Reino W. Hakalal Howard University Washington, D. C.
Dimensional Analysis and the Law of Corresponding States
Physical chemistry students are ordinarily taught the applications of dimensional analysis to the checking of equations resulting from derivations, and to the conversion of nnits from one system to another. But these are not the only important applications of dimeniional analysis. The application of dimensional analysis to the theory of models has long been an important part of the intellectual equipment of physicists and engineers, particularly civil and mechanical engineers, and would also be of considerable value to physical chemists. The present article is intended to serve as an introduction to those aspects of dimensional analysis which are useful in the theory of models. To illustrate the basic technique that is involved, we shall employ i t to derive the ideal gas law, the law of corresponding states, and the virial equation in reduced form. Fundamenbls of Dimensional Analysis What is actually meant by a dimension has been the subject of much philosophical speculation. (See, for example, the discussion by Bridgman in his "Encyclopaedia Britannica" article on Dimensional Analysis.) The present author finds it useful to regard a dimension as a generalization of a unit of measurement. That is, the dimensions of a physical quantity are concerned with the nature and not the amount of the quantity. For example, Angstron~s, centimeters, inches, miles and light-years are all nnits expressing "length-ness." The nature of the physical quantity is expressed in din~ensionalanalysis by means of a dimensional formula which shows how the given quantity is related to various other physical quantities which are regarded, for one reason or another, as being fundamental. To avoid any unnecessary and possibly misleading philosophical connotations, the present author uses the name dimmsional variables for the quantities which are assumed to be fundamental. Although the choice of dimensional variables is conlpletely arbitrary in a mathematical sense, mass ( m ) , length ( I ) , time (1) and temperature (8) are usually chosen for convenience. (There is no standard notation. The syn~bolsfor time and temperature are sometimes interchanged, and capital letters are often used.) Various other quantities have been regarded as dimensionally fundamental, but we shall not need to consider any of these. (Interested readers can refer to the excellent section on Units and Conversion Factors a t the end of the "Handbook of Physics," edited by Condon and Odishaw.) The easiest way to derive the dimensional formula for a given physical quantity is from any one of its units. By way of illustration, we cite the familiar Present address: SyracuseUniversity, Syracuse, N. Y., 13210.
380
/ Journal of Chemical Education
dimensional formula of velocity, it-', which is derivable on sight from any one of the units of velocity, such as cm sec-', mile hr-', or even megaparsec century -'. It may be mentioned a t this point that the ancient Greeks (perhaps not all of them) were aware that the dimensions of volume are 13. A physical eqwltion, such as the van der Waals equation of state, for example, is a mathematical relationship which connects various physical quantities. If all of the terms in the physical equation have the same dimensions, the equation is said to be dimensionally homogeneous. Physical equations which are not dimensionally homogeneous can arise in either of two ways: by adding together two or more dimensionally homogeneous equations of different dimensions; or by omitting a proportionality constant which is not dimensionless, known as a dimensional constant, such as the gas constant, R. Equations obtained the first way are obviously not fundamental and are in fact ridiculous. Those obtained the second way may he fundamental but are not complete. A properly formulated physical equation is dimensionally homogeneous because, as the familiar saying goes, "you can't add apples and oranges and get all oranges or all apples." A word of caution is necessary here: it sometimes happens that two physical quantities, the addition of which would be nonsensical, formally possess the same dinlensions. If each term of a dimensionally homogeneous physical equation is divided by any one of the terms, each term of the resulting equation will evidently be dimensionless; that is, the dimensional formula of each tern1 of the resulting equation will be n ~ ~ l " t ~ 8 ~ . If all of the terms in the original equation are linear terms, then all of the terms in the resulting equation will be di?nasionless power products. (If A, B, C, . . . represent physical variables, a power product of these variables is denoted by AaBbCc. . . where a, b, c, . . . are pure numbers of any magnitude and of either sign. A familiar example of a dimensionless power product is the compressibility factor, P V I n R T , or, in the above notation, PVn-'R-IT-' .) It is the fundamental theorem of dimensional analysis that any dimensionally homogenous equation, whether linear or not, can be reduced to a form in which the mathematical variables are all dimensionless power products of the variables in the original equation. A number of rigorous proofs of this theorem have been given, but they are all somewhat abstruse. That such a reduction is always possible should, however, become clear later on. (Every physical chemistry student, hopefully, has carried out such a reduction of van der Waals equation.) This theorem has two very important consequences. First, the reduced equation is independent of any
particular system of units. Second, the reduced equation contains fewer mathematical variables, the dimensionless power products, than did the original equation, for there are fewer mathematically independent dimensionless power products of a set of physical variables than there are physical variables in the set. For instance, the physical variables pressure, volume, number of moles, the gas constant, and absolute temperature constitute a set of five members, yet they can be combined into a single dimensionless power product, PVn-'R-'T-'.
products. We shall now learn how to find such a set. (The mode of presentation is to a large extent original with the present author.) Formation of a Complete Set of Dimensionless Power Products First of all, it is necessary to have a complete set of the important physical variables. Theory, even if not accurate, is helpful here. Suppose that the important physical variables are A, B, C, . . . . Then each of the dimensionless power products will be of the form
Fundomentals of the Theory of Models This reduction to a smaller number of n~athen~atical variables is of practical importance because an eqnation of four or more variables cannot be plotted in the form of a single curve or even a family of curves, but if it can be reduced to a form containing, say, only three mathenlatically independent dimensionless power products, it can then be plotted as a family of curves, also known as a chart. It is not necessary to know the physical equation to set up a dimensionlesspower product chart. Experimentally determined values of the various dimensionless power products can be plotted to form the chart. If all of the important physical variables have been accounted for, and all possible mathematically independent dimensionless power products are formed from these, then the chart of the dimensionless power products must haue universal validity even {fits equation is unknown. A dimensionless power product chart is useful for the comparison of the characteristics of widely different physical systems because a single value of a particular dimensionless power product can result from very wide ranges of values of the physical variables which constitute the given dimensionless power product. Consider PVn-'R-IT-', for example: It has the same numerical value, for a constant mass of material, at (1 atm, 1000 ml, 300°K), (lo-' atm, 10" ml, 30,000°K), and (lo7 atm, ml, 3'10. Engineers use dimensionless power product charts for the scaling-up of models. This application is central to the theory of models. Experimentation with models is considerably less expensive than experiments on the "real thing." Dimensionless power product charts are developed empirically on the basis of the behavior of small-scale models. Then the predicted behavior of the "real thing" can be read off the chart. I n many engineering situations, such as the design of aircraft, the systems involved are so complicated that no adequate physical theory exists. The theory of models is then the only possible approach. As was indicated earlier, if all of the important physical variables have been thought of, and if all possible mathematically independent diiensionless power products have been formed from these physical variables, then the procedure of the theory of models is no less valid than is the use of equations based on accurate physical theories. (Airplanes fly faster, ships sail more smoothly, and highways stand up better under heavy traffic-all because the theory of models is valid.) In order to develop a dimensionless power product chart, or a family of such charts, if need be, it is necessary, as was mentioned earlier, to have a complete set of mathematically independent diiensionless power
=
AaBbCo
_ ..
where a represents a dimensionless power product, and a, b, c, . . . are any pure numbers, integral or fractional and positive or negative. We shall denote the dimensions of a quantity other than mass, length, time and temperature by the name or the symbol of the quantity enclosed in square brackets. Thus, the dimensional equation for the various dimensionless power products is [TI = [ A I o [ B ] ~ [ C.].= . We now replace [ A ] , [ B ] , [Cl, . . . by the equivalent dimensional formulas in terms of m , 1, t, and 8. Suppose that these formulas are given by the dimensional equations [ A ] = mdOtW [ B ] = m4O'tV@6' [Cl
...
=
rn="lO"W@n
where ar, p, y, 6, or', p', y', a', a", p", y", 8"). . .are pure numbers. Then, upon substitution, we obtain IT] = ( m ~ 1 0 t ~ @ ~ ~ ( m o ' l B ' t 7 ' 8 a ' ) h ( m = r 1 B. ." 1 ~ " ~ 6 . = nz(wz+~'b+="c+ . . . ) 1 ( B a + B'b + B"c+ . . .) X t(ro+7'b+rxc.. .) e(6a+6'b+6"e+ . . .) By
''T
is dimensionless," we mean that
[TI
=
rn0lot%'
Equatimg the powers of m , 1, t, and 0 in the last two dimensional equations results in the following set of exponenGsum equations: rn: ma + a'h + m"e + . . . = 0 B"c + . . . = 0 1: @a+ B'b t:
+ ra + r'b + rvc + . . . = 0
0:
6a
+ 6%+ 6"c + . . .
=
0
This system of exponent-sum equations is to be solved simultaneously in order to find values for a , b, c, . . . If one or more of the exponent-sum equations is a linear combination of other exponent-sum equations, then simultaneous solution will lead only to an identity. To obtain a useful result, it is necessary that each of the exponent-sum equations employed be linearly independent. We therefore need a criterion of linear independence. I n discussing this criterion, it will he convenient to refer to the matrix of the coefficients of a, b, c, . . . in the exponr~~t-sum t!quntions a tht! dimmsirmnl inal,%r. It ma\. hc recalled thut \\.hen a nin of 3 drtwminant " is a linear combination of any other rows, the value of the determinant is zero, and conversely. It follows that if the exponent-sum equations are not linearly
.
~~~
~
Volume 41, Number 7, July 1964
/
381
independent, then all possible determinants, of order equal to the number of rows in the dimensional matrix, formed by selecting columns from the dimensional matrix, will be equal to zero. As a corollary, if even one of these determinants is nonzero, then the exponentsum equations are linearly independent. Consequently, the number of linearly independent exponentsum eqtiations i s equal lo the order of the nonzero determinant of highest order obtainable from the dimensional matrix by selecting rows and columns. This number is known as the rank of the dimensional matrix. The number of exponents a, b, c, . . . minus the number of linearly independent exponent-sum equations is equal to the number of independent sets of values of a, b, c, . . which will satisfy the exponent-sum equations. Since there are as many exponents as there are physical variables assumed to describe the physical system completely, and since each set of values of a, b, c, . provides one and only one dimensionless power product AnB'CC . . . , it follows that the number of muthe~wtically i~td~pendentrli~nr~~siimless pouvr nnrlfrrtai n u comnlrte set 7,~~~111nl10 the nurnbrr ofvhmicul " variables minus the rank of the dimensional matrix. This is the basic theorem in the applications of dimensional analysis to models of physical systems. It supersedes the "T-theorem" of Buckingham, according to which the total number of independent dimensionless power products in a complete set is equal to the number of physical variables minus the number of dimensional variables, which is not always true, as was first pointed out by Bridgman. The first proof of the correct theorem was given by Van Driest in 1946 and independently at around the same time by several other authors. The proof presented above is different from and simpler than, though equivalent to, that of Van Driest. It resembles, but is simpler than, a proof due to Langhaar. The fundamental theorem of dimensional analysis, that any dimensionally homogeneous equation can be put into reduced form, is a corollary of the above basic theorem of the theory of models, which really has to do with the general limitations on the form of a dimensionally homogeneous equation. The two are often lumped together as a single theorem, sometimes known as the "corrected T-theorem." The algebraic development in this section, while used to provide a proof of the basic theorem of the theory of models, is not needed to find the exponent-sum equations and the matrix of their coefficients. The same matrix is obtained by tabulating the numerical values of the exponents (ar, 0, y, 6, a', P', y', sf, a", P", yU, a", ...) occurring in the dimensional formulas of the physical variables. The physical variables are listed a t the top of the table, the dimensional variables at the left, and the values of the exponents in the body of the table. Each column thus correspouds to a dimensional equation and each row to an exponent-sum equation. It is also not necessary to solve the exponent-sum equations simultaneously, as the values of a, b, c, . . . can be found by iuspection of the dimensional matrix. It is, of course, necessary to find the rank of the dimensional matrix, for only in this way can the correct number of independent dimensionless power products in a complete set be ascertained. To find the rank of the dimensional matrix, it is
necessary to evaluate determinants, to see whether or not they equal zero. One of the most convenient ways to find the value of a determinant is to expand it in terms of minors. It may be recalled that the minor of a n element is the determinant formed by deleting from the original determinant the row and column in which the element occurs. The original determinant is equal to the algebraic sum of the producta formed by multiplying together the elements of any column or row of the determinant by their minors, the sign of each product being taken as either plus or minus depending upon the location of the element in the original determinant in accordance with the following checkerboard pattern:
.
. .
".
382 / Journal of Chemical Education
Another convenient way to evaluate a determinant is to transform it into either one of the triangular forms: a, az
as
...
0 0 0 bl 0 0 ba c3 0 bd
c4
...
b~ ct dl b~ ca dl
a,
or
d4
0 0 0
.. .
.. .
0 ca dz O O d 4
The original determinant is equal to the product of the elements along the principal diagonal, a,bzcad4. . . , of either one of the determinants in triangular form. This transformation is readily accomplished by means of the theorem that any multiple of the elements of any column (or row) can be added to the elements of any other column (or row) without changing the value of the determinant. We shall now illustrate these general principles by applying them to the particular case of a real gas. Application to a Gaseous System
For an ideal gas, the important physical variables are known to be the pressure, volume, number of moles, and the absolute temperature of the gas. These are related inaphysical equation, the ideal gaslaw, to a dimensional constant, R : PV/nT
=
R
The units of R depend on the units chosen for P , V, and T . I n dimensionless form, the ideal gas law is: PVInRT
=
1
I n the case of a real gas, however, PV/nRT is not a constant but depends upon P and T and on the various molecular parameters of the gas. The latter are related to the critical constants of the gas, whence PV/nRT is a function of P,T and the critical constants. We shall determine the general nature of this function by meam of dimensional analysis. As before, the dimensions of the assumed fundamental qnantities mass, length, time, and temperature shall be denoted by the symbols m, 2, t, and 8, respectively, while the dimensions of other physical quantities shall be denoted by the names or the symbols of the quantities enclosed in square brackets. Pressure is a force divided by an area, and energy is given by the product of a force and a displacement, so that we shall have to consider the dimensions of force. We can get this from the knowledge that force is mass times acceleration, acceleration is the rate of change of
velocity, and velocity is the rate of change of position. We thus have the following system of dimensional equations: [velocity] = [length]/[time] = 11-' [aceelerrttion] = [velocity]/[time]= 1 f P [force] = [mass][acceleration] = mWa [pressure] = [force]/lareal = mltP/l2 = m1-ft-P
[energy] = [force][distance] = rnlPtP
Sometimes it is a bit tricky to determine the dimensions of a given physical quantity. For example, we might, offhand, think that the number of moles of a substance is a dimensionless quantity. This is not the case. The numher of moles is found by dividing the number of grams of tbe substance by its molecular weight. Thus the dimension of the number of moles is given by the dimension of the number of grams, namely, mass, divided by the dimensions of molecular weight. The latter quantity, being a relative mass (twelve times the mass of a molecule divided by the mass of a carbon 12 atom), is dimensionless. Therefore the dimension of the number of moles is mass, [n]
=
m
The units of the gas constant R are, for example, lit atm deg-'mole-', whence its dimensions are given hy
The molar volume a t the critical point, VG,does not have the dimensions of volume alone, but of volume divided by the number of moles of gas: [V,] = [volume]/[moles]= lam-' I t is convenient to tabulate the dimensional equations for the physical variables P, V, a, R, T, P,, I , 7', which we consider to be important: T' n R T P. V, Tr P nr
1
1
-1 -2
1
e
I
0
0 :i 0 0
0 2
l 0 0 0
-2
0 0 0
-1
I
1 -I -2 0
-1
:i
0 0
0 0 0 1
The physical variables are listed a t the top of the table and the dimensional variables a t the left. Each column of this dimensional matrix, it may be noted again, is equivalent to a dimensional equation. The numerical entries in a given column are the powers of the dimensional variables in the dimensional equation for the physical variable whose symbol appears a t the head of that column. We recall that this matrix is also the matrix of the coefficients of the exponentsum equations, each row correspondiig to an exponentcsum equation, and that the rank of this matrix is therefore equrtl to the numher of linearly independent exponent-sum equations. We shall now determine the rank of the dimensional matrix. The determinant of the first four columns of thc dimensional matrix is 1 -1
-2 0
0 3 0 0
1 0 0 0
0:
i 1(
The choice of columns is largely arbitrary. Any four nmidentical columns (the columns of P and P. are identical, as are those of T and T,) may be chosen. Four columns must be selected as there are four rows in the dimensional matrix, and a determinant always contains the same numher of columns as it does rows. Identical columns cannot he selected because a determinant having two identical columns equals zero; since columns and not rows are involved in the vanishing of such a determinant, this does not prove that the rank of the matrix is less than the numher of rows in the matrix. We are to determine whether the determinant of the first four columns is equal to zero. We shall evaluate it by expansion in terms of elements in the same row or column and their minors. In this way we find that
Since the determinant does not equal zero, and its order is four, the rank of the dimensional matrix is four. Thus the four exponent-sum equations are all mathematically independent. (If this determinant had been found to he equal to zero, another determinant of the same order would have been evaluated also. If all determinants of order four were found to vanish, then at least one of the equations could be obtained by linear combinations of two or more of the other equations, whence it would have been necessary to eliminate a t least one of these equations from the set.) We thus have four independent equations and eight, unknowns, therefore 8 - 4 = 4 degrees of freedom. That is, there are four diierent, mathematically independent sets of values of the exponents of the physical variables in the dimensionless power products which will satisfy the exponent-sum equations, and therefore four diierent, mathematically independent, dimensiouless power products in a complete set. We are now ready to find these four d s . An efficient and physically realistic way in which to proceed is to begin by a consideration of the most important physical variables first, then to include the rest one a t a time until they are all accounted for. Thus we shall begin by examining the submatrix
The corresponding cxponeut-sum equations arr
The solution of this system of equations is readily found to be whence r =
(PVInRT)..
Volume 47, Number 7, July 1964
/
383
The latter result could have been readily obtained directly irom the dimensional matrix by inspection. Since only one T occurs, it must he equal to a dimensionlrss constant, as must any root of a. Thus, PI'/nRT = a nniversd, dimensionless constant
We have thus derived the general form of the i d a l gas law by meam of dimensional analysis. We shall now consider P,. The only dimensionless combinations which it can form with the variables previously considered are PIP, and P,V/nRT. Adding V , to the set, we find the additional a's: VlnV,, /'V,/RT and P,V,/RT. Consideration of T , leads to four more: TIT,, PVInRT,, PVJRT,, and P,V,/RT,. An infinite number of other a's can be formed from these ten by taking products of these a's as well as products of the powers and roots of these r's. But, as we discovered earlier, a complete set of a's contain only four members. Which four n's shall we choose? As the choice is entirely arbitrary mathematically, we shall have to base our decision on convenience. The compressibility factor, PVInRT, is an important quantity, because of the ideal gas law, whence we shall include it in our set. Its value a t the critical point, P,V,/RT,, is kuown for many substances, so we shall include it also. The simplest remaining a's are PIP,, VlnV,, and T I T , of which we must omit one since the complete set contains only four members, whence any one of these r's is not mathematically independent, as is also evident from the relationship
Experimental convenience dictates the choices of PIP, and TIT,. Thus we have proved by means of dimensional analysis that, provided P , V , n, R , T , P,, V,, and T , are the only important physical variables, the functional relationship PV n i =f
P.V. P T P: 'P.)
is universally valid. Dimensional analysis cannot decide the mathematical form of the function. We can, however, proceed as follows: Let us assume that we can expand thr unknown function, with the aid of Taylor's theorem in several variables, as a power series in (Z, - Z,'), where Z, P,V,/RT, and ZCo is a constant. Letting P, PIP, and T , T I T , also, the result can br written
---
--
Suppose that we adopt some particular value of Z,' as a standard. If argon, for example, is to be the standard, then Z,' = Z, (argon), whence fu(Pn,Ta) = Z(argm)
That is, we can use the P,V,T, behavior of some reference substance such as argon as a standard. Then f?(P,,Z',J(Z, - Z,0)2, the terms ,fi(P,,T,J(Z, - ZCo), . . .are corrections to the standard P,V,T, data. Consequently, all substances hailing thc same Z, values will have the same P,-17,-T, behavior. This is the law of corresponding states in a more general form than it is nsually given. 384 / Journal of Chemical Education
AIeissner and Seferian were the first to suggest that all substances with the same value of Z, obey the same principle of corresponding states. This concept was used by Lydersen, Greenkorn, and Hougen to prepare a n extensive correlation of P-V-T data and thermodynamic properties of gases and liquids. For substances having the same 2, valne, wc may note that the result of the dimensional analysis can hr expressed as PV P T n i =f(~:
T.)
which is the law of corresponding states in its usual form. According to the theory of models, the T-chart (plotting,say, PVInRT) asa function of P/P,forvarious values of TIT,) will have universal validity for all substances having the same value of P,V,/RT,. If we expand the original unknown function, by means of Taylor's theorem, as a power series in PIP,, keeping P,V,/RT, constant, and make use of the experimental fact that lim(P 0 ) PV/nRT = 1, we obtain the uirial equation i n reduced form. Evidently, the coefficients of the powers of P/P, must be functions of T I T , and P,V,/RT,. The five-sixth powers (square roots times cube mots) of the surface tensions of numerous liquids decrease linearly with an increase in temperature. (The author has found this relationship useful in estimating critical temperatures.) The reader may wish to try putting this relationship into dimensionless form, using either P, or V , as an auxiliary parameter, in addition to the gas constant, R , and the critical absolute temperature, T,. The result will be a law of corresponding states for the temperature variation of surface tension.
-
Acknowledgmen?
The author is grateful to Dr. Erwin Fishman for his useful criticism of the initial draft of this article. The basic concepts were developed during the tenure of a Petroleum Research Fund grant a t Howard University, and the article was cast into its final form while holding a National Science Foundation Science Faculty Fellowship a t Syracuse University. Literahre Cited ( I ) AITKEN,A. C., "Determinants and Matrices," Oliver and
Boyd, Edinburgh, 1954. (2) BIRKAOFF, G . "Hydrodynamics," Princeton University
Press, Princeton, 1950 (theory and useful historical notes). ( 3 ) BRIOCMAN. P. W.. "Dimensiond Analvsis." Yale Universitv
Press, ~ e ~wa v e n 1922 , (the original treatise, now out df date, but still useful if read with care). (4) RRIDQMAN,P. W., "Dimensiond Andy+" in "Encyclopaedia Britannica," E. B., Inc., Chicago, 1953, vol. 7, p. 387 (general introduction and philosophical aspects: erroneous =-theorem). ( 5 ) CONDON, E. U., AND ODISHAW, H., Editors, "Handbook of Phvsics." McGraw-Hill. New York. 1958. ~nnendixon nits and Conversion ~itctors." ( 0 ) LANQHAAR, H. L., J. Franklin Institute, 242, 459 (1947) (theory). H . L., "Dimensional Anrtlysls and the Theory (7) ~~ANanAAR, of Models," John Wiley, New York, 1951 (best theoretirnl treatment, with engineering applications). ( 8 ) LYDERSEN, A. L., GREENHORN, R. A,, AND HOUCEN, 0. A,, "Thermodynamic Properties of Pure Fluids from thc Theorem uf Corresponding States," Report No. 4, University of Wisconsin Engineering Experiment Station, 1955 (9) MEISSNER, H. P., AND SEPERIAN, R., Chcm. Eng. P70@7.,47 679 (19.51)(correspondingstates).
