be noted that they all are measured in units of energy (Joules, in the SI system of units). The different terms are used to help clarify the role each plays in the energy flow process.

There will be two stages of energy transfer here.

• Heat Pump: Work $W$ will be done on a heat pump to change temperatures in heat reservoirs. (See Fig. 4a.) In this case, $W$ represents energy we would have to pay for.
• Heat Engine: A temperature difference will cause heat flow and do work on a resistor. (See Fig. 4b.) In this case, $W$ represents the energy generated by already existing temperature differences, like geothermal heat. No payment required!

Figure 4: Energy flow in a heat pump and a heat engine

If a resistor $R$ carries a current $I$, then the power dissipated in the resistor is $P=I^2 R$. Since $P=W/t$, it follows that $W=I^2 Rt$. This expression is true, provided $I$ does not vary with $t$. If $I$ does vary with $t$ (as will be the case for the heat engine part) then

For the heat pump, we will use a Peltier (pronounced, pell-tee-yay) device. The Peltier is a semiconductor material sandwiched between two thin ceramic plates. When an electric current passes through the Peltier, one side becomes cold while the other side becomes hot. The cold and hot faces of the Peltier are cemented to small blocks of aluminum, each with mass 0.019 kg, which act as heat reservoirs.

The Peltier is mounted on a circuit board for ease of making connections. The apparatus shown in Fig. 5 will be used for both the heat pump and heat engine parts.

Figure 5: Electrical connections for heat pump/heat engine

With the knife switch in the heat pump position, the circuit board and associated components reduce to the circuit shown in Fig. 6. The digital meter should be set to measure DC volts. The output from the Peltier Current jacks will measure the voltage across the 1 ohm resistor. Using Ohm’s law, the current is just $I=V/R =V/(1~\Omega)$. So, for example, if the voltage you measure is 2.3 V, then the current is 2.3 A.

Figure 6: Heat pump circuit

In the heat engine mode, the internal energy stored in the hot aluminum block will flow to the cold aluminum block through the Peltier. In turn, the Peltier will send current through the 10 ohm load resistor and perform work on the resistor. The circuit for the heat engine mode is shown in Fig. 7.

Figure 7: Heat engine circuit

Connect the circuit as shown in Fig. 5. Note the connecting wire which you should place as indicated at the bottom of the board. This connection will set the load resistor to 10 ohm for the Heat Engine part. Note also that, in the Heat Engine part, the current will reverse direction relative to the Heat Pump function.You will need to gather data for the heat pump and heat engine parts of the experiment continuously, so plan ahead!

Before you turn on the power supply, briefly touch the aluminum blocks to feel and comment on their temperatures. Open the Logger Pro configuration file named, Heat. Place the knife switch in the vertical (neutral) position.

The LQM may have a slight voltage offset, so it is necessary to set its reading to zero with no voltage applied to its input for the channel which measures the Peltier current. To do so select Zero from the Experiment menu. Make sure that only the box for LabQuest Mini: 1 CH3: Voltage (+/- 10V) is checked, and click OK.

Turn on the power supply and set it to about 5 V (as read on the power supply voltmeter). In this part you will use the DVM (digital voltmeter) to measure the voltage across the Peltier. Note that the current is measured indirectly by measuring the voltage across a 1-ohm resistor which is in series with the Peltier. The temperatures are measured indirectly, by determining the resistance of the thermistors embedded in each aluminum block. The conversion to temperature is done in software, using Eq. (5).

Click the Collect button, wait a few seconds then place the knife switch in the Heat Pump position. Record the initial Peltier voltage read on the DVM. Plots of the temperatures of the hot and cold aluminum blocks should appear on the computer. Leave the system in the Heat Pump mode for about 1 minute, and then record the final values of the Peltier voltage shown on the DVM. Briefly touch the aluminum blocks again and note how their temperatures have changed. Quickly flip the knife switch to the Heat Engine mode, while continuing to collect data.

In this mode, the computer will gather Peltier current and temperatures data as before. Note the changing Peltier current and the power delivered to the 10 ohm load resistor. Let the system run for the remaining time, about another minute. The program will stop automatically. Turn off the power supply and return the knife switch to the vertical position.

If you need to repeat the experiment, wait 2 or 3 minutes to allow the aluminum blocks to return to room temperature.

Save the data on the computer using a new filename. Take a screenshot of the plots and mark the regions of each plot which pertain to the Heat Pump and Heat Engine parts of the experiment.

Questions/Tasks:

• Determine how much work (energy) you put into the heat pump. This quantity is shown as $W$ in Fig. 4a.) To do so…
  • Calculate the average voltage across the Peltier during the Heat Pump portion of the cycle from the starting and ending voltage values.
  • Calculate the average power into the Peltier from $P_{avg} = IV_{avg}$.
  • How long was the heat pump turned on? Using this time, calculate how much work (energy) you put into the heat pump from $W = P_{avg}t$.
• From the data plots on the computer, determine the temperature changes in each aluminum block while in the heat pump mode.
  • To measure from the plots, pull down the Analyze menu and select Examine. This feature provides a cursor with which you may obtain measured values from any point on the plots.
• Use Eq. (1) to calculate the energy transferred into or out of each aluminum block.
  • You may assume that the specific heat of aluminum at room temperature is 900 J/kg $^{\circ}$C, and that the mass of each aluminum block is 0.019 kg.
• Make a sketch of Fig. 4a. Using your data from the Heat Pump part, label the arrows with the values of energy flow you measured: $W$, $Q_H$, and $Q_C$.
• Demonstrate if any energy has been lost in this process. If so, where do you think it went?

In physics, efficiency is generally defined as output energy divided by input energy and has values ranging from 0 to 1. Performance, on the other hand, includes an assumption of a desired outcome. For example, the heat pump can be used either as a refrigerator or as a heater. Let us consider how well your Peltier performs each function. We may define the coefficient of performance (COP), analogous to efficiency, as

If we consider heating as the objective, then the relevant heat transfer is into the hot reservoir:

Since Fig. 4a shows that $Q_H$ is generally greater than $W$, it follows that $COP > 1$.

If, on the other hand, we consider cooling to be the objective, then

Questions/Tasks:

• Using your data, calculate the coefficient of performance for the heat pump, acting as a heater.
  • For every unit of energy $W$ that we would have to pay for, how many units of energy would go into heating?
• Calculate the coefficient of performance for the heat pump, acting as a refrigerator.
  • For every unit of energy $W$ that we would have to pay for, how many units of energy would go into cooling?

Note that the larger the $COP$, the more efficient the heat pump. Practical heat pumps can have a $COP$ of about 3 or 4.

Questions/Tasks:

• From the data plots, determine the temperature changes for the hot and cold reservoirs (Al blocks) during the Heat Engine part of the cycle.
• Using Eq. (1), determine the heat lost from the hot reservoir and the heat gained by the cold reservoir.
• From the power vs. time plot, use Eq. (8) to find the work done $W_R$ on the 10 ohm load resistor.
  • The integral of the $I^2 R$ vs. $t$ plot may be obtained by selecting that plot and then clicking on the integrate button . The value of the integral is shown in the small box on the plot.
• Sketch a diagram like Fig. 4b showing the hot and cold reservoirs and the load resistor. Label the arrows with the values of heat flow you measured.

It is traditional for heat engines to evaluate efficiency, $\eta$, rather than $COP$, although the concept is similar:

Note from Fig. 4b, that this ratio ranges from 0 to 1.

Question:

• Calculate the efficiency of the heat engine.

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Write your conclusion in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

• What do your data tell you?
• How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
• Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
• Do your results lead to new questions?
• Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.