Using Rubber-Elastic Material-Ideal Gas Analogies ... - ACS Publications

In the Classroom

Using Rubber-Elastic Material–Ideal Gas Analogies To Teach Introductory Thermodynamics Part I: Equations of State Brent Smith Department of Textile Engineering, Chemistry, and Science, North Carolina State, Raleigh, NC 27695

This paper is a supplement to the study of introductory thermodynamics in undergraduate physical chemistry. It is not intended to be a description of polymer chemistry, for which purpose it is incomplete. Part I describes equations of state, and Part II describes the laws of thermodynamics (1). Traditional physical chemistry textbooks introduce the laws of thermodynamics by using solids, liquids, and especially gases as examples of thermodynamic systems for computing work, heat, energy, enthalpy, and so forth. For example, introductory texts typically define work in terms of action through a distance and then proceed immediately to develop an example for reversible isothermal work for a volume change by an ideal gas. dW = ᎑PoppdV

and they do not follow the logical development of thermodynamics, beginning with equations of state, followed by the laws, thermochemistry, and so forth. Many introductory thermodynamics and physical chemistry texts and, in some cases, entire undergraduate plans of study, ignore REM examples as a tool to introduce the laws of thermodynamics (12–14). Ordinary solids and liquids are often uninteresting examples because they do not change size significantly when temperature or pressure changes, leading to trivial results in many types of problems. Non-ideal gas equations of state often lead to computational difficulties such as difficult integrals or expressions that are difficult to evaluate. Even solving the van der Waals equation of state for volume can be a challenge, because it is a cubic equation when written explicitly as volume.

W = ᎑nRT ln(Vf兾Vi)

PV 3 + (᎑Pb – RT )V 2 + aV + (᎑ab) = 0

Continually using the ideal gas as the only example eventually can confuse the distinction between general concepts and the special case for the ideal gas. Alternative examples, questions and problems, in addition to the usual ideal gas examples, are very useful to create a broader understanding of the general case. One particularly suitable system is rubber-elastic material (REM), for many reasons. First, REM offers many strong analogies to ideal gas concepts. These are particularly enlightening in the study of thermodynamics at the introductory level. Second, REM provides non-trivial but manageable examples to improve understanding of the general case. Third, polymeric REM is very important in today’s world and exhibits behavior quite different from other materials. Studying these materials at the fundamental level sets a solid foundation for advanced studies of polymers and REM. Finally, REM behavior is very easy to demonstrate in the classroom or laboratory, and therefore can be used to good advantage for demonstrations and exercises (2–5). Mathias, for example, published a collection of 83 demonstrations and experiments suitable for lecture or lab use (6). Measuring and containing gases requires more complex apparatus and arrangements than handling REM. For these reasons, it is worthwhile to study REM to introduce the basic concepts of thermodynamics. During about 30 semesters of teaching junior-level undergraduate physical chemistry, I have successfully used REM examples and problems to enhance students’ understanding of general thermodynamic concepts. Some of these are presented here. Several published sources cover parts of this material, but these are generally advanced treatments based on statistical mechanics, and are generally unsuitable for introductory undergraduate thermodynamics (7–11). Also, they generally fail to fully develop the striking analogies of REM and ideal gases,

The ideal gas is a very useful example because it has a simple equation of state with large temperature and pressure dependence. Likewise, REM has the same features. When heated at constant pressure, gases (unlike condensed phases) exchange significant amounts of energy with the surroundings in the form of work. The integrals and expressions involved are relatively easy to work with for the ideal gas. This allows one to formulate fairly simple but non-trivial introductory thermodynamic questions and problems. REM examples presented here provide an alternative system to supplement the ideal gas for examples, questions, and problems to more clearly elucidate fundamental thermodynamic issues.

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Historical Perspective The remarkable properties of REM have long been known, and were recognized but never explained by the great physicists of the 19th century, like Thompson and Joule. They recognized that these materials, which they referred to as “jelly” or “rubber”, were unique because they have great elasticity like gases but are incompressible. They have great deformability like fluids but recover their initial shape when unstressed, like solids. In fact, these 19th century physicists could easily conceive of a liquid or gaseous fluid, but a REM solid fluid was beyond their comprehension. William Thompson (Lord Kelvin) wrote about these unique properties saying: “A vulcanized India-rubber band, for instance, is capable of being stretched, again and again, to eight times its length, and returning nearly to its previous condition when the stress is removed.” He recognized that REM undergoes “great changes of shape but slight changes of bulk. They have in fact…the same compressibility as water.” He also knew that REM contracted when heated, unlike other solids and liquids, which expand when heated (15). In his Baltimore Lec-

Journal of Chemical Education • Vol. 79 No. 12 December 2002 • JChemEd.chem.wisc.edu

In the Classroom

tures on Modern and Theoretical Physics in 1884, he presented various molecular explanations for the behavior of elastic materials based on various mechanical arrangements including springs and ball-and-socket connections between molecules and sub-molecular particles. He did not conceive that REM could be polymeric. In fact, in his Lecture XIII he conceded: “I am afraid this problem of the molecules in the elastic solid presents enormous difficulties to us” (16). Likewise James Joule studied the cooling and heating of REM, noting that these materials readily heated when stretched and cooled upon contraction. He also studied contraction of REM when heated under constant tension. He determined the heat capacity of India rubber to be 0.415 cal/g K. He studied the bulk thermal expansion and recognized this as a separate effect from rubber-elastic behavior, that is, contraction when heated. He also studied “set” and other non-ideal REM behaviors. But he, too, had no molecular explanation for these behaviors (17). The Basis of REM Behavior Many descriptions of REM take a statistical mechanical (molecular) approach. The emphasis in this document, on the other hand, is in molar, not molecular thermodynamics. Nevertheless, knowledge of the causes of REM behavior at the molecular level provides a sound basis for entropy and microstructure concepts that are useful in teaching the second and third laws. One useful conceptual microstructure model is that the polymer is a three-dimensional zigzag structure with freely rotating bonds that allow the polymer to change its end-to-end length by coiling and uncoiling. When not crosslinked, polymers can melt or dissolve to form liquids. But when crosslinked, polymers can become REM due to the entropy-driven coiling and uncoiling of the polymer chains between the crosslinks. In this sense, REM is essentially a crosslinked liquid, or solid fluid polymer network with complete segmental mobility of monomer units. To understand the basis for the similarity of REM and ideal gases, it is useful to look beyond the macroscopic material and to examine the behavior in terms of microstructures. For ideal gases, this often is done by studying Maxwell’s distribution, the kinetic theory of gases, and by deriving the ideal gas law from Newton’s laws for point particles. Ideal REM provides excellent additional examples for these and other studies, notably the mechanics of rubberlike elasticity, lattice models of polymer solutions, entropy, and microstates, as well as some excellent possibilities for distribution function problems in terms of molecular weight distributions. Non-ideal gases are often studied in terms of such topics as intermolecular potentials and critical behavior of fluids. These non-idealities are, in some cases, manifestations of Joule’s law violations, which can also contribute to non-ideality in REM. A useful demonstration of crosslinking is to mix dilute aqueous solutions of 5% polyvinyl alcohol (PVA) and 2% borax. Prior to mixing, the PVA solution is a viscous liquid, but adding a small amount of borax solution forms crosslinks and converts the solution to a pseudo solid with unique creep and elastic properties (18). This material is not an ideal REM because PVA does not have freely rotating bonds, but nevertheless the crosslinking demonstration is useful. Another simple demonstration is to attempt to dissolve a rubber band

in toluene. Toluene swells the rubber band to many times its normal size, but the crosslinks prevent the polymer molecules from dissolving (19). If not crosslinked, rubber dissolves. This is easy to demonstrate with a soft rubber eraser (not crosslinked), a rubber band (crosslinked), and toluene (solvent). Treloar reviews findings of Gee and others, showing that crosslinked rubber behaves more like ideal REM after it has been solvent-swollen, due to reduction of inter- and intra-molecular interactions of polymer chains (11). REM can conveniently be viewed as a crosslinked structure of polymers. These polymers in turn can be viewed as a series of connected monomer repeat units, as shown in Figure 1. Conceptual microstates of such a polymer are shown in Figure 2.

Extended: long end-to-end distance

Coiled: short end-to-end distance

Figure 1. Polymer and REM configurations.

Fully extended: end-to-end length is 5

Coiled: end-to-end length is 3

Coiled: end-to-end length is 1

Figure 2. Some conceptual microstates of a polymer with five monomer units.

JChemEd.chem.wisc.edu • Vol. 79 No. 12 December 2002 • Journal of Chemical Education

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In the Classroom

Comparison of REM to “Ordinary” Materials Solids resist deformation of size or shape due to large changes in energy with molecular (or atomic or ionic) displacement from equilibrium positions in a three-dimensional intermolecular potential. These materials have essentially no change in entropy with small deformations. Their molecules are held together in a rigid fixed three-dimensional configuration by strong short-range intermolecular forces arising from an intermolecular potential energy well with a pronounced minimum. When solids are heated, the increased thermal energy allows for greater thermal motion around the equilibrium positions in the intermolecular potential well, causing expansion of the bulk material. Thus, for solids, the effects of force (or stress) and temperature are a consequence of intermolecular potential energy. The undeformed rest state (size and shape) of a solid is defined by the three-dimensional intermolecular potential. Liquids, like solids, resist compression because of the same intermolecular potential energy effects, but they have no resistance to deformations at constant size because the molecules are not constrained to a precise three-dimensional position, only to a specific separation. Therefore they flow freely, but do not compress. The rest state size of a liquid depends on molecules being at their equilibrium distances as defined by the intermolecular potential energy. The reststate shape is defined by the shape of the container (or by surface tension). The effect of heating is similar to solids. Thus behavior of liquids is also interpreted in terms of the intermolecular potential energy. Pressure (or force) for an ideal gas results from the cumulative effect of many molecular impacts on the walls of the container. Intermolecular potential energy between molecules is not a factor in the pressure of an ideal gas (but is important in the study of non-ideal gases). Increasing the temperature of an ideal gas leads to increased momentum of the individual molecules, thus increasing the pressure (if system size is constant) or increasing size (if opposing pressure is constant). The effect of temperature is interpreted in terms of individual molecular motion, not intermolecular potential energy. The effect of deforming a gas (changing its size) at constant temperature is interpreted in terms of entropy, not energy. The undeformed rest state in the absence of any opposing pressure is infinite size. This is the state of maximum entropy. When compressed at constant temperature, the pressure of the gas rises. This is not because of any change in the intermolecular potential energy nor change in individual molecular kinetic energy, but rather due to the confinement of the gas to a smaller region of space (smaller size), with correspondingly fewer possible configurational microstates that correspond to the compressed macrostate. Table 1 summarizes the conceptual origin of forces that arise upon deformation of materials. Ideal REM (like all ideal Joule’s law substances, including ideal gas) has no change in energy with change in shape at constant size. Thus the restoring force arises exclusively from the change of entropy with shape, which (for REM) is negative and much smaller in magnitude than the energy changes seen in solids for comparable deformation. Therefore a relatively large extension of REM causes the development of a relatively small restoring force. For REM, like ideal gas, the rest state is the length

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Table 1. Conceptual Relative Contributions to Force in Idealized Materials Material

Deform Shape at Constant Volume

Deform Size (Volume)

(∂E/∂l ) T,V

(∂S/∂l ) T,V

(∂E/∂V ) T

(∂S/∂V ) T

Solid

large

~0

large

~0

Liquid

~0

~0

large

~0

Gas

~0

~0

~0

small

REM

~0

small

large

~0

which allows the maximum entropy (the maximum number of microstates). This will be discussed in more detail later. The first law can be used to illustrate that entropy is the origin of pressure in an ideal gas dE = dQ + dW dE = TdS – PdV P = ᎑(∂E兾∂V )T + T(∂S兾∂V )T P = T(∂S兾∂V )T Since (∂E兾∂V )T is zero for ideal gas (Joule’s law), P has entropic origin. Since (∂S兾∂V )T is positive, pressure acts in the same direction as increasing volume. Likewise, for REM deformation in length at constant volume and temperature, the first law can be used to show the origin of force dE = dQ + dW dE = TdS + f dl f = (∂E兾∂l )T,V – T(∂S兾∂l )T,V f = ᎑T(∂S兾∂l )T,V As for an ideal gas, (∂E兾∂l )T is zero for REM (Joule’s law), thus f has entropic origin. Unlike ideal gas, (∂S兾∂l )T is negative for REM, thus force acts in the opposite direction as increasing length. In all of the above, P is pressure, V is volume, E is energy, Q is the heat for the deformation process, W is the work for the process, T is temperature, l is REM length, and f is the force that develops as a result of stretching. REM provides an outstanding and directly observable effect that can easily be seen in the classroom or laboratory as a consequence of entropy. As REM stretches, the polymer chains uncoil to a state of higher entropy (fewer microstates are associated with longer end-to-end distances). As REM contracts, the chains coil up and the system increases in entropy due to the increased number of microstates associated with short end-to-end polymer chains. This leads to the high stretch and recovery of REM. When stretched, entropy “pulls back” in a way that can be sensed directly by touch, without instrumentation. The above indicates why the force required to extend REM (at constant volume) or compress a gas is vastly less than the force required to compress a solid or liquid. For example, 1% elongation requires thousands of Newtons of force for a steel wire, but only hundredths of a Newton for REM,


In the Classroom

a difference of five or more orders of magnitude. Although REM is an incompressible solid material, its mechanical and thermodynamic behavior bears striking similarities to an ideal gas because the main effect of REM deformation at constant temperature is entropic, not energetic. The maximum-entropy rest state of REM in the absence of any deforming force is its unstretched length. As REM is stretched, an inward restoring tension develops as the polymeric chain network is extended to states of lower entropy—a direct analog to the entropic outward pressure of a compressed ideal gas that has been confined to a small space, allowing fewer microstates.

tracts in the other dimensions in such a way that its volume remains constant so that

The Guth–James Equation of State

Just as in the case of an ideal gas, REM develops a restoring force, f, arising from entropic effects

Many polymeric materials exhibit REM behavior above their glass transition temperatures (20). The Guth–James REM equation of state can be justified conceptually to elucidate the nature and behavior of REM, in comparison to other materials (21). As in the case of an ideal gas, it is possible to arrive at the REM equation of state through statistical mechanics (10, 11). For the introductory teaching of physical chemistry, the following concepts can lead to an intuitive understanding of the REM equation of state: REM comprises a network of many crosslinked polymer chains with freely rotating bonds, allowing extension of the network with no change in energy REM bulk stretching of the network causes affine deformation of individual polymer chains between the junction (crosslink) points REM bulk response arises from molecular configuration entropic effects, not intermolecular energy effects

ex e y e z = 1 ex = e ey ez = 1兾ex = e᎑1 ey = ez = e᎑1/2

This necking down of cross-sectional area is easy to observe when stretching an ordinary rubber band. The entropy change for the stretch is ∆S = ᎑1/2nk(e 2 + e᎑1 + e᎑1 – 3) = ᎑1/2nk(e 2 + 2e᎑1 – 3)

f = ᎑T(∂S兾∂l )T leading to the following expression for the restoring force f = ᎑T(∂S兾∂l )T = ᎑T(∂S兾∂e)T (∂e兾∂l )T f = ᎑T [᎑1/2nk(2e – 2e᎑2)]兾(lo) = (n兾lo)kT(e – e᎑2) Switching from molecular units to molar units, the number of crosslinked polymer network chains per unit length (n兾lo) transforms into the REM molar linear density (D), and Boltzmann’s constant (k) becomes the molar gas constant (R). The above leads to the Guth–James equation of state for REM f = DRT(e – e᎑2)

Using statistical mechanical methods, the entropy, S, of a crosslinked network of n extended polymer chains can be shown to be (10, 11) S = ᎑1/2nk(ex2 + ey2 + ez2) where n is the number of chains, k is Boltzmann’s constant and ex, ey, and ez are the extensions in the various dimensions. In each dimension, e is the stretched length divided by the relaxed length so l = elo. When unstressed, REM relaxed (in its rest state) with ex = e y = ez = 1 ex ey e z = 1 In this rest state, REM has maximum entropy. Stretching decreases the REM entropy by reducing the number of possible configurational microstates that correspond to the stretched macrostate. The entropy change associated with a deformation is ∆S = Sdeformed – Srelaxed ∆S = [᎑1/2nk(ex2 + ey2 + ez2)] – [᎑1/2nk(1 + 1 + 1)] ∆S = ᎑1/2nk(ex2 + ey2 + ez2 – 3) To characterize REM elongation in one direction, let us consider elongation where ex = e. In that case, the REM con-

(1)

This form of the REM equation of state is often very useful for problem solving. In order to extend the analogy of REM with ideal gases, it is possible to bring the Guth–James equation of state and the ideal gas law into very similar forms. For ideal gas, the pressure (deforming force per unit area) is P = (n兾V )RT

REM restoring force generated by n (many) chains is n times the restoring force of a single chain bulk volume remains constant when REM is stretched

initially stretch in the x-direction consequence of constant volume isotropic response

P = ρRT where ρ is the ideal gas density (molecules per volume). Similarly for REM, the stress (deforming force per unit cross-sectional area) is σ = f 兾ao = [n兾(lo ao)]RT(e – e᎑2) σ = ρRT(e – e᎑2) where lo is the unstretched length, ao is the unstressed crosssectional area, and ρ is the density of crosslinked rubber elastic polymer network chains (molecules per unit volume). It is striking that two such different physical systems are described by such similar equations of state. This will be discussed in later sections of this paper, in particular comparing the two in terms of action (pressure or stress) and size (extension or volume). Useful Examples Examples for teaching introductory physical chemistry are presented below and in Part II of this paper (1). These examples extend the analogy between ideal gases and REM, pointing out similarities and differences in many aspects of their behavior. Through an understanding of such analogies, students can have an opportunity to learn about thermodynamics at a deeper level—knowledge in one case can contribute to understanding of the other.


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In the Classroom Table 2. Integer Solutions to the Guth–James Equation of State

Table 3. Physical Variables for Ideal Gas and REM Variable

Ideal Gas

REM (Stress Form)

REM (Force Form)

Temperature

Temperature (K)

Temperature (K)

Temperature (K)

1.7500

Size

Volume (m3)

Extension (dimensionless)

Length (m)

3

2.8889

Action

Pressure (Pa)

Stress (Pa)

Force (N)

4

3.9375

Gas Constant

R (J mol᎑1 K᎑1)

R (J mol᎑1 K᎑1)

R (J mol᎑1 K᎑1)

5

4.9600

6

5.9722

Amount

ρ (mol m᎑3)

ρ (mol m᎑3)

D = density (mol m᎑1)

7

6.9796

Equation of State

P = ρRT

σ = ρRT(e – e᎑2)

f = DRT(e – e᎑2)

1

0

2

Equation 1 is trivial to solve for f or T, but is cubic if written explicitly with elongation as the independent variable. This may seem difficult to solve, but it is not if kept in a form similar to its original form. (e – e᎑2) = f 兾DRT One estimates a trial value of e = f 兾DRT (or slightly larger than 1, if f 兾DRT < 1) as a first guess. Then various values of elongation are tested using the above equation to see if the equation is satisfied. This is very easy using a spreadsheet or calculator. With a little experience, this can be solved readily by trial-and-error, or graphical methods. Also, it helps for problems to be written with integer solutions, typically between 1 and 8, as shown in Table 2. A physical equation of state is a mathematical relationship between amount of material, temperature, size, and action. A comparison of the variables and dimensions for REM equation of state compared to ideal gas is shown in Table 3. Plotted REM isotherms appear quite different from ideal gas, as shown in Figure 3. Increasing amount (n, ρ, or D) or temperature increases the action (P, σ, or f ). The Guth–James equation of state has been used as the basis for demonstrations and laboratory exercises that can easily be performed with minimal equipment (2, 6). A simple demonstration of hanging weights from a rubber band, and measuring length as a function of applied force, shows that force and elongation are not directly proportional. It is easily shown that the REM rest state in the absence of applied action is unit elongation, not zero elongation as it would be if eq 1 were f = DRTe. It is possible to gain an intuitive understanding of REM physical behavior by correlating a simple classroom demonstration with eq 1 and the information in Table 3 and 4, without doing numerical calculations at all. As e and f become very large, f ~ DRTe. For the ideal gas, P = nRT兾V. This shows that, although there is similarity between the two equations, there are two very important differences. First is the rest state, and second is the fact that f increases with increasing e, while P decreases with increasing V at constant temperature (7). P and V have a reciprocal relationship in which dP兾P = ᎑dV兾V, whereas f and l do not. When f becomes small (approaches zero) the limiting value of elongation is unity. The limiting slope (∂f兾∂e)T as elongation approaches unity is 3DRT. From the demonstration suggested above, it is possible to gain an intuitive understanding, and to sketch an isotherm plot of f (ordinate) versus e (abscissa), 1448

14 12

Pressure / Pa

e – e᎑2

10 8

ideal gas

6 4 2 0 0

5

10

15

Volume / m3 3

Force / N

e

2

rubber-elastic

1

0 0

1

2

3

Elongation Figure 3. Physical isotherms of ideal gas and REM.

at constant temperature as shown in Figure 3. The effect of changing T or D in eq 1 can easily be realized. The relationship between the direction of change in size and the direction of action is different for the two systems. The opposite relationship for action and size for REM versus an ideal gas has a direct implication in terms of calculations of work for these systems because of the relative directions of action compared to size. dW = f dl dW = ᎑PdV

for REM for ideal gas

The sign convention is that work done on the system is positive.


In the Classroom

The REM length is l = elo = a兾Vo, so

Table 4. Useful Derivatives e→1

e→∞

REM General

Ideal Gas Equivalent

(∂e/∂T) f

0

᎑e/T

᎑(e/T) (e3 – 1)/(e3 + 2)

+V/T

(∂f/∂T) e

0

σ/T

+σ/T

+P/T

(∂f/∂e)T

3ρRT

ρRT

+(f/e) + (3ρRT/e3)

᎑P/V

Derivative

DRT(e – e᎑2) = nRT兾loe (e 2 – e-1) = n兾lo D = constant not a function of temperature in this case e = constant not a function of temperature in this case Therefore e is constant, and the piston stays in the same position regardless of temperature because the REM elongation does not depend on temperature in this experiment. The REM tension and the pressure of the gas, of course, increase as temperature is raised. Mathematical Analogies REM offers good alternative opportunities for introductory exercises involving derivatives. Only a few very simple integrals and differentials are needed to study REM. The partial derivatives are all easily obtained directly or by the chain rule, reciprocal rule, and cyclic rule. For REM

Piston Ideal Gas

(∂f 兾∂e)T = DRT(1 + 2兾e 3) = f 兾e + 3DRT兾e 3

REM

(∂f兾∂T )e = DR(e – 1兾e 2) = f 兾T (∂e兾∂T )f = ᎑(e兾T)(e3 – 1)兾(e 3 + 2) Figure 4. An ideal gas under a piston restrained by REM.

The first two are easy to obtain directly. The third can be found by several methods, including the use of the reciprocal or cyclic rule. A comparison of REM and ideal gas derivatives is shown in Table 4. Note that (∂e兾∂T )f is always negative and, in the limit of high elongation, approaches (᎑e兾T ). This is similar to the ideal gas case but note the difference in sign, where (∂V兾∂T )P is always positive, and equal to (+P兾T ). In both cases, the result is action divided by temperature. When heated at constant pressure, an ideal gas expands but an ideal REM contracts. This is a fundamental difference between REM and ordinary materials. The second of the three derivatives shows that (∂f兾∂T)e=f兾T. Similarly for an ideal gas, (∂P兾∂T)V = P兾T. In each ideal case

Elongation e = l/lo

5

4

3

2

1

[∂(action)兾∂(temperature)]size = action兾temperature

0 150

200

250

300

350

400

Temperature / K Figure 5. REM elongation versus temperature at constant force.

Without making any numerical calculations, it is instructive to look at plots such as: Action (P, f, or σ) versus size ( V or e) at constant temperature (T ) as shown in Figure 3 Action (V or e) versus temperature (T) at constant size (V or e)

An Equation of State Problem This problem combines both REM and ideal gas equations of state. An ideal gas is confined at equilibrium beneath a massless piston in a cylinder as shown in Figure 4. The piston is supported by the pressure of the gas, and is simultaneously restrained by REM tension. When the system is heated, does the piston move up, move down, or remain stationary? At equilibrium, the forces are balanced, so the REM force is equal to the area of the piston (a) times the pressure of the gas, so f = aP. Using the equations of state, DRT(e – e᎑2) = a(nRT兾V) = nRT(a兾V )

Size (V or e) versus temperature (T) at constant action (P, f, or σ)

The latter (e versus T at constant f ) has significant consequences with respect to the ratio of heat capacities to be discussed later, therefore it deserves close attention. The form of this function can be realized by solving the equation of state for T to get T = f 兾[DR(e – e-2)] = ( f 兾DR)[e 2兾(e 3 – 1)] Since f is a constant, the term ( f 兾DR ) is a constant, so the relationship of T and e is governed by the function [e 2兾(e 3 – 1)]. The shape of this function can realized by sketching the lines e 2 versus e, and e 3 – 1 versus e. Then on


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In the Classroom 4

Force / N

3

2

1

0 1

1.5

2.0

2.5

3.0

Extension e Figure 6. Deviations from ideal REM behavior.

another graph, the ratio can be plotted by looking at the individual graphs of e 2 and e 3 – 1. Where is the ratio zero? Where is it negative? Where is it positive? Where is it large? Where is it small? Where is it infinite? What is the limiting form when e is very large? These ideas help to see the form of the e versus T plot at constant f, as shown in Figure 5. It is also useful to consider the above results qualitatively in terms of the equation of state, physical measurable or observable changes that occur in the system, and common sense. Also note what happens when the “constant” parameter varies. For example, how does force f on a rubber elastic material depend on its elongation e at constant temperature? What happens if that temperature is higher or lower? This can be determined by comparing the above graphs and the equation of state with common sense and experiences. Non-Ideal Behavior Usually, non-ideal gases are introduced by some discussion of intermolecular potentials, attractive and repulsive forces, dissociation, and so forth. These discussions are related to the macroscopic behavior of gases in terms of their microscopic structure. Deviations from ideality also exist in REM (see Figure 6) and, like non-ideal gases, these can also be explained in terms of microscopic structure of the system. In addition, this lays the groundwork for concepts of conformation, configuration, and entropy. Specific types of non-ideality include three types (11). Non-ideality for both REM as well as gases can result from Joule’s law violation. In REM this occurs at large extension (typically e > 8), when REM fails to correlate ideally with the Guth–James equation of state due to crystallization, restricted bond rotation (for example, due to steric effects), bond angle distortion, or bond stretching. The result is higher-than-predicted force at high elongation. After bond rotation has provided for the maximum end-to-end length possible, other modes of extension may come into play. These include bond angle bending, bond stretching, and bond breaking. Each in turn represents orders of magnitude additional energy requirement. Thus at high elongation, the force 1450

required for further REM extension rises dramatically. This can easily be felt by stretching a rubber band to its “elastic limit”. This is analogous to compressing a gas to such a high density that intermolecular potentials significantly affect its behavior. In both REM and ideal gas cases, the energy of the system becomes dependent on its size, Joule’s law is violated, and non-ideality is observed. Other types of nonidealities of REM, not analogous with gases, include “creep”, “set”, and “stress decay”. “Set” or hysteresis, seen at low elongation when recovery from stretching is incomplete due to permanent deformation of the material. Misalignment, entanglement, or variations in REM morphology (such as crystalline/amorphous structures), flaws in the material, or incomplete or defective crosslinking are seen as viscoelastic nonidealities, that is, “creep” or “stress decay” in real materials. Creep is a slow continuation of elongation when REM is held under constant force and temperature. Stress decay is a slow decrease in force when REM is held at constant elongation and temperature. Barometric Distribution Law Analog The behavior of an ideal gas in an isothermal atmosphere in a gravitational field has a direct analog in the mechanical problem of a REM hanging from a rigid support under the load of its own weight. For the ideal gas of molecular weight M, the ground-level pressure is Po, and gas is contained by the weight of the gas above it. As altitude increases, the pressure decreases so that the drop in pressure equals the decrease in weight of the gas above, dP = ᎑␳gdz, where P is pressure, z is height, g is the gravitational field constant, and ␳ is the gas density. Substituting the gas density ␳ = MP兾RT and integrating pressure from Po to P and height from 0 to z, the pressure as a function of height is P = Po exp(᎑Mgz兾RT). Similarly, at the bottom of the REM, there is no tension and no elongation because there is no weight hanging below that point. At higher positions, the weight of REM hanging below that point causes increasing tension, thus increasing REM elongation. When the REM is elongated, the local linear density decreases. The tension (upward force) at any point is given by the equation of state f = DRT(e – e᎑2 ). Force and elongation change with position. The unstretched linear density is D (taken in this example to be unity), and the local (variable) linear density is d = D兾e. The temperature is constant. The incremental weight added in a element of the REM at position z is dw = (gd )dz, where g is the gravitational constant. The increment of tensional force change due to this added weight is df = ( ∂f兾∂e)T de. This increment of tension equals the increment of weight. So, setting df = dw (∂f兾∂e)T de = (gd )dz DRT(1 + 2e-3)de = gddz Separating the variables and integrating from 0 to z for height, and from 1 to e for elongation, gives the functional relationship between the position and elongation. z = (RT兾g)(e 2兾2 – 2兾e + 3兾2) Although this can not be easily solved explicitly for e as a function of z, it is easy to compute values of z, d, and f for various values of e. Once known, various plots can be made


In the Classroom

Table 5. Elongation, Position, Density, and Tension in Free-Hanging REM

A very long REM hangs from a rigid support in a gravitational field

Tension/N

1.0

0

1.000

0

1.5

219

0.667

1755

2.0

424

0.500

2910

2.5

649

0.400

3891

3.0

905

0.333

4803

3.5

1197

0.286

5684

4.0

1527

0.250

6548

4.5

1897

0.222

7400

5.0

2308

0.200

8248

5.5

2759

0.182

9091

6.0

3252

0.167

9931

as shown below for tension, elongation, and density versus z. Table 5 shows values for e, z, d, and f, and plots are shown in Figure 7. It is also possible to develop a conceptual polymer solution analog to the barometric distribution law (but it is imperfect due to buoyancy effects). In that analog, the gas concentration as indicated above by pressure P = Poexp(᎑Mgz/RT) is conceptually analogous to concentration (C) gradients in centrifuged polymer solutions C = Coexp(᎑Mgz/RT ). This is easy to demonstrate by observing the viscosity of centrifuged polymer solutions like the PVA mentioned previously. The centrifuge is necessary to make g large enough to develop an observable effect. Other Analogous Situations Many alternative problems comparing REM and ideal gas can be formulated to improve the breadth of students’ experiences in thermodynamics. Several of these will be presented in Part II of this paper (1). In addition to those presented there, many other opportunities are possible. For example, the study of thermodynamics involves the kinetic theory of gases and, in particular, Maxwell’s distribution of molecular velocities (or energies) in a gas. Distribution functions have wide applicability, and a useful supplemental example of a distribution function is polymer molecular weight distributions (22). REM heat capacity at constant force (Cf ) and at constant length (Cl ) are analogous to ideal gas heat capacity at constant pressure (CP ) and at constant volume (CV ). The constant-size heat capacity reflects the kinetic energy of motion within the system. These are conceptually related to the Einstein equation that is presented in some texts to explain the increasing heat capacity of gases as increasing temperature activates more and more modes of motion in the gas molecules. For polymers, that conceptually includes in particular the segmental mobility of monomer units in the pseudo-liquid crosslinked polymer structure. Further information on heat capacity is covered in Part II of this paper (1).

Tension

z

Weight

z=0 6

Elongation of REM

Local Density/ mol m᎑1

5

4

3

2

1 0

1000

2000

3000

2000

3000

2000

3000

Height / m 10000

Tension of REM / N

Height (z)/m

8000

6000

4000

2000

0 0

1000

Height / m

Relative Linear Density of REM

Elongation (e)

z=h

1.0

0.8

0.6

0.4

0.2

0.0 0

1000

Height / m Figure 7. A free-hanging REM stretches under the load of its own weight.


1451

In the Classroom

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Wiley and Sons: New York, 1978. 14. Jefferson, A.; Phillips, D. N. J. Chem. Educ. 1999, 76, 232– 236. 15. Thompson, W. Mathematical and Physical Papers, 1st ed.; C J Clay and Sons: London, 1890. 16. Kargon, R.; Achinsrein, P., Eds.; Baltimore Lectures on Modern and Theoretical Physics–Historical Perspectives, MIT Press: London, 1987. 17. Joule, J. P. The Scientific Papers of James Prescott Joule, Physical Society of London, 1887, reprinted by Dawson’s Hall of Pall Mall; London 1963. 18. Casassa, E. Z.; Sarquis, A. M.; Van Dyke, C. H. J. Chem. Educ. 1986, 63, 57–61. 19. Hill, T. L. Thermodynamics for Chemists and Biologists, 1st ed.; Addison-Wesley: Menlo Park, CA, 1968. 20. Beck, K. R.; Korsmeyer, R.; Kunz, R. J. Chem. Educ. 1984, 61, 668–670. 21. James, H. M.; Guth, E. J. Chem. Phys. 1943, 11, 455–461. 22. Ward, T. C. J. Chem. Educ. 1981, 58, 867–869.


Using Rubber-Elastic Material-Ideal Gas Analogies ... - ACS Publications

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