A mnemonic for Maxwell's thermodynamic relations


min egg at Berkeley, California (near sea level) [M. Bean, private communication]. Extrapolating this egg from Lake Tahoe to Pike's Peak by means of t...
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min egg a t Berkeley, California (near sea level) [RI. BEAN, private communication]. Extrapolating this egg from Lake Tahoe to Pike's Peak by means of the iscnthalpic/iscntropic model, one calculates that a 10min egg a t sea level corresponds to an 85-min egg a t 14,000 ft. It might also be relevant to note that hadboiling an egg is not primarily a process of albumin denaturation, for the rate-determining step is solidification of the yolk. While the white of an egg contains a lot of albumin, the yolk contains a lot of other stuff [XI. BEAN,ZOC.cit.]. Furthermore, chemical Binet,ics are of doubtful applicability to hard-boiled eggs. Since the yolk is usually near the center of the egg, its solidification rate is mainly determined by the rate of heat transport through the egg, which probably varies widely with egg size, shell thickness, etc. Perhaps the fluffiness of omelets is an egg parameter better suited to precise scientific investigation of the effects of altitude. CAROLER. GATZ

as long as the concentration of acid C A prevails, ~ a nearly linear part of the curve with slope S near unity is obtained. Though of no use for determining K, it is a good control of the reliance of measured absorhances. (In this region thc methods discussed by Ramette (J. CHEM. EDUC.,44, 647, 1967) give unreliable values. Because a small difference in ahsorptivities (E - EAR)is multiplied by a relatively large factor CH+ experimental errors become significant.) Gradually the slope of the curve will become smaller with increasing p H and finally tend towards zero. By drawing a straight line through any set of two neighboring experimental points on the curve (not too far from Christian's "50% point") one finds the slope S for a distinct p H value where logAE S = dd pH

This pH value will he different from the unknown pK by the distance 6 on the p H scale. Inserting pH = pK 6, and S from eqn. (3) into eqn. (2) we get

+

1 0 - i ~ K + 61 +~ O - P K

(4)

= 1 0 - i ~ K+6i

From this we derive 8 = log S - log ( 1

Graphical Method for Acidity Constants

and finally

To the Editor: A recent paper by Christian (J. CHEH.EDUC.,45, 713, 1968) describes a rapid graphical method for determining complex formation constants. With minor differences, for instance using activities instead of concentrations, I have been applying an analogous procedure for the determination of acidity constants of very weak acids (pK, > 10) with good success. There is one point worth mentioning. For the acid dissociation

pK = pH

- log S

- S)

+ log (1 + S )

Uncertainties in pK values obtained by this method are roughly half the differencebetween the pH values of the two experimental points chosen for reading the slope

AH = A - + H +

with the constant K

=

A Mnemonic for Maxwell's Thermodynamic Relations

C*-CH+/CA,,

(let me use concentrations this time to conform to Christian) my expression corresponding to Christian's eqn. (4) is

where EAHand E A - are molar absorptivities of the species AH and A-, respectively, and E is an apparent abso~ptiuity,calculated from the measured absorbance ex, the analytical concentration of acid CK+' and cell length 1by ex = ECH+"1. The logarithmic form of eqn. (1) may be written lag AE = log K + log (Ea-- EAR)- log (CH+ K) Calculating the derivative of log AE with respect to p H will give

+

10-pn dlog AE dpH lo-*" 10-2"

+

By this procedure the unknown En-will he dropped and there rests only pK to he determined. The plot of values log AE, calculated from spectroscopic measurements, against pH will give a curve like that represented as Figure 1 in Christian's paper. At smaller pH values,

To the Editor: This letter proposes a method which will facilitate the handling of the mathematical equalities among the mixed partial-second-derivatives of the thermodynamic potentials. These mathematical equalities are called Rlaxwell's relations. The thermodynamic potentials are considered to he the internal energy U , the Helmholtz free energy F, the Gihhs free energy G, and the enthalpy or heat energy H. Each of these last three potentials is a partial Legendre transformation of the function U , and so all are closely related. Each of these potentials is a function of 1 parameters and will give rise to l(1-1) pairs of mixed second derivatives, or that many Maxwell relations, where 1 is any positive integer. As examples of the Maxwell relations, consider the following

where

Volume 46, Number 8, August 1969

/

535

and

Therefore,

Similarly,

Diffusion from Gels

To the Editov: Recently J. C. Dennis [J. CHEM.EDUC.,45, 432 (1968)] proposed that the solution for the problem of diffusion of electrolytes from a gel column, of height a (0 < x < a ) , and initial concentration unity, to a wellstirred fluid a t x = 0 (zero initial concentration and height a) is

where

and

Hence

Each of the eqns. (1) and (2) is one of Maxwell's relat,ions. The Maxwell rclations may be seen to possess a certain symmetry which suggests the following method to remember them. The convention used here is to treat S and T, V and P, and N and p as pairs of conjugate variables. The extensive parameters (X) are S, V , and N; the corresponding intensive parameters (I) are T, P, and p. Then

+

where p = (2n 1 ) r / 2 a , X = Dp2, D = diffusion coefficient (I) f: D(x, C ) ) , and u ( x , t ) = concentration in gel. However this solution appears to be in error. The derivation presented by Dennis requires the solution to the following differential equation

where ru (x, t ) = u (x, t ) - v(t),and v(t) is the concentration in the well-stirred fluid. The expression given for w ( z , t ) is not a correct solution to the differential eqn (2). The correct solution for the concentration profilc in the gel is

where p cot p

= -

1.

Thc general solut,ion (dcrivntion of t,his and other solutions t,o be published elsewhere) for the case of gcl height a , and fluid height vr, and initial conccntratio~~s Cl and Cz in the gel and well-stirrcd fluid respectively, can be shown t,o be I n the above relations, one may notice that The conjugate variables, of the numerator and the denominator on t,heleft, appear, respectively, a4 t,hedenominator and numerator on the right. On d h e r side of the equation, the variable held constant is that which is the eonj~gatevariable of the nwnerxtor. R represents any other specified vaiahles, and is common on both sides. The nnmbers in parentheses index t,hc eollpled extensive and intensive variables. The relations eqns. ( 4 ) and (;)) are basically the same.

With further reference to Maxwell's relations, choose the upper sign if N or p does not appear as a differential; choose the lower sign if N or p does appear a~ a differential; but if, in addition to the presence of N or p as a differential, P or V appears as a differential, choose the upper sign. It is of interest to note that this formalism, with the exception of the sign convention, may be extended to include additional sets of variables. This set includes the electro-magnetic extensive and intensive variables, as well as the stress and strain variables.

536 / Journal of Chemical Education

u(z,t) =

wherc

+ a l m + C1

CP- CI

-

1

p cot p

= -

a/m.

We wish to thank the U.S. Air Forcc Ctmbridge Rescarch Laborat,ories for partial financial support of this work.