introduction to physical chemistry: John T. Yates, Jr.
Antioch College Yeilow Spr~ngs,Ohio
Thermodynamics, Kinetic Theory, and Statistical Mechanics
It is clearly evident that the standard undergraduate senior physical chemistry course of a few years ago has been moved forward in the curricula of many colleges and universities. Certainly a primary reason for this trend toward earlier presentation of physical chemistry is the realization that advanced courses in analytical, inorganic, and organic chemistry can be taught more effectively with students having some early training in physical chemistry. An alternative to moving the entire course in physical chemistry ahead is to extract certain basic subjects from physical chemistry and to present these subjects in an introductory course. Since physics is basic to physical chemistry, the topics in the introductory course which I am about to describe all bear a strong relationship to physics, or perhaps better to chemical physics. This course is presently offered at the sophomore or junior level to both chemistry and physics Presented before the Division of Chemical Education at the 141st Meeting of the American Chemical Society, Washington, D. C., Mt~rch,1962.
majors a t Antioch. One of the objectives is to strengthen the training of undergraduate rhemistry students in physics. The course is divided into three major sections: the thermodynamics of pure substances, the kinetic theory of gases, and elementary statistical mechanics. The design is to proceed from a macroscopic level to a microscopic level of appreciation of the properties of matter. I shall not attempt to relate in detail the sequence of topics covered in this course; instead I will mention those topics which best illustrate the structure and the unique features of this course. Thermodynamics
We begin the first section on thermodynamics with the usual subjects dealing with the description of thermodynamic systems. We consider the concept of temperature and the use of various types of thermometers. The relationship of the international temperature scale to ideal gas thermometry is emphasized. It is considered important here to show
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beginning students some of the experimental difficulties involved in precision temperature measurement.' We next consider the general concept of the equation of state of a substance, and here the van der Waals equation is introduced to illustrate an empirical attempt to describe departures by gases from ideality. The concept of thermodynamic work is next introduced, with particular care to include types of work other than expansion work and electrical work. Then coefficients of cubical expansion and compressibility for ideal and imperfect gases and particularly for solids and liquids are presented. It is then feasible to calculate for a solid or liquid the expansion work for various processes other than the usual constant pressure process. The first law is now stated and its consequences are examined. In this connection the various methods for measuring the heat capacity of a substance are discussed with particular emphasis on the Nernst adiabatic vacuum calorimeter for low temperature heat capacity studies.= Having studied the internal energy as a thermodynamic state function, we invent and employ the enthalpy function as an aid to understanding topics such as the Joule-Thompson effect, the thermodynamics of an open system such as a turbine, and the Bernoulli effect where the pressure of a flowing incompressible fluid is related to its velocity under constant energy conditions. At this point, we examine phase equilibria in one component systems, critical phenomena, and the law of corresponding states. Special emphasis is placed on Bridgeman's contribution in the area of high pressure phase equilibria studies. In this connection we discuss the P-V-T surface of water over a wide range of pressure, rather than the often used projection on the P-T plane.3 The second law of thermodynamics is then introduced as a generalization of our experience, and the equivalence of the Clausius and the Kelvin statements of the second law is demonstrated using reversible engines. This naturally leads us to the Kelvin thermodynamic temperature scale. One of the most important applications of the second law is the Clausius-Clapeyron equation which we derive both in the exact and approximate forms. Application is then made to phase diagrams and vapor pressure problems. The entropy function is now deduced in the usual thermodynamic manner and entropy changes for various reversible and irreversible processes are considered. The Helmholtz and Gibbs free energy functions are devised and utilized in appropriate fashion. One problem here which is always of interest involves the calculation of the equilibrium vapor pressure above tiny spherical droplets where the influence of surface tension is important. This subject is intimately related to supersaturation in a mist containing very tiny An excellent treatment of this subject may he found in PAEL, M. A., "Principles of Chemical Thermodynamics," pp. 1 4 3 , MeGraw-Hill, New York, 1951. * Experimental aspects of this subject are well-treated in WHITE, G. K., "Experimental Technique8 in Low-Temperature Physics," Oxford University Press, New York, 1959. a ZEMANSKY,M. W., "Heat and Thermodynamics," McGrawHill, New York, 1957, pp. 203-6.
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liquid droplets. Also, the superheating of boiling liquids may be understood quite easily when one considers the vapor pressure above the concave liquidvapor surface in a vapor bubble under a boiling l i q ~ i d . ~ I n every case where possible, we introduce the thermodynamic functions by demonstrating their particular utility in understanding various k i d s of physical problems. Finally, by using the Euler criterion for exact differentials, the Maxwell relationships for one component systems are formulated. These equations are used to derive a large number of representative thermodynamic diEerentia1 relationships. Consider the general differential expression dz
=
M(z,y)dz
+ N ( z , y)dy
If there is a function z(x,y), then dz is an exact differential and,
Now, since order of differentiation is immaterial,
which is the Euler criterion for an exact differential, dz, in two independent variables, x and y. Now the differentials of all thermodynamic state functions are exact differentials,since the state function is completely defined except for a constant of integration by the values of the independent variables, i.e., 2 =
z(x, y)
+ constant
From the general differential forms of the state functions for one component systems, where the only work done is expansion work, dE dH dA dG
= = = =
T d S - pdV Vdp 1dS -pdV - S d T V d p - SdT
+
we obtain the four Maxwell relations by cross differentiation.
The two other possible differential entropy expressions are obtained from the definitions of C , and
c,.
With these six equations, a large variety of useful thermodynamic relationships may be formulated. 'PAUL,M. A,, "Principles of Chemical Thermodynamics," McGraw-Hill, New York, 1951, pp. 274-6.
Kinetic Theory of Gases
At this point the basic principles of thermodynamics have been introduced and examined in some detail. The students have become aware of some of the types of problems which may be handled by means of thermodynamics. This is an appropriate time therefore to introduce the kinetic theory of gases, so sharply contrasted to thermodynamics in method and scope. Some of the topics which we cover are Maxwell's derivation of the velocity distribution, the use of distribution functions for averaging, intermolecular potential energy functions, molecular beams, transport properties, equations of state for gases, and the heat capacity of gaseous polyatomic molecules. The topics dealt with by means of kinetic theory are related back to thermodynamics wherever possible, and a deeper understanding of the properties of matter is therefore achieved via molecular theory. Stdistical Mechanics
The stage has now been set for the third and h a 1 major discipline to be covered in this course-dementary statistical mechanics. Here we are concerned with calculat,ing the macroscopic equilibrium properties of a system from a knowledge of the microscopic situation. We begin with classical Maxwell-Boltzmann statistics and the fundamental assumption of statistical mechanics that all conceivable microstates of an assembly are equally probable. The general expression for the thermodynamic probability of a given macrostate is then statistically formulated, the most probable macrostate is selected, and the general Boltzmann distribution function is derived. The entropy function is now connected to the thermodynamic probability of this most probable macrostate. The internal energy function can then be expressed in terms of the partition function for the assembly. At this state of the development the students are able to see the real power of statistical mechanics in relation to thermodynamics, since it is now possible in principle to calculate the magnitude of the thermodynamic state functions from a knowledge of the microscopic situation. Previously, using thermodynamics alone, only differences in the magnitude of state functions could be obtained. We apply classical statistical mechanics to an ideal monatomic gas, evaluate the translational partition function by integration, and arrive a t the MaxwellBoltzmann distribution function for molecular velocities. The same function had been obtained earlier by different methods in the kinetic theory section. An interesting problem which can be handled quite nicely a t this point is the Boltzmann distribution of particles in a force field; one particular example of this gives the barometric formula for an isothermal atmosphere. The transition from classical statistical mechanics to the statistical mechanics of systems containing diatomic molecules which have rotational and vibwtional degrees of freedom is easily accomplished if the quantum mechanical expressions for the allowed energy levels are accepted. The last lectures in this
course are therefore concerned with the calculation of the internal energy and the heat capacity of gaseous diatomic molecules and the relationship which these thermodynamic quantities bear to molecular spectroscopy. This introductory course5 forms the base upon which an additional two-term senior physical chemistry sequence is built. With students having a strong foundation in thermodynamics, the transition to chemical thermodynamics is swiftly and easily accomplished. The economy of time thus achieved in senior physical chemistry makes possible a rather thorough treatment of subjects such as statistical thermodynamics and quantum chemistry, in addition to the usual subject matter. Evaluation of the Course
What are the advantages in this approach in teaching physical chemistry? Foremost is the fact that this arrangement permits the beginning chemistry student to perceive the flavor of physical-chemical endeavors before his last or next-to-last year in college. I n addition, it is also felt that the three basic disciplines of this course are treated in somewhat greater depth than is normally possible in two terms of senior phys:cal chemistry. Finally, the introductory course produces some of the advantages which are proclaimed by those who would move the entire course in physical chemistry to the early years. At the present time there is no regular laboratory associated with the introductory course. By the time the students have completed this course they are a t a theoretical level sufficient to enable them to do rather sophisticated experimental work in certain areas of thermodynamics and the kinetic theory of gases. An elective laboratory course for chemistry and physics majors has therefore recently been instituted. At the present time the plan is to offer experimental work in areas such as: low temperature calorimetry; vapor pressure of solids by effusion studies; crystal energy of solids from the vapor pressure; the thermodynamics of physical adsorption; thermal conductivity of gases; thermal diffusion; gas viscosity. This elective laboratory program will enable students who are particularly interested in experimental chemical physics to undertake laboratory problems which are designed to precede and supplement the required laboratory in senior physical chemistry. The introductory course in thermodynamics, kinetic theory, and statistical mechanics has been offered to chemistry majors at Antioch College for the past five years, and it is a required course for all B.S. candidates in chemistry and physics. The course is taught alternately by members of the chemistry or physics departments. Chemistry majors who have taken this course seem to be more mature scientifically during their college days than those trained in the conventional fashion. q t the present we are using as a text SEARS,F. W., "An Introduction to Thermodynamics, The Kinetic Theory of Gases, and Statistical Mechanics," Addision-Wesley, Reading, Mass., . , , L., 1953. Another text with a similar approseh is ~ ~ h ~ W. H. ~ F~ ~~ sari~ ~~ h ~~~ 1962.~ ~A~ i syllabus is available upon request.
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