In the Classroom edited by
JCE DigiDemos: Tested Demonstrations
Ed Vitz
Kutztown University Kutztown, PA 19530
Similarity and Difference in the Behavior of Gases: An Interactive Demonstration submitted by:
checked by:
Guy Ashkenazi Department of Science Teaching, The Hebrew University, Jerusalem 91904, Israel;
[email protected] James S. Gordon Division of Science & Mathematics, Central Methodist University, Fayette, MO 65248
Jason D. Hofstein Department of Chemistry and Biochemistry, Siena College, Loudonville, NY 12211
When presenting the topic of gases in a general chemistry course, the two main sub-topics are the ideal gas law and the kinetic molecular theory. The ideal gas law describes the macroscopic relations between observable properties of gases: pressure, volume, temperature, and amount; the kinetic molecular theory offers a microscopic molecular mechanism that explains this macroscopic behavior. Research has documented that while most students can perform well on algorithmic questions requiring the use of the ideal gas law, many of them fail miserably when presented with conceptual questions based on the kinetic molecular theory (1–4). It is therefore important to teach both sub-topics in an integrated manner and help students form connections between the observed macroscopic relations and the underlying microscopic mechanism. This article describes a lecture demonstration that can serve as a unifying context for both aspects of gas behavior. Through the years, many lecture demonstrations that concern gas behavior have been described in the literature. There are demonstrations that deal with the ideal gas laws: Avogadro’s law (5–9), Boyle’s law (5–7, 9), and Charles’s law (5–7, 10, 11). These demonstrations measure observable gas properties (pressure, volume, temperature, and mass) in a closed system at equilibrium. Other demonstrations pertain to the kinetic molecular theory, through discussion of Graham’s laws of effusion (5–7, 12). These demonstrations measure the rate of change of observable properties with time in an open system. The demonstration described in this article combines both aspects by measuring the rate of change of pressure with time in a closed system that reaches equilibrium. The theoretical framework that motivated this specific mode of presentation is diSessa’s theory of knowledge in pieces (13). This perspective holds that non-expert knowledge is a set of loosely connected ideas about the world, rather than a tightly connected, logically organized structure. These ideas can be used to generate explanations in particular situations and in response to particular questions or cues. diSessa calls these ideas phenomenological primitives or p-prims (14). They can be understood as simple abstractions from common experiences. They are phenomenological in that they are responses to experienced and observed phenomena. They are linked to, and activated by, those phenomena, rather than being general or abstract. They
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are primitive in the sense that they are generally self-evident and need no explanation; they are simply regarded as an expected event. Relevant examples of such p-prims are “heavier is slower” and “heavier is stronger”, which intuitively arise from playing (or watching) contact sports. Such primitives are productive in the context in which they were formed and are also applicable in some scientific contexts, as will be shown in this article. Because they are limited and specific in their range of application, they cannot be considered as being robust scientific knowledge. However, these primitives serve as resources, or building blocks, from which robust scientific knowledge is constructed. The process of becoming an expert is not only a process of acquisition of new knowledge elements, but also a process of coordination of existing knowledge elements and the extension or constriction of conditions under which particular elements may be applied productively. Experts often rely, in their thought processes, on the same knowledge elements as novices do, but the experts know where these knowledge elements are applicable and where their application should be avoided. For example, an expert chemist will employ the concrete “heavier is slower” p-prim when thinking about relative molecular speeds of different gases. However, the expert knows that this simplification is only valid for the average speed of gases at the same temperature and knows which abstract mathematical relations to use in case of gases at different temperatures. Under this theoretical framework, students’ misconceptions are not considered as mistakes that need to be uprooted, but rather “faulty extensions of productive prior knowledge” (15). By presenting a demonstration that exposes multiple aspects of a phenomenon, we provide the students with multiple opportunities to construct productive extensions of prior knowledge. They are required to consider diverse knowledge elements, apply them in a specific context, and assess the productivity of their application. Specific knowledge elements will prove productive for some aspects, but will fail to explain others. Robust scientific knowledge is constructed by realizing which knowledge elements relate productively to which aspects of the phenomena, and how these specific facets can be interrelated and generalized to produce a more complete and abstract understanding of the system.
Journal of Chemical Education • Vol. 85 No. 1 January 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom
A suitable pedagogical framework for carrying out such a process is an interactive teaching method called POE—predict, observe, explain (16). The first step in a POE task is to ask students to predict the outcome of some event, such as what might happen during a demonstration. They are then asked to describe what they observe and finally asked to reconcile any conflict between what they predict and what they observe. The prediction phase activates students’ prior knowledge as they try to make sense of the phenomena at hand. The observation phase triggers a process of self-assessment, testing the productivity of specific elements of knowledge. In the explanation phase, the students need to reconstruct their knowledge and generalize their observation in a way that takes into account possible extensions and impeding limitations of their prior knowledge. The POE method was employed in delivering the demonstration, and the knowledge in pieces theory was used to interpret student responses in the prediction and explanation phases. Relevant Physical Concepts The basic question underlying this demonstration is in what ways different gases are similar and in what ways do they differ? If we consider two equimolar samples of different gases, occupying equal volumes at the same temperature, the following properties differ:
• Particle mass.
• Density—the total mass per unit volume, which is directly proportional to the particle mass.
• Average particle speed—inversely proportional to the square root of the particle mass. This is the determining factor of flow rate in effusive flow (Graham’s law).
• Mean free path—larger molecules are more likely to collide with themselves and therefore travel a shorter distance, on the average, between collisions. The average length of the free path between collisions is an important factor in transport phenomena, namely diffusion (mass transfer), heat conductance (energy transfer), and viscous flow (momentum transfer). The size of the molecules is calculated from experimental measurments of the above transfer rates by assuming that gas molecules are hard spheres that undergo random, elastic collisions. The molecular size calculated from such experiments is called the collision diameter.
• Intermolecular forces—the cohesive forces between molecules determine the temperature below which the gas particles cease to move independently and form a condensed phase.
• Internal modes of motion—in addition to continuous translational motion, polyatomic gas particles also perform rotational and vibrational motions. This affects the heat capacity of the gas, because kinetic energy is distributed between all modes of motion, each contributing to the total kinetic energy.1 When the supplied heat is distributed over more modes of motion, it has a smaller effect on temperature, which measures the average kinetic energy per mode of motion, and therefore the heat capacity is larger.
The two samples are similar on two accounts:
• Concentration—the number of moles in volume (n∙V ).
• Average translational kinetic energy—at a temperature T, each molecule holds, on the average, a translational kinetic energy equal to 1/2kT in each of the three dimensions, where k is Boltzmann’s constant. This quantity is independent of other modes of motion, or the mass of the particle.
A gas exerts pressure on the walls of its container. According to the kinetic molecular theory, the pressure is a result of impulsive collisions of the moving particles with the walls. The pressure is therefore proportional to the rate of collisions times the average impact of each collision:
constant P at volume
u impact of a molecular collision
× rate of collisions
The average impact of each collision is proportional to the average momentum of the molecule (mv, where m is the mass of the molecule and v is the average velocity)—a faster or heavier molecule will collide more forcefully.2 The rate of collisions is proportional to the average velocity times the concentration of the molecules (n∙V )—faster molecules will collide with the walls more frequently:
constant P at volume
u mv v
n V
mv 2
n V
The expression mv2 is proportional to the average kinetic energy of translation, Ek, and therefore
constant P at volume
u Ek translation
n V
The last step shows that the pressure of the gas is not independently affected by particle mass and average velocity, which are different for the two samples, but is affected by the product mv2, which is the same in both samples, regardless of mass and internal structure. It also shows that the pressure is proportional to the concentration of gas particles, which is the same in both samples, and not to the density, which is different. This gives the ideal gas law, which states that the pressure of a gas only depends on its temperature and its concentration and is independent of molecular differences between gases. The only two differences not accounted for so far are in the mean free path and intermolecular forces. Intermolecular collisions tend to increase the frequency of collisions with the wall because the molecules have less free space to move in. Intermolecular forces tend to decrease the impact of each collision because the molecules no longer move independently. The attractive forces cluster individual molecules together momentarily, so effectively they behave like a slightly smaller number of slightly heavier particles, on the average. However, for gases under standard temperature and pressure, the mean free path is 100–1000 times bigger than the collision diameter (17). This means that a gas particle spends most of its time far enough from other particles so the effect of intermolecular collisions and forces on total pressure is negligible, and the actual pressure converges to the ideal value.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 1 January 2008 • Journal of Chemical Education
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In the Classroom 120
injected gas
Pressure / mbar
100
three-way valve pressure sensor
80 60
He CO2
40 20 0
vacuum pump
0
10
20
30
Time / s Figure 1. Experimental setup.
FIgure 2. Sample results.
Procedure
on the volume of the filter flask and ambient atmospheric pressure and is reproducible within ±1 mbar when using the same experimental setup. The results of the experiment are projected in real time from the instructor’s computer on a video screen, so the students see how the pressure changes with time. Typical data for the first run are shown in Figure 2 (circles). To prepare the system for the second run, the syringe is pulled out of the septum and the needle is capped. The three-way valve is set to connect all three tubes, evacuating the filter flask and the pressure sensor.
Preparation A solid rubber stopper that fits a 250 mL filter flask is bored to accommodate a glass tubing (~15 mm diameter, ~10 cm long), which is inserted through the stopper (Figure 1). The glass tubing is tightly fitted with a rubber septum. A three-way plastic valve is connected using Tygon vacuum tubing to the sidearm of the filter flask, to a vacuum pump, and to a pressure sensor (tested with both Vernier’s sensor and Leybold Didactic’s CASSY sensor). The pressure sensor is connected to a computer for real-time recording of data (using Vernier’s Logger Pro or Leybold Didactic’s CASSY Lab, respectively). The three-way valve is set to connect all three tubes, and the vacuum pump is turned on. When the pressure reaches its minimum value, the pressure sensor is recalibrated to show 0 mbar. The software is set to automatically record data every 1.0 second when the pressure rises above 1 mbar (trigger signal). The plunger of a 50 mL plastic syringe is prepared to be locked at a constant volume, by pulling it to the 50 mL mark, and then drilling a hole through the plunger, into which a nail can be inserted, just above the barrel. Two gas bags are prepared; one is filled with helium and the other with carbon dioxide. Demonstration The syringe is filled with ~55 mL of helium and fitted with a 25 gauge hypodermic needle (0.5 × 16 mm, orange). The plunger is then locked in place by inserting a nail into the drilled hole, and pushing the plunger down to the 50 mL mark until the nail is squeezed tightly against the barrel. The excess helium clears away any air trapped in the hypodermic needle. To begin the experiment, the three-way valve is set to connect only the filter flask and the pressure sensor, blocking the vacuum pump. The plastic cap of the hypodermic needle is removed, and the needle inserted into the rubber septum. Gas flows from the syringe into the filter flask, triggering data recording on the computer. The nail keeps the plunger from being pushed down by the atmospheric pressure. Data recording is stopped manually once the pressure has stabilized (in 30–40 seconds). The final pressure should be 100–150 mbar, depending
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Predict Before carrying the second demonstration, the students are presented with the data in Table 1. They are asked to predict what will be the final pressure if the above procedure is repeated using the same molar amount of CO2 instead of He and encouraged to discuss their prediction with their neighbor. After ~3 minutes of discussion, their predictions are polled using the following options:
1. CO2 would produce appreciably less pressure than He.
2. CO2 would produce about the same pressure as He.
3. CO2 would produce appreciably more pressure than He.
Several students are asked to explain the reasoning leading to their prediction. Observe In the second run, the exact same procedure is followed, except for using CO2 instead of He. The graphing option is set to overlay the new data over the previous data set, so the students can compare the two runs in real time. The new graph progressively appears on top of the old graph, so the rates of change and final pressure can be easily compared. Typical data for the second run are shown in Figure 2 (squares). Students who predicted that the CO2 pressure will be lower are very happy (and vocal) in the first 10 seconds of the experiment, but lose heart as the two graphs approach and eventually converge. A student is then asked to describe the results and prompted, if needed, to explicitly state the similarities and differences between the two runs—while the CO2 pressure rises more slowly at first,3 it converges into the same final pressure as helium at equilibrium.
Journal of Chemical Education • Vol. 85 No. 1 January 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom Table 1. Properties of Helium and Carbon Dioxide Property
He
CO2
Molar mass/(g mol‒1)
4.0
44.0
Density at STP/(g L‒1)
0.18
2.0
Collision diameter/pm
215
453
Boiling or sublimation point/°C
‒269
‒78.5
Heat capacity/(J mol‒1 K‒1)
20.7
37.1
Table 2. Number Distribution of Students’ Coincident Prediction and Explanation and Total Percent Prediction
Note: The collision diameter is calculated from measurements of gas viscosity (17).
Explain The students are asked to discuss the result of the second demonstration with their neighbor and formulate an explanation that fits the experimental result, reconciling any conflicts between what they predicted and what they observed. After ~3 minutes of discussion, their predictions are polled, using the following options:
1. The average kinetic energy of the particles in both gases is the same.
2. The average velocity of particles in both gases is the same.
3. The rate of collisions with the walls of the container in both gases is the same.
4. All of the above.
Wrapping Up Following the students’ explanations, the instructor leads a Socratic dialog in which the following questions are addressed:
• What does the rapid rise of pressure in He, as compared to CO2, signify on the molecular level? (He atoms move faster.)
• If He atoms move faster, what can we say about the rate of collisions with the walls of the container? (He atoms collide more frequently.)
• If the rate of collisions with the walls of the container is greater in He, why is it that both gases exert the same pressure after pressure equilibrium is achieved? (Each collision has less impact, because He atoms are lighter.)
• How does the difference in collision diameter affect the rate of collisions between gas particles? Between gas particles and the walls of the container? Do collisions between gas particles have an appreciable effect on gas pressure? (A higher collision diameter means more collisions between particles. CO2 molecules collide more frequently between themselves, but apparently, this has no appreciable effect on gas pressure.)
• How does the greater attraction between CO2 molecules, as judged by the much higher sublimation point, affect the pressure of the gas? Do intermolecular interactions have an appreciable effect on gas pressure? (Apparently, intermolecular
Explanation 1
2
3
4
Total (%)
1
12
1
5
4
11.9
2
51
5
20
15
49.5
3
28
1
24
18
38.6
Total (%)
49.5
3.8
26.6
20.1
100
Note: The header numbers refer to the statements in the text. The correct statements are in bolded font.
interactions have no appreciable effect on the pressure of the two gases under the current conditions. This is true for all gases under moderate pressure.4)
• If CO2 molecules store more kinetic energy than He atoms at the same temperature, as judged from its higher heat capacity, how come the average translational kinetic energy of both gases is the same? (The overall kinetic energy in CO2 is distributed between more modes of motion. Apparently, the translational kinetic energy does not depend on molecular structure, but only on temperature.)
These questions address specific knowledge elements that appear in students’ explanations, as exposed in the prediction phase (detailed in the next section). The instructor builds upon students’ partial conceptions and leads them to acknowledge certain relations that coordinate the separate elements into a coherent picture of gas behavior. Hazards Syringe needles are sharp and potential sources of puncture wounds—they should be capped when fitted to the syringe and after injection is complete. The filter flask is subject to implosion when evacuated. Never apply reduced pressure to a flatbottomed flask unless it is a heavy-walled filter flask designed for that purpose. Check the flask for cracks before evacuation. Results and Discussion It is informative to judge the value of the proposed demonstration in light of actual students’ conceptions regarding the targeted concepts. The above procedure was carried out for two consecutive years in a general chemistry course for first-year biology majors at this university. The students were taught about the kinetic molecular theory of gases, by the same professor, four days before the demonstration. Attendance in the classes was 139 and 121 students. Students’ predictions and explanations were polled using electronic response pads (InterWrite PRS). The polling system is not 100% reliable, and not all students’ responses get recorded. Table 2 summarizes the results for 184 students (99 in the first year and 85 in the second) who had their answers to both prediction and explanation questions recorded. Students also recorded their prediction and reasoning on a sheet
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 1 January 2008 • Journal of Chemical Education
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In the Classroom
of paper, which was collected at the end of the lecture. This procedure was adopted from Sokoloff and Thornton’s interactive lecture demonstrations (18). Looking at the total percent of students choosing each prediction, only half the students (49.5%) predicted correctly that the CO2 pressure would be about the same as that of the He. The common reasoning for this prediction was that since n, V, and T are the same for both gas samples, P should also be the same, according to the ideal gas equation. Approximately onethird of the students (38.6%) predicted that the CO2 pressure will be appreciably higher than that of He. There were several lines of reasoning leading to this prediction: CO2 molecules are heavier; CO2 is more dense so there are more particles in the same volume; and CO2 molecules have more kinetic energy because of their higher heat capacity. Each explanation can be interpreted as employing a relevant p-prim that originates intuitively from everyday experience: “heavier is stronger” (contact sports), “density is crowdedness” (a jam-packed crowd), and “more capacity more effect” (rich is powerful), respectively. A smaller percentage of the students (11.9%) predicted a lower pressure for the CO2. The reasoning was that CO2 molecules are heavier and thus move slower or that the kinetic energy of CO2 molecules is distributed between many modes of motion and only part of it goes into translation. Again, these can be interpreted by assigning relevant p-prims: “heavier is slower” (contact sports) and “more is less” (more partakers means a smaller share). It is important to note that all these alternative paths of reasoning are partly correct: the heavy CO2 molecules move slower than the He atoms and collide less frequently with the walls, but the collisions are more forceful; CO2 molecules have more kinetic energy because of its larger heat capacity, but this energy is distributed between more modes of motion; and the pressure of a gas increases when its density is increased—by increasing the number of particles or decreasing its volume, but not by changing the particles’ mass. However, none of them yields the correct prediction, because they only consider part of the relevant factors. Students who failed to predict correctly should not just give up their prior knowledge. Rather, they need to identify the range of applicability of this knowledge and its limitations and coordinate it with other pieces of knowledge to create a more inclusive understanding of the subject. The relationship between students’ predictions and explanations was tested using χ2 statistics, and was found non-significant at the 5% level (χ2 = 8.0, df = 6, p = 24%). This means that students who predicted correctly did not perform significantly better than their peers on the explanation question. Only 56.0% of the students who predicted correctly also provided a correct molecular level explanation. This shows that a large part of the correct predictions were the result of an algorithmic approach, based on the ideal gas equation and do not reflect conceptual understanding. It is interesting to note that even though the demonstration clearly shows different rates of flow for the two gases, 23.9% (= 3.8% + 20.1%) of the students stated that the average velocity of particles in both gases is the same. These students concentrated on the similarity at equilibrium and ignored the divergent kinetics. This indicates that just showing a demonstration is not
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enough because a novice might attend only to some aspects of the demonstration, and not to others, or interpret the results differently from what is expected. Only through further questioning and discussion can the lecturer draw students’ attention to the points an expert would regard as important. Students’ explanations also fail on the account of internal consistency. One-fifth of the students (20.1%) stated that both the average velocity and the average kinetic energy in both gases is the same. This is mechanically impossible since the masses of the particles are different. One-fourth of the students (26.6%) stated that the rate of collisions with the walls is the same for both gases, while the average velocity of the particles is different. This is again mechanically impossible since the concentration of both gases is similar and average particle velocity thus determines the rate of collisions. These incoherent explanations show that many students do not have a firm theory to back up their explanations, but rather make explanations on the spot by attending to specific cues and activating specific p-prims. While experts try to tie as much knowledge into a general and coherent explanation, students are often content with explanations that fit just a specific aspect of the phenomena, the one that catches their attention. In this case, a reasonable p-prim, activated by the similarity in the final state, is “same effect same cause”. The expert explanation, on the other hand, ties together the divergent kinetics with the similarity in final state and regards the final state as resulting from a combination of opposing causes. The divergent kinetics originates from the difference in particle mass between the two gases. The lower velocity and higher mass of CO2 molecules affect the pressure in opposing directions but offset each other exactly to give the same final effect. Each effect is understood intuitively by the expert, but the exact offsetting is not intuitive because the velocity effect is squared—it affects both collision rate and collision strength. It is only through the mathematical association of pressure and the kinetic energy of translation that this offsetting becomes intelligible. Conclusion The above results support the view of students’ knowledge in pieces. On the one hand, many students’ reasoning and explanations are not indicative of a coherent alternative theory. Some of them can be interpreted as the activation of p-prims—intuitive elements of knowledge that are used to explain specific aspects of a phenomenon in a limited, fragmented context. Others can be interpreted as the application of algorithmic knowledge in situations where a macroscopic result can be obtained by substituting values into a formula but with no understanding of the underlying microscopic mechanism. On the other hand, these fragmented knowledge elements cannot be simply classified as mistakes because they work well in many cases. Many of them are still used by experts to provide quick and simple understanding of a complex situation, by looking at it as an interconnected set of simple elements. It is not the existence of these knowledge elements that distinguishes between expert and novice, but rather the expert’s ability to coordinate the different elements in a general and abstract way and to know where to apply them and what are their limitations.
Journal of Chemical Education • Vol. 85 No. 1 January 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom
A POE-based interactive lecture helps students become aware of their prior knowledge and judge the validity and applicability of these knowledge elements. Through discussion of their predictions and explanations, the lecturer can help students to see the relations that exist between their isolated knowledge elements in a specific context. Instead of discarding unproductive knowledge as mistaken, the lecturer takes it apart and uses its elements as building blocks for the construction of a coherent knowledge structure. Such coordination of previously unrelated pieces of knowledge can be generalized and abstracted using mathematical relations, ultimately leading to the construction of robust scientific knowledge. The demonstration described in this article concurrently exposes both the divergent and convergent behavior of two different gases. It provides a context for classroom discussion that touches upon many subtle points that connect the kinetic molecular theory with the ideal gas law. As such, it offers an opportunity for students to activate their prior knowledge, test it, and engage in meaningful construction of relations between their fragmented knowledge elements. Acknowledgments I would like to thank Calvin DeLano and Matityahu Altshul from the Hebrew University for their technical help in the lab and Heather Sklenicka from Rochester Community and Technical College for her helpful comments. Notes 1. The calculation of heat capacity is not as straightforward as this sentence might imply. At room temperature, many vibrational modes, and some rotational modes, are not fully active because of energy quantization and hold less energy than classically expected. In addition, vibrational modes also have a potential energy component that should be taken into account. The translational motion behaves classically at all temperatures and holds exactly 3/2 kT (1/2 kT per mode of motion). 2. Students do not need formal training in mechanics to know this. Everyday experience provides many opportunities for students to construct the idea that the impact of a collision is proportional both to the mass and the velocity of the colliding object (American football is the author’s favorite example for this). 3. This experimental setup cannot be used for a quantitative demonstration of Graham’s law because the passage of gas through the hypodermic needle is a combination of effusion and viscous flow (5)— the needle is too long and too wide for inter-molecular collisions to be neglected. Since helium atoms are smaller and have a longer mean free path, they lose more momentum to the stationary walls of the needle, and the gas exhibits higher viscosity. The rate of pressure change in the helium sample is therefore lower than expected based on purely effusive dynamics. 4. “Moderate pressure” is a relative term. Gases with weak intermolecular forces (like helium) can be compressed to higher pressures than gases with strong intermolecular forces (like carbon dioxide), before deviations from ideal behavior become significant. The extent of the deviation can be estimated by referencing the second virial
coefficient of the gas (19). Using the virial equation for 25°C and 1 atm, the densities of helium, carbon dioxide, and butane are about 0.9995, 1.005, and 1.03 times the density expected based on the ideal gas approximation, respectively. This means that while butane shows a small (3%) deviation from ideal behavior at room temperature and atmospheric pressure, carbon dioxide will only show a similar deviation at 6 atm, and helium at 60 atm. The deviation is reduced at higher temperatures: at 125 °C and 1 atm, the deviations for helium, carbon dioxide, and butane are 0.04%, 0.2%, and 1%, respectively. At lower temperatures, the intermolecular forces ultimately bring about a complete breakdown of the ideal gas approximation, when the gas liquefies.
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