Leo Mandelkern Department of Chemistry and Institute of Molecular Biophysics Florida State University Tallahassee, 32306
Macromolecular Principles in Teaching Undergraduate Physical Chemistry
Macromolecules are now recognized as an important class of substances whose function and properties are vital to the sustenance of mankind and to its matekal comfort. Although studies of polymers are rapidly losing what might be termed second-class citizenship in the world of chemical compounds students are still rarely afforded the opportunity to learn about them, except at advanced levels at a few institutions. The reasons for this apparent paradox have been recently discussed ( I ) . I t is an important matter which must be resolved in the very near future. It is not the purpose of the present paper, however, to argue the pros and cons for the development of formal courses in ~ o l v m e rchemistw at the undergraduate level. We would rather like to focu; our attention on a quite different matter which apparently has not been generally recognized. In the development and maturing of polymer science some very important principles have been established, which are not restricted to polymers, hut have applications to many other areas of chemistry. New concepts were introduced and older ones generalized.~Studies of polymers can provide a more general understanding and keener insight into certain portions of . nbvsical . chemistrv. Our maim obiective is to show that not only do these concepts make specific principles clearer, but they are demonstrably more interesting and stimulating to the undergraduate student. A large number of examples are available, and a selected few will be discussed in detail. I t is important that we recognize that these principles can be introduced in a very natural way without the sacrifice of any rigor. I t is not necessary to set aside any traditional subjects, nor to artificially introduce sections on macromolecules and by necessity, a t this level, superficially describe their properties. We hope that someof the ideas and examples presented will be used b y others in the teaching of the firstcourse in physical chemistry. We have deliberately excluded consideration of solutions which are discussed elsewhere in the Symposium (2). Problems in chemical kinetics have been omitted because of space and time limitations. General Themodynamics When polymeric systems are included in the discussion of the First Law generalized sets of intensive-extensive variables are automatically introduced rather than being limited to just pressure-volume. These variables, such as force-length, surface tension-surface area, are sometimes mentioned in passing, hut are rarely used in text examples except perhaps as advanced problems. Thus in an operational sense, the formulation of the Thermodvnamic Laws is usuallv cast in terms of P-V variables. students are thus restrictedto examples involving gases and simple liquids. Tbis procedure is restrictive in terms of learning and interest. Moreover, because of constant usage and human frailty serious error can develop. Although most often the correct condition is stated initially, the idea that AH = q,, i.e. for a process carried out at constant pressure the heat absorbed is equal to the change in enthalpy, is carried along. This is, of course, not correct unless one only considers pressure-volumework variables. This is not a matter of practical concern unless one is dealing with reactions occurring on fibers or membranes subiect to external forces. ~any;mportant reactions, particulahy of biochemical and biolo~icalinterest, are in this cateeorv. In these cases the enthalG tables do n i t directly y i e ~ d t h kheat changes. Appro~
priate corrections must be made for the other work terms and when not, serious errors have been known to be made. A Carnot cycle is a very convenient and U S P ~ I Idevice ~ to pr~~\~ideadefinifiveexample(,f the limitatiunsof the First Law A d also serves as a very good introduction to the Second Law. The usual working substance is an ideal gas, or perhaps some other gas, and one obtains a closed loop in p-V space because the adiabatics of the working substances are steeper than the isuthernmals. An ideal gas is used because it3 equation of state simplifies calculafions. Pedagogicdly, this is restrwtive since the same cycle and principlescan be demonstrated with a rubber hand, using force-length variahles, since the adiabatics are also steeper than the isothermals in this case (. 3 4 .) . Moreover, thianalog of a steam engine can be illustrated with a rubber band operatine a t a phase transition (see below) a t two different temperatires (3).At the expense of some'redundancv, the . point is made that a Carnot cvcle does not re.~ quire a gas as a working substance and that piysical chemistry is not based upon the properties of gases. The properties of rubber-like substances also yield interesting and illuminating consequences of the Second Law. They illustrate the relationship between thermodynamic formalism, macroscopic properties, and give clues to molecular structure. Gough (5-6) observed in the early 1800's that rubbers possessed thermal properties that were different from other substances. He found that on rapidly stretching (adiabatic) a rubber band the temperature rises. This phenomenon can be demonstrated very simply by rapidly extending a rubber hand and immediately placing it to one's lips. More sophisticated measurements show that there is about a ten degree temperature rise for a 4-5 fold extension. I t follows, therefore, that there must be a decrease in entropy with extension (6). From this observation and the Second Law a unique thermal expansion coefficient is expected and is observed. A normal thermal expansion is displayed by virtually all simple suhstances and by underformed polymers. However, stretched rubbers possess negative thermal expansion coefficients, contraction being observed on heating, on . expansion . cooling. These two observations are self-consistent as indicated by the followine analvses. Thev also teach us that deformation of rubber-lice substances is unique in that the entropy decreases. The chanee in the Gihbs Free Enerw. --. in terms of the independent variables, pressure, temperature, and length, can be written as (6) dG
= Vdp
- SdT + fdL
(1)
so that
from which we obtain the appropriate Maxwell relation
Thus the thermodynamic equation of state for elasticity becomes
~~~
Tbis expression is completely analogous to the equation of state of a gas which is the one usually presented in textbooks. Volume 55, Number 3, March 1978 1 177
The neeative thermal expansion coefficient is thus exoected ) ~positive. ,~ since the modulus ( a f b ~ is From these very simple experiments, which are easily experienced by a student, some very important principles can be demonstrated. These include the properties of partial derivatives, as used in thermodynamics, cross-secondderivatives and Maxwell's relations, and the origin of force, or tension develooment. in a formal hut exact sense. rather than merelv a mechanical concept. This latter point is very important for students who studv or related subiects (7).These ..nhvsioloev . observations clearly point outthat there mustbe a difference in molecular structure between rubbers and other substances. However, we should note, with some humility, and as a lesson in scientific endeavor, that about a century elapsed between the observations of the unique thermoelastic qroperties of rubber and a molecular understanding of its elastic prooerIn addition to the usual analysis of equilibrium between two states, which leads to the Clapeyron equation, macromolecular systems in phase equilibrium can be subject to a uniform tensile forceat const& pressure. It can be shown under these conditions that (8-9)
Here A S and AL are the changes in entropy and length that occur uoon transformation of the entire fiber a t constant T. P and f: For this process we note that As=
-
AF-fAL
Equation (9)is analoeous to the Clapevron equation and there is icorrespondence between -f withp, and V with L as intensive-extensive variables. One can derive theoretically and demonstrate cxpcrimentally sets ot coexistence nlrves fur crvitul-liqu~dequ~libriu~n when the force is plotted against e the lengthat constant temperature (8-9). ~ h e s correspond to coexistence curves for vapor-liquid condensation. As a consequence of this type phase transition, large tensions can he developed a t constant length or anisotropic dimensional chanees observed at constant force. These chanees will occur sharply because they accompany a phase transition and will he very sensitive to variation in temperature and chemical environment. This mechanism of tension development and shortenine can serve as the basis for natural and synthetic mechanoc~emicaldevices (8-11). Since phases can exist in equilibrium when suhject to a complete set of intensive variables, the Gibbs Phase Rule is best expressed, even in elementary presentations, as
-
F=C-P+I (10) Here I is the number of intensive variables that describe the system. Usually I is replaced by 2 giving the erroneous impression that temoerature and oressure are the onlv intensive iariables of concern. ~ i s t a k e s a r obviously e avoided by first considerine" the most eeneral exoression. which can be verv " useful in analyzing problems involving polymeric fibers; ~articularlvthose involvine ohase transitions (12). One of the most challen&g problems that faces society today is the conversion of free enerev -.sources into useful work. ~rreipectiveof any practical applications that can he made, macromolecular systems effectivelv illustrate the princioles involved and the nnalyses th-t nee2 tu I I made. ~ The devices that have tren dcicr~hed,and which operate qdically, divide 178 1 Journal of Chemical Education
into non-isothermal and isothermal categories. We have discussed the basic Carnot cycle in which a rubber-like substance, which does not undergo a change of state, serves as the working substanre. , \ r t ~ a ldevicesbased on this pri~~ciplv are found in the pendulum described hy Wiecnnd (1.3, and on a wheel, whosespokes are stretched rubber bands, and one side of which is heated (4,14).In these devices the center of gravity is displaced on heating and returns to its initial posi60n on cooling. Thus advantage is taken of the thermal expansion coefficient of rubber-like substances. In another non-isothermal example, the working substance is a fibrous macromolecular svstem. It ooerates between two temneratures so that a cooperative phase transition takes place between the hiehlv ordered cwstalline state and the disordered liquid state:^ large anisotropic diminution in length occurs whch is reversed on cooline bv virtue of covalent intermolecular cross-links introduced inthe highly ordered state (8,11,15). A similar type of phase transition can he regulated isothermally hy cl;e~~nica~intrm
