Estimation of Avogadro’s Number

Name: _________________________ Partner(s): _________________________

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THERMODYNAMICS & HEAT ENGINES

In the previous investigation, you examined the equation of state of an ideal gas and

explored how pressure, volume, and temperature are related for a fixed amount of gas. In

particular, you examined the isothermal process (T = 0), isometric process (V = 0), and

the isobaric process (P = 0). The purpose of this session is to explore the adiabatic

process and then employ this process as you move an (assumed) ideal gas through a

complete cycle of expansions and compressions. By doing so, you will be able to make

the gas do useful mechanical work. Such a mechanism is called a heat engine.

A heat engine is a device operating between two temperatures in which a working

substance (a substance that changes its volume as heat is added or removed) is moved

through a cycle in order to extract some energy to do useful work from the heat that is

naturally flowing from the hot side to the cold side of the device. In this session you will

construct and operate two simple heat engines.

You will then measure the work output in comparison to the heat input in order to

determine the efficiency of the engines. The efficiency (denoted e), is a number between

zero and one and describes how well the engine converts the heat into work. A perfect

engine is would have an efficiency of 100% or e = 1. While this would not violate the

law of conservation of energy, it will be found that it is impossible for a heat engine to

convert heat completely into work while operating in a cycle. In other words, a real

engine will always exhaust waste heat back into the environment and have e < 1.

Part I: The First Law of Thermodynamics

In the previous investigation, you explored the relationships among pressure, volume and

temperature for a fixed amount of an ideal gas. Such systems are used extensively in the

study of thermodynamics (the study of thermal energy and its conversion into other forms

of energy). In particular, an enclosed gas in a cylinder with a movable piston is the usual

starting point for examining heat transfer and its relationship to work.

Let us take a moment and recall some of the observations you made in the previous

investigation. Recall the experimental set-up used for the isothermal process (Fig. 1):

Fig. 1: Experimental set-up to investigate an isothermal (T = 0) process.

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Question: For a fixed amount of ideal gas, what is the relationship between the pressure

and volume for an isothermal process?

Question: When you compressed the piston of the syringe, you and the (external

atmosphere) performed work on the gas. While the gas is being compressed, was the

work you and the atmosphere did on the gas positive or negative? How do you know?

Question: During the compression, did the gas do work on you? If so, was the work

positive or negative? Explain.

Question: When the gas expanded, the external atmosphere pushing inward on the piston

again does work. Was this work by the environment on the gas positive or negative?

How do you know?

Question: During the expansion, did the gas do work on its environment? If so, was the

work positive or negative? Explain.

Question: Using the relationship that W = Fdx, show the work done by a gas in cylinder

of cross sectional area A with a movable piston is given by W = PdV.

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Recall the experimental set-up for the isometric process (Fig. 2):

Fig. 2: Experimental set-up to investigate an isometric (V = 0) process.

Question: For a fixed amount of ideal gas, what is the relationship between the pressure

and temperature for an isometric process?

Question: When you immersed the sealed flask in the ice water, did heat enter or leave

the system as the system came to equilibrium? What was the evidence?

Question: After being immersed in the ice water, the system came to equilibrium. Did

the gas do work on its environment? If so, was the work positive or negative? Explain.

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Question: When you immersed the sealed flask in the hot water, did heat enter or leave

the system? What was the evidence?

Question: After being immersed in the hot water, the system came to equilibrium. Did

the gas do work on its environment? If so, is the work positive or negative? Explain.

Recall the experimental set-up you used for the isobaric process (Fig. 3):

Fig. 3: Experimental set-up to investigate an isobaric (P = 0) process.

Question: For a fixed amount of ideal gas, what is the relationship between the volume

and temperature for an isobaric process?

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Question: When you immersed the sealed flask in the ice water, did heat enter or leave

the system as the system came to equilibrium? What was the evidence?

Question: As the system came to equilibrium after being immersed in the ice water, did

the gas do work on the environment? If so, was this work positive or negative? Explain.

Question: When you immersed the sealed flask in the hot water, did heat enter or leave

the system? What was the evidence?

Question: As the system came to equilibrium after being immersed in the hot water, did

the gas do work on the environment? If so, was this work positive or negative? Explain.

Now that you have had the opportunity to review the findings of your previous

investigation, it should not come as a surprise to you that the gas can exchange heat with

its surroundings as well as do work on its surroundings (depending on the process). The

energy that enters the gas (either by heat entering or work being done on the gas) adds to

the total store of energy that is already in the gas. This total store of energy already in the

system is called the internal energy (denoted Eint). Of course, if heat leaves the gas or the

gas does work on its surroundings, then the internal energy of the gas must decrease in

accordance with energy conservation.

This leads to the first law of thermodynamics:

Eint = Q – W.

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Important Note: In the equation above, Q is the heat added to the system and W is the

work done by the system on the surrounding environment.

The first law of thermodynamics is just conservation of energy with the inclusion of

thermal phenomena. Notice that the first law relates the heat added and the work done by

the system to the change in internal energy. In other words, the first law states that what

energy enters because of temperature differences minus the energy that leaves because of

the work the system does is equal to the change of the total energy that was in the system

at the start.

Since an ideal gas law ignores any interactions (other than the “billiard ball” collisions)

among the gas molecules, there is no potential energy between any pair of molecules.

(There may still be potential energy with the atoms of a single molecule, however.)

Thus, the internal energy is the sum of the individual the kinetic energies and potential

energies of all the molecules in the system. As a result, it can be shown that the internal

energy of an ideal gas depends only on the absolute temperature of the system. That is,

Eint = Eint(T) only.

Question: Based on the first law of thermodynamics, is it possible for a gas to undergo a

process in which the internal energy remains constant, even while work is done either on

or by the gas? If so, describe such a process?


process in which the internal energy can change, but no work is done? If so, describe

such a process?


process in which the internal energy can change, but no heat enters or leaves the system?

If so, describe such a process?

Checkpoint: Consult with your instructor before proceeding. Instructor’s OK:

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Part II: Adiabatic Processes

As you probably have deduced, a system can under a process in which no heat enters or

leaves the system. Such a process is called an adiabatic process. That is, for an adiabatic

process, Q = 0. In accordance with the first law of thermodynamics, the change in

internal energy of a system is the negative of the work done by the system. That is,

Eint = –W

This makes sense. If a system does work on its surroundings (W > 0), then energy is

transferred to the surroundings and comes at the expense of the system’s internal energy.

Of course, if work is done on the system, then the system (W < 0), the system increases

its internal energy.

In this activity, you will examine the behavior of an ideal gas undergoing an adiabatic

process. In particular, you measure the final temperature of a gas before and after it

undergoes an adiabatic compression.

Your group will need the following materials/equipment for this part:

1 fire syringe

1 metric ruler and/or caliper

Small bits of tissue paper, cotton, or match head

Safety glasses & safety gloves

Recall that when you compressed a fixed amount of gas isothermally (Fig. 1), the gas was

compressed slowly. Since you did positive work to compress the gas, the gas must have

done negative work as the piston moved inward. (The gas molecules exert a force to

push the piston outward. Thus, if the piston moves inward, the gas must do negative

work.)

Question: According to the first law of thermodynamics, a compressed gas does negative

work on its surroundings. However, if the process is isothermal, the internal energy does

not change. What must you therefore conclude about the heat added to the gas during an

isothermal compression?

Question: If instead the gas is compressed adiabatically, what do predict will happen to

the temperature of the gas if you now compress the gas? Explain you reasoning.

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Procedure

1. Examine the fire syringe. The fire syringe is a cylinder with a movable piston that is

housed inside of a Plexiglass tube. A small piece of combustible material can be

placed at the bottom of the inner cylinder and piston inserted.

2. Place a small piece of tissue paper or match head in the cylinder and insert the piston.

3. Place the cylinder in a vertical position and slowly push the piston down as far as it

can go.

4. Allow the piston to come back out and reset it to the starting point again.

5. Measure as best as you can the length of the air column when the piston is at the

starting position and at maximum compression. List these values in Table 1 below.

6. Use the calipers to measure the inside diameter of the tube and then calculate the

initial and final volumes of the gas. Complete Table 1.

Table 1:

Initial tube Length (cm)

Final Tube Length (cm)

Inside Diameter of Tube (cm)

Initial Volume of Tube (cm3)

Final Volume of Tube (cm3)

7. As rapidly as possible, compress the piston and observe the piece of tissue paper.

When you compressed the gas slowly, the process was essentially isothermal. When you

compressed the gas rapidly, the process was essentially adiabatic.

Question: Why do you suppose the isothermal compression and the adiabatic

compression produced such drastically different results?

While it is likely not obvious at this point in your physics career, there is a relationship

between the absolute temperature and volume for an adiabatic process such as the one

you just witnessed. In the case of air, this relationship is

TiVi0.4 = TfVf

0.4,

(where the exponent is on the volume only). In other words TV0.4 = constant.

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8. Assuming that room temperature is the initial temperature, calculate the final

temperature at the maximum compression of the adiabatic process. Show your work.

Tf = K = °C

Note: Ignition temperatures of tissue paper are typically in the range of 215°C-250 °C.

(Values depend on a variety of factors such as thickness, density, composition, humidity,

oxygen concentration, etc.)

Question: Does the temperature you just calculated seem consistent with the behavior of

the tissue paper after the adiabatic compression? Why or why not?

Question: Using the ideal gas law: PV = nRT, and the relationship between the absolute

temperature and volume for an adiabatic process: TV0.4 = constant, show that the

relationship between the pressure and volume for air undergoing the adiabatic process

shown is given by PV1.4 = constant. (That is, PiVi1.4 = PfVf

1.4.)

Question: Assuming that atmospheric pressure is the initial pressure, calculate the final

pressure at the maximum compression of the adiabatic process. Show your work.

Pf = Pa = atm

Question: Recall that for an adiabatic process, Q = 0. Now explain why the temperature

of the gas in the cylinder must increase in terms of the first law of thermodynamics.


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Part III: Cyclic Processes & Heat Engines

In this part, you will construct two mechanisms that will do mechanical work while

operating in a cycle. The first will involve the thermodynamics of a rubber band.


1 hanging mass (1-kg)

1 ring-stand and clamping hardware

1 electric hair dryer or heat gun

1 thick rubber band

1 centimeter ruler

1 stopwatch

Prediction: Suppose you have a mass hanging at rest from a rubber band. If you were to

uniformly heat the rubber band, what do you think will happen to the hanging mass?

Why?

Procedure

1. Using the apparatus provided, hang the 1-kg mass vertically from the rubber band

next to the ruler. Note the initial location of the hanging mass.

2. Using the stopwatch, record the heating time while using the hair dryer (or heat gun

on a low heat setting) to warm the rubber as uniformly as possible. (Be sure to keep

the heat source moving and do not get closer than 10 cm so that the rubber band does

not melt!) Observe the motion of the hanging mass. Note the final height. Record

the warming time and the change in position y below.

Heating time t = s y = m

Question: What happened to the hanging mass as the rubber band was warmed? How

does your observation compare with your prediction?


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Suppose you needed to lift a series of masses one-by-one from a conveyor belt to another

conveyor belt at a higher elevation using a larger scale version of the rubber band

apparatus that you just examined. To do this, you would need to operate your apparatus

in a cycle in order reset the rubber band so that it would be ready to lift the next mass.

Question: As soon as you attach the mass, what do you expect to happen to the rubber

band? Why?

Question: Based on your observation earlier, what do you need to do to the rubber band

in order to lift the first mass from the lower level to the higher level?

Question: After you remove the first mass, what do you expect to happen to the rubber

band? Why?

Question: What would you now need to do to the rubber band so that it can be made

ready to pick up the next mass approaching on the lower conveyor belt?

Question: Outline a complete cycle that the rubber band would have to undergo in order

to repeatedly lift masses from the lower belt to the upper belt. Be sure to describe each

step in the cycle. (Hint: Revisit the four questions above.)

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Question: Consider the steps you outlined in the previous question. For which one does

heat enter the rubber band?

Question: Did all the heat from the hair dryer go into doing the mechanical work

necessary to lift the mass? If not, where did some of the heat go?

Question: Did all the heat that actually went into the rubber band go into doing the

mechanical work necessary to lift the mass? If so, then how do account for resetting the

rubber band to pick up the next mass after the previous mass was removed from the

rubber band? If not, where did the wasted heat go?

How efficient is your rubber band engine? In short, the efficiency is “what you get”

compared to “what you pay.” For your rubber band engine, what you get is the work

done to lift the mass. What you pay is the electric bill to run the hair dryer. As stated

earlier, the efficiency of any heat engine describes how well the engine coverts the heat

input into useful work. That is,

e =Woutput

Qinput

3. Calculate the heat input from the power rating of your hair dryer and the warming

time in Table 2 on the next page.

4. Calculate the work output in lifting the 1-kg mass to the height y.

5. Calculate the efficiency of your rubber band engine.

Table 2:

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Warming Time, t (s)

Power Rating of Hair Dryer, P (W)

Qinput = Pt (J)

Woutput = mgy (J)

Efficiency = |Woutput| / |Qinput|

Question: Is the efficiency what you expected? Why or why not?


Part III-B: Ideal Gas Engine

In this activity, you will construct a different engine. In this case, you will manipulate a

gas (assumed ideal) through a cycle in order to lift a mass.


1 computer with LoggerPro™ software installed

1 universal laboratory interface (ULI) box

1 pressure sensor

1 glass syringe (10 mL) with Luer lock connector

1 digital thermometer or temperature sensor

1 ring-stand and clamping hardware

1 boiling flask (50 mL)

1 one-holed rubber stopper to fit flask

2 beakers (500 mL)

Flexible tubing with Luer lock connectors

Access to hot water/ice water

1 50-g mass

Procedure

1. See handout…

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Estimation of Avogadro’s Number