ARTICLE pubs.acs.org/jchemeduc
The Statistical Interpretation of Classical Thermodynamic Heating and Expansion Processes Stephen F. Cartier* Department of Chemistry, Warren Wilson College, Asheville, North Carolina 28815, United States
bS Supporting Information ABSTRACT: A statistical model has been developed and applied to interpret thermodynamic processes typically presented from the macroscopic, classical perspective. Through this model, students learn and apply the concepts of statistical mechanics, quantum mechanics, and classical thermodynamics in the analysis of the (i) constant volume heating, (ii) constant pressure heating, (iii) isothermal expansion into vacuum, (iv) adiabatic irreversible expansion, and (v) adiabatic reversible expansion of an ideal gas. Whether used as a tool in lecture or as a hands-on computational exercise, the model described herein is a powerful quantitative and visual tool that enables students to more readily grasp the microscopic basis for macroscopic events. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Quantum Chemistry, Statistical Mechanics, Thermodynamics
O
ne of the great challenges in teaching physical chemistry is presenting the many, seemingly disparate concepts that comprise the curriculum in a cohesive, fluid manner. In an attempt to address this challenge and present a more unified approach to one aspect of the curriculum in particular, a statistical model has been developed and applied to interpret thermodynamic processes typically presented from the macroscopic, classical perspective. Through this model, students learn and apply the concepts of statistical mechanics, quantum mechanics, and classical thermodynamics from a more integrated perspective. Some textbooks1,2 do provide introductory molecular-level interpretations of classical thermodynamic processes that can serve as valuable introductions to the more quantitative model presented here. The primary goal of the current work is to provide a statistical model to interpret the (i) constant volume heating, (ii) constant pressure heating, (iii) isothermal expansion into vacuum, (iv) adiabatic irreversible expansion, and (v) adiabatic reversible expansion of a perfect gas. Whether used as a tool in lecture or as a hands-on computational exercise, the model enables students to more readily grasp the microscopic basis for changes that take place upon the heating (or cooling) or expansion (or compression) of a gas. Kozliak3 and Novak4 have suggested using alternative quantitative molecular approaches to entropy and the second law as one means to bridge the gap between the “thermodynamics first” and “quantum first” perspectives of the physical chemistry curriculum. Furthermore, several articles recently appearing in this Journal5,6 have presented ways in which to integrate the teaching of spectroscopy and statistical mechanics. To emphasize the significance and elucidate the statistical basis of the second law, several unique approaches to introducing statistical mechanics have also been described.7,8 Consistent with a number of articles Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
that have elucidated the nature of entropy in terms of the dispersal of energy,912 these approaches provide students the ability to conceptualize the statistical molecular behavior of matter from the microscopic to the macroscopic. The Boltzmann distribution represents the most probable distribution among states and corresponds to the configuration with the greatest number of microstates (W). Statistically, the Boltzmann equation (S = kB ln W) provides the quantitative link between the number of ways a particular configuration can be achieved (W) and the associated entropy (S) of the system,13 where kB is the Boltzmann constant. In other words, the greater the number of microstates associated with a particular distribution of particles among energy levels, the greater the probability of its observation and the greater its entropy. The average number of accessible states depends quantitatively on both the spacing of energy levels and the energy available to the system and is given by the “molecular partition function” or the normalization constant of the Boltzmann distribution. The molecular partition function “contains all the information needed to calculate the thermodynamic properties of a system of independent particles”.1 Implicit in this statement is the intrinsic importance of probability in general and the partition function in particular.1416 Here, this grand concept is applied to understanding the expansion of gases from a microscopic perspective. In elucidating the statistical nature of macroscopic changes, the goal is to provide students the ability to independently grasp this significance and make connections between quantum mechanical and thermodynamic behaviors. For example, an Published: July 22, 2011 1531
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understanding of the fundamental principles governing the Carnot, Otto, Diesel, and Stirling thermodynamic cycles17 and atmospheric phenomena such as adiabatic lapse18,19 and Chinook (foehn) winds20 provides students with a more quantitative, microscopic basis for interpreting observed macroscopic effects. Such cognition ultimately helps establish a higher level of quantitative, molecular intuition that can be applied to questions across the range of chemistry disciplines.
The entropy of the system (in three dimensions) under the different sets of conditions studied is also calculated. To determine the molar translational entropy, ST, the SackurTetrode equation, " # e5=2 3 ST ¼ R ln qtransð1DÞ ð5Þ N
’ THE MODEL A graphical model employing Microsoft Excel has been developed to statistically model thermodynamic processes typically treated in the physical chemistry curriculum from a classical perspective. The model system consists of one particle of a perfect gas with a mass of 1 amu initially confined to a volume of 1 nm3 for each of the processes studied.a To simplify the analysis without losing instructional value, the system is analyzed in one dimension (1D). This simplification also eliminates the necessity of accounting for degenerate states and thereby permits the use of the terms “states” and “levels” interchangeably. Allowed translational energies, εtrans(1D), are calculated from the one-dimensional particle in a box model,
is applied, where N is the number of particles in the gas and R is the gas constant. This expression is derived from the Boltzmann equation and ultimately relates the average number of accessible states as given by the translational partition function to the degree of dispersal of energy as measured by the entropy.
εtransð1DÞ ¼
n2 h2 8mL2
ð1Þ
in which L represents the cube root of the total volume, m is the mass of the particle, n is a positive whole number, and h is Planck’s constant. The translational contribution to the molecular partition function, qtrans(1D) (in one dimension) is calculated, qtransð1DÞ ¼
ð2πmkB TÞ1=2 L h
ð2Þ
where T is temperature, and subsequently, the fractional populations, pi, as defined by the Boltzmann distribution, of the allowed energy states are determined from pi ¼
eβεtransð1DÞi qtransð1DÞ
ð3Þ
in which β = 1/(kBT). The molar internal energy of a perfect gas from a classical perspective is given by U3D = (3/2)RT and the one-dimensional analog is U1D = (1/2)RT. Statistically, the average internal energy, ÆUæ, of a small number of gas particles is given by ÆUæ ¼
∑i pi εi ¼ ∑i
εi eβεi q
ð4Þ
This sum is a general expression for the weighted average of energies based upon the fractional population of states (pi) of different energy (εi). When applied to a large number of particles, it can be shown that this expression simplifies to the classical expression. Similarly, under conditions of higher temperature and less confinement, the weighted average defined by this discrete sum leads to the same prediction of energy as the classical expression. Using this statistical expression for the average internal energy of the model system, the average internal energy is calculated and then superimposed on the population distribution and density of states graphs.
’ CONSTANT VOLUME HEATING OR COOLING In the teaching of the gas laws and the first law of thermodynamics, one of the simplest processes discussed is the constant volume heating of a perfect gas.21 Under constant volume conditions, the change in molar internal energy is given by the equation, ΔU = CV,mΔT, where CV,m is the molar constant volume heat capacity of the perfect gas, equal to (3/2)R, and each translational degree of freedom contributes U1D = (1/2)RT to the molar internal energy. Qualitatively, it can be described as a process in which the translational kinetic energy of the gas particles increases upon heating, thereby increasing the internal energy of the system. From a classical perspective, the increase in internal energy can be rationalized in the context of the kinetic theory of gases and the associated increase in kinetic energy of the particles. In the context of quantum mechanical properties, however, the classical interpretation provides no insight and a statistical interpretation must be relied upon. As a perfect gas is heated under constant volume conditions, the density of translational energy states (number of energy states within a given energy range) does not change. According to the one-dimension particle in a box model, a particle’s allowed energy states and their spacing do not depend upon temperature. However, as seen in the expression for the Boltzmann distribution (eq 3) introduced earlier, the fractional population (pi) of these states does depend upon temperature. As is described in most physical chemistry textbooks, at higher temperatures (as the exponent decreases in magnitude), a greater number of exponential terms contribute to the sum (more states are populated). At lower temperatures (as the exponent increases in magnitude), a smaller number of exponential terms contribute to the sum (fewer states are populated). For this reason, as a perfect gas is heated (or cooled), its internal energy changes. The changes in the fractional populations of states upon heating the model system are summarized in Figure 1. As seen in the figure, when the temperature is increased, the population of higher energy states increases, the distribution broadens, and the average internal energy increases. At the highest temperature tested, 2400 K, there is a negligible discrepancy between the statistical predictions and the (one-dimensional) limiting value, (1/2)RT, as predicted by classical equipartition.22 Applying the statistical model to a larger number of particles (and thereby using the canonical partition function) or increasing the temperature, mass, or dimension of confinement would also result in a statistical prediction evermore consistent with this value. Furthermore, molar translational entropies are calculated and display an increase with temperature, which is consistent with a 1532
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Figure 1. Relative populations under constant volume heating conditions plotted as a function of energy at temperatures of (A) 300 K, (B) 1200 K, and (C) 2400 K. The distribution gets progressively broader as the temperature increases. The average internal energy (ÆU1Dæ = 1/2kBT) corresponding to each temperature is also indicated.
Table 1. Classical and Statistical Predictions of Internal Energy for the Model System in One-Dimension under Constant Volume Conditions Constant Volume
a
Classical Molar Internal
Heating
Energy/
Statistical Molar
qtrans
Strans/
Temperatures
(J mol1)
Internal Energy/(J mol1)
(1D/3D)
(J 3 mol1 K1)
300 K
1247
1239 (0.66)a
9.91/9.75 102
78.0
1200 K
4989
4983 (0.12)
19.83/7.80 103
2400 K
9977
9974 (0.03)
28.04/2.20 104
95.3 104
Percent deviations from the classical prediction displayed in parentheses.
greater number of populated states (a greater partition function), and therefore more corresponding microstates at higher temperatures. The predictions of internal energy (including deviations from the classical values) and entropy for the system described here are summarized in Table 1.
’ CONSTANT PRESSURE HEATING OR COOLING When a perfect gas is heated under constant pressure conditions, its temperature (internal energy) and volume both increase.21 As seen in the previous example, under constant volume conditions, an increase in temperature leads to an increase in the population of higher energy levels. Under constant pressure conditions, as predicted by Charles’ law, an increase in volume must accompany the increase in temperature. From the particle in a box model, as the volume increases, the energy level spacing decreases. In response to this increase in both temperature and density of states, the population distribution shifts accordingly toward higher levels. The model system of interest is heated at constant pressure from an initial volume of 1 nm3 and a temperature of 300 K to a final volume of 8 nm3 and a temperature of 2400 K. The system is then modeled in one dimension, corresponding to an increase in the size of the one-dimensional “box” from 1 to 2 nm. The fractional populations as a function of energy are plotted in Figure 2. Clearly, to accommodate the additional stored energy at the higher temperature the distribution shifts to higher energies leading to a greater internal energy and corresponds to a higher translational energy state. The translational states are compressed as a result of the increase in volume and (in addition to the increase in temperature) this affects the translational entropy of the system. From a qualitative, spatial perspective, the larger, higher temperature system would be considered to be more “disordered”
and therefore in a state of greater entropy. However, interpreting the system from a statistical molecular perspective provides a quantitative, accurate interpretation of the entropy of the system as a function of the dispersal of energy among states rather than a dispersal of particles in space. As the volume increases, the density of states increases leading to a greater population of levels at the same temperature. However, in this particular case, in addition to the increase in volume, the temperature is also increasing and the additional energy provided goes into populating higher levels compared to the lower temperature case. As more levels are populated and the partition function increases, the weight (number of microstates) of the configuration increases. As predicted by the Boltzmann equation, if the predicted distribution has a greater number of equivalent microstates, its entropy will be greater. Furthermore, as seen in the previous example of constant volume heating, when the temperature increases, the population of higher energy states increases, the distribution broadens, and the average internal energy increases. Again, at 2400 K, there is a negligible discrepancy between the statistical predictions and the (onedimensional) limiting value, (1/2)RT, as predicted by classical equipartition.22 All of this can be quantitatively verified and the results are summarized in Table 2.
’ ISOTHERMAL EXPANSION INTO VACUUM Virtually all physical chemistry textbooks treat in great detail the work done in isothermal expansions of perfect gases. An isothermal process is one in which the temperature does not change and, for a perfect gas, the internal energy does not change. Upon the isothermal reversible or irreversible expansion of a perfect gas, work is done by the gas on its surroundings, and therefore, an equivalent amount of energy in the form of heat must flow into the system to maintain the constant internal 1533
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Figure 2. Fractional populations under constant pressure heating conditions plotted as a function of energy at temperatures of (A) 300 K and (B) 2400 K. The distribution gets progressively broader as the temperature increases. The average internal energy (1/2kT) corresponding to each temperature is also indicated.
Table 2. Classical and Statistical Predictions of Internal Energy for the Model System in One Dimension under Constant Pressure Conditions
a
Constant Pressure Heating Initial
Classical Molar Internal
Statistical Molar Internal
and Final Conditions
Energy/(J mol1)
Energy/(J mol1)
qtrans (1D/3D)
Strans/(J mol1 K1)
Ti = 300 K; Vi = 1 nm3 Tf = 2400 K; Vf = 8 nm3
1247 9977
1239 (0.69)a 9974 (0.038)
9.91/9.75 102 56.09/1.77 105
78.0 121
Percent deviations from the classical prediction displayed in parentheses.
Figure 3. Fractional populations for isothermal (300 K) expansion into vacuum plotted as a function of energy (and labeled by quantum state) for three different volumes (A) L, (B) 2 L, and (C) 4L. The distribution gets progressively broader as the volume of the system increases. The average internal energy, ÆUæ1D, in one dimension for the system is the same for each volume but corresponds to a different point in each distribution due to the compression of states.
energy of the system.23,24 The combination of the flow of heat into the gas and the work done by the gas on the surroundings leads to no net increase in the internal energy of the system but does result in an increase in the density of states due to the increase in volume. To clarify the statistical nature of isothermal expansions, consider the case of such an expansion into vacuum. Although no work is done in this particular case, it does not affect the statistical modeling of the expansion and the same model can be applied to reversible and irreversible expansions as well. As the standard thermodynamic treatment clarifies, in the particular case of an isothermal expansion of a perfect gas into vacuum no work is done, ΔU = 0 and no heat flows. (In fact, the same conclusions can be drawn for the adiabatic expansion of perfect gas into vacuum and this model applies to this case as well.) To complement this macroscopic interpretation, a statistical analysis of a perfect gas initially confined to 1 nm3 and
Table 3. Internal Energy, Partition Functions, and Molar Translational Entropy of the System Undergoing Isothermal Expansion into Vacuum Isothermal Expansion Statistical Molar Initial,
Internal
Intermediate, and Final
Energy/(J
qtrans
Entropy/(J
Conditions
mol1)
(1D/3D)
mol1 K1)
1239
9.91/9.75 102
78.0
1245.6 1245.8
19.8/7.80 103 39.7/6.24 104
95.3 113
T1 = 300 K; V1 = 1 nm3 3
T2 = 300 K; V2 = 8 nm T3 = 300 K; V3 = 64 nm3
expanded to 8 times and then 64 times its original volume is presented. To simplify the analysis, the expansion is considered in one dimension so that the length of the confining “box” (1 nm) 1534
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Figure 4. Adiabatic irreversible expansion: Fractional populations plotted as a function of energy (and labeled by quantum state) for the model system undergoing adiabatic irreversible expansion against a constant external pressure equal to 1/16 its initial pressure under the conditions specified in Table 4.
increases to 2 times (2 nm) and then 4 times (4 nm) its original size. The same concepts are applied in this example as in the previous isochoric and isobaric expansions. As this is an isothermal expansion, the internal energy of the perfect gas does not change but the density of states increases. As a result of the compression of states, a greater number of higher energy levels must contribute to the discrete sum that yields the average internal energy and consequently, the fractional populations of lower levels decrease relative to their initial values. To maintain constant internal energy, the population distribution shifts toward higher quantum states (n). As a result, as the temperature remains the same and the volume increases, the average internal energy remains constant but corresponds to increasingly larger values of n. The increase in density of states upon expansion is clearly shown in Figure 3. Each set of data is plotted on the same energy scale. Clearly, as the volume of the box increases, the density of states increases. This is reminiscent of the correspondence principle that states that as the degree of confinement of a quantized system is relaxed, the system behaves more classically until a continuum of states is achieved. These plots convey the general trend toward correspondence of quantum mechanical behavior with classical behavior as the degree of confinement is reduced and the compression of states increases. As a result of state compression and the corresponding increase in the number of populated states (the partition function) upon expansion, the number of microstates corresponding to each Boltzmann distribution also increases. The Sackur Tetrode equation accounts for this increase in the number of microstates and quantitatively predicts an increase in the translational entropy of the system. These results are summarized in Table 3.
’ ADIABATIC IRREVERSIBLE EXPANSION In an adiabatic irreversible expansion, no heat flows and the amount of work done by the gas on its surroundings is not a maximum.2527 In this case, work is being done by the gas on its surroundings and the change in internal energy is manifested as a change in temperature of the gas. However, because this is an irreversible process, the amount of work done by the gas is not a maximum and the change in temperature of the system does not attain its maximum value. Additionally, as this is an expansion, the volume change manifests itself in an increase in the density of states of the system.
Table 4. Internal Energy, Partition Functions, and Molar Translational Entropy of the System Undergoing Adiabatic Irreversible Expansion Adiabatic Irreversible Expansion Initial and Final Conditions
Statistical Molar Internal Energy/(J mol1)
qtrans (1D/3D)
Entropy/ (J mol1 K1)
Ti = 300 K; Vi = 1 nm3 Tf = 188 K; Vf = 10 nm3
1239 778.0
9.91/9.75 102 16.9/4.79 103
78.0 91.3
Graphically, the same model applied to the system undergoing heating at constant pressure applies here. In that particular case, there is a change in temperature as well as a change in volume. However, under the current conditions, although the volume is increasing, the temperature is decreasing. To apply our model to this case, once the changes in temperature and volume have been classically determined, they can be statistically treated to visualize both the shift in occupation numbers of higher levels and the compression of states. The results of such an analysis are presented in Figure 4 and Table 4. There are two competing effects present in this particular case. The temperature is dropping, but the density of states has increased. Thus, despite the fact that more states would be populated at the same temperature for the expanded system, the decrease in temperature leads to a lower average internal energy. Even though the average internal energy of the expanded system corresponds to a higher translational state, it is actually lower in magnitude because of the compression of the energy scale upon gas expansion. This is evident in the density of states plots in Figure 4 and summarized in Table 4. Despite the decrease in temperature that accompanies the adiabatic irreversible expansion of a perfect gas, the entropy of the system increases in the course of this expansion. Examining the SackurTetrode equation permits us to conclude that the decrease in temperature (which alone would lead to a decrease in entropy) has less of an effect than the increase in volume. From a statistical perspective, the effect of the increase in the density of states resulting from the increased volume outweighs the competing effect of the decrease in temperature. The net effect is observed both in the increase of the translational partition function and the absolute entropy as calculated using the SackurTetrode equation and summarized in Table 4. As the values indicate, these processes are spontaneous. When a gas under pressure expands quickly, it can be assumed to occur adiabatically and the temperature drops. As a result, high-pressure gas expanding from a gas cylinder or aerosol can feels cool upon expansion. 1535
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Figure 5. Adiabatic reversible expansion: Fractional populations plotted as a function of energy (and labeled by quantum state) for the model system undergoing adiabatic reversible expansion between the (A) initial and (B) final conditions specified in Table 5.
’ ADIABATIC REVERSIBLE EXPANSION The adiabatic reversible expansion of a perfect gas is conceptually the most challenging and complex example of gas expansion to grasp and is typically the last of the expansion processes to be introduced.16,2830 The subtlety of reversibility is challenging in and of itself and additionally, this is typically the first point at which the heat capacity ratio (γ = CP/CV) is encountered.31 Treating this type of expansion from a statistical perspective provides an additional element of subtlety, but also provides another valuable opportunity to integrate and discuss the aspects of quantum mechanics, statistical mechanics, and classical thermodynamics. The “adiabatic principle” of quantum mechanics is the basis of the BornOppenheimer approximation and states that a molecular system initially in some designated energy level will remain in that same level if it undergoes sufficiently slow changes in internuclear separation. By extension, if we imagine the increase in volume of the perfect gas system to occur very slowly, this perturbation cannot induce transitions between energy levels and is considered to be an “adiabatic perturbation”. Because the expansion is occurring so slowly, the gas is always in an equilibrium state and the process is considered to be “reversible” in the thermodynamic sense. Thus, a quantum mechanical adiabatic perturbation corresponds to a thermodynamic adiabatic reversible process.2,32 As a result, in an adiabatic reversible expansion, energy is only crossing the system boundary in the form of work, which leads to a decrease in the average internal energy of the gas, but with the distribution of particles over the energy levels remaining unaltered. Consider the model system initially at 300 K to undergo an adiabatic reversible expansion to five times its initial volume. As this is an adiabatic expansion, accompanying the pressure decrease is a drop in temperature. Consequently, there is an associated increase in the density of states for the expanded system, but because this is an adiabatic perturbation, the decrease in temperature statistically compensates for this increase and the fractional population distribution does not change. The only way to achieve the same distribution when the densities of state differ is to remove some energy. In this particular case, kinetic energy is removed in the form of expansion work resulting in a decrease in internal energy and temperature. Upon analysis, because there is no shift in the Boltzmann distribution, the magnitude of the partition function does not change. The density of states increases in the course of this process, but the fractional populations remain the same. Therefore, the average internal energy is
Table 5. Internal Energy, Partition Functions, and Molar Translational Entropy of the System Undergoing Adiabatic Reversible Expansion Adiabatic Reversible Expansion Initial
Statistical Molar
and Final
Internal
qtrans
Entropy/
Conditions
Energy/(J mol1)
(1D/3D)
(J mol1 K1)
Ti = 300 K; Vi = 1.0 nm3 Tf = 102 K; Vf = 5.0 nm
1239 419.0
9.91/9.75 102
78.0
9.91/9.75 102
78.0
3
obtained from the sum of the same number of terms, all of which are smaller in magnitude resulting in an overall smaller value. This decrease in internal energy manifests itself as a drop in temperature. The Boltzmann distributions are identical under the constraints of the initial and final conditions when the process is carried out adiabatically and reversibly. As shown in Figure 5, the populations of respective levels are identical, but correspond to different internal energies as a result of the change in density of states (i.e., note the different scales on the y axes). Thus, for any set of initial and final conditions of a gas that has undergone an adiabatic, reversible expansion, the fractional populations are identical even though they correspond to different average internal energies. In comparing this process to the adiabatic, irreversible expansion previously addressed, it is evident from Figure 4 that when the process occurs irreversibly the density of states shifts accordingly, but the initial and final fractional population distributions are quite different from each other. Furthermore, the drop in temperature is greater in the reversible case because the maximum amount of energy, in the form of work, has been removed from the system. From the same set of initial conditions, if the system expands adiabatically and irreversibly to the same final fraction of its initial pressure, the drop in temperature is less than the corresponding drop when it occurs reversibly corresponding to less of a change in internal energy. Because the adiabatic reversible expansion is considered to be an adiabatic perturbation, the fractional population plots of Figure 5 show no shift in the Boltzmann distribution. Consequently, as a result of how the average internal energy is statistically defined, this figure demonstrates that the internal energies are different, even though they correspond to the same quantum state. However, this particular point on the y axis 1536
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Journal of Chemical Education corresponds to a different energy under each set of conditions due to the compression of states upon expansion. To intuit from a qualitative perspective, one may be led to conclude that the expanded system has a greater translational entropy due to greater “spatial disorder”. Alternatively, one may conclude that the system under its initial set of conditions has greater translational entropy because it is at a higher temperature and the gas particles have greater kinetic energy imparting to them greater “randomness” of motion. Neither of these conclusions would be correct and this particular example is a powerful argument for emphasizing the statistical nature of entropy. The entropy of the system under the two sets of conditions is identical. Because this particular expansion is considered to be an adiabatic perturbation in which the population distribution among states does not change, the two macrostates, as defined by their respective temperatures, have the same degree of “energy dispersal”, have the same number of associated microstates, and consequently, have the same entropy. These results are summarized in Table 5.
’ CONCLUSION A statistical, graphical model has been developed to integrate the teaching of statistical mechanics, quantum mechanics, and classical thermodynamic processes. The statistical molecular interpretation of various heating and expansion processes occurring under different sets of conditions that are typically treated from a classical thermodynamic perspective provides one avenue through which the physical chemistry curriculum can be presented in a more unified, cohesive manner. Through the use of these models, students have the opportunity to simultaneously apply their knowledge of concepts such as the Boltzmann distribution, the partition function, the particle in a box, internal energy, and entropy. From such an integrated approach, students acquire greater molecular-level intuition into physical as well as chemical processes that strengthens their ability to rationalize and predict observed behaviors. ’ ASSOCIATED CONTENT
bS
Supporting Information Description of the Excel model; raw data. This material is available via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ADDITIONAL NOTE a Modeling a system containing only one particle and then calculating the populations, molar energy and molar entropy is perhaps challenging to those being introduced to the field. An obvious and interesting question that could arise is the following: How can one particle simultaneously be in so many different states? Modeling a system of a larger number of would facilitate the visualization of populated states. Conceptually, the author finds that students are more cognitively receptive to a large number of particles being distributed among states rather than envisioning the distribution as representative of the accessible states of a single particle. However, the disadvantage of basing the discussion on a large number of particles confined to 1 nm3 at the
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temperatures considered in this model is that for 1000 particles this corresponds to a pressure on the order of kilobars (kbar). Despite the fact that the model treats an ideal gas, these pressures are wholly unrealistic. To treat the system of more particles at more reasonable pressures, a larger volume could be incorporated but this leads to loss of resolution of states on a reasonable scale. The behavior of an ideal gas expanding on a larger scale would mirror the behavior demonstrated here. However, at such a realistic lower pressure, representation and visualization of discrete states becomes exceedingly difficult. In using this model as a pedagogical tool, it is important to emphasize that the distributions represent the probability of a single particle accessing any of the states and that every particle in the system would be characterized by the same probabilistic behavior. This thereby permits the visualization of how any large number of particles would distribute themselves among states.
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