Computer Program Correction
To the Editor: I wish to bring to the attention of those who requested a copy of the computer program referred to in the article which appeared in the December, 1968 issue of THIS JOURNAL (p. 767) the following correction: The tenth card shondd read as follows: PS12S = 2.06*RaFACT2
The author apologizes for this inadvertent error in the list,ing. No additional copies of this program are available for distribution though other programs are planned a t a later date.
It should also be pointed out that pure water boils considerably below 91°C on Pike's Peak. If the atmosphere is assumed to be isothermal, the barometric pressure at altitude h is given by P = Poe-M'h'nT. With Po = 760 torr, M = 28.8, T = 1 5 T , and h = 14,000 ft (just below the summit of Pike's Peak), this equation gives P = 460 torr. At this pressure, water boils at 86.5% However, an isothermal model does not represent the atmosphere accurately over an altitude range of 14,000 ft. A better simple model is an isenthalpic/isentropic atmosphere of constant heat capacity, for which P CP log - = - log 1 Po R
(
With Po = 760 tom, T o = 20°C, M
Egg Boiling
To the Editor: Trotman-Dickenson has recentlv challenged a textbookstatement that "atop ~ i k e ' sPeak, where water boils at 9l0C, it would take 12 hr to hardboil an egg (without a pressure cooker). . .about 10 min are required a t 100" ," stating that it takes less than twice as long to boil an egg a t 91°C as at 100°C [J. CHEM. E ~ u c . 45, , 537 (1968)l.
534
/
Journcrl of Chemical Educofion
=
28.8, and
CI.=
5 R, the air pressure at 14,000 ft for this model at2 mosphere is 303 torr, and the boiling- temperature of water is 76.2'C. To check these theoretical methods, calculations were made with h = 6500 ft: for the isothermal model. P = 600 tom, T, = 9 3 . 5 0 ~ ;for the isenthalpic/isenkopic model, P = 565 torr, T B = 92.0°C. A telephone call to a home economist who owns one-fourth of a cabin a t Lake Tahoe, California (altitude about 6300 ft), elicited the information that tap water there boils at 93'C (one can reconcile this datum with the isenthalpic/isentropic model by invoking a slight amount of pollution) and that a 20-min egg a t Lake Tnhoe is equivalent to a 12-
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
