letters pH Range - 4 7 to +47?
To the Editor: My recent article: "Preparation of Solutions in the pH Range of Approximately -47 to +47," J. CHEM. E ~ u c . 49. . 776(1972). ~rovideseven more negative instruction '(and'p~)''tilan I had originally k e n d e d . The lowest (most acid) pH one could obtain by my procedure is not -46.8 as I indicated, but 14 - 46.8 = 32.8 (approximately -33). Also, the 100-fold dilution of the 10-6 M acid solution leads to [H30+] = M and not lo-' A4 as stated. I had planned to suggest that general chemistry teachers refer their students to this article and ask them to locate the flaw in the reasoning. n'ow they can be asked to find two flaws and a typographical error. I trust this will triple everyonc's enjoyment.
More Certointy about Uncertainty
To the Editor: The Heisenberg Uncertainty Principle is generally discussed as an atomic scale phenomenon; e.g. a nucleus undergoing exchange between two chemically and hence magnetically different sites, will show a broadening of the resonances if the frequency of exchange (in Hz) is comparable to the difference in chemical shift (in Hz). I n electronics the analogous concept is introduced, somewhat more convincingly, from the point of view of the properties of the electromagnetic wave alone. A wave of frequency 1 Hz (a cycle per second) needs the order of me second for its wavelength (frequency) and hence its energy to be determined. Similarly a band of frequencies centered at 1 kHz of width 1Hz needs the order of one second for the "beat profile" to be established and hence the bandwidth (and uncertainty in energy) t o be determined. Thus, if the recorder on the nmr spectrometer is required to have a bandpass of x Hz (i.e., resolve structure where the highest frequency component of the spectrum is x Hz) then it cannot have a time constant greater than 1/x seconds. On a more sophisticated level, the relationship between pulse shape and frequencies required is demonstrated by means of Fourier Transform pairs and the result that short pulses require a large number of frequencies and extended wave trains require few frequencies is called the principle of reciprocal spreading. A practical example is the transmission of TV pulses which must cover 525 lines on a screen, repeated 30 times per second with 360 pulses per line, and this indicates apulse length of l = 1/(525
x
30
x
and demands a bandwidth for transmission of 6mHz. The exact form of the relationship (h/2a, h/4u) depends on the definition of A in one case and bandwidth in the other. I n the operator treatment of uncertainty in quantum mechanics it is sometimes pointed out that if one wants t o write the wave function in terms of momentum (p) rather than position (x) then the relationship between the two wavefunctions is a Fourier transform. It seems that a lot of the aura of mystery about uncertainty could be reduced by starting with the simplest of explanations.
B. K. SELINGER THEAUSTRALIAN NATIONAL UNIVERSITY Box 4, P.O., CANBERRA, A.C.T. 2600, AUSTRALIA
A Theorem for Simplification of Rote Equation Derivation
To the Editor: I n the article by Donald Shillady on "A Theorem t o Simplify the Derviation of Certain Rate Equations," [J. CHEMEnuc., 49, 347 (1972)l the theorem he expounded for simplifying integrals obtained for kinetic relations may be alternatively derived (or proved) using the Taylor's expansion series. Since the Taylor series is as useful to physical chemists as the mean-value theorem is to mathematicians, this proof is presented in the following paragraph. Given a function f(x) which may be represented as the ratio of polynomials p(x)/q(x) where p(x) and q(x) have no common factors (relatively primed), p(x) is of lower degree, and q(x) = j(x) (x - a) [where (x - a) is a non-repeating factor], then
A Taylor's expansion of g(x) about the region x
=
a
=
1
gives
Thus, application of eqn. (2) to eqn. (1) gives
Equation (3) is obviously true for j(x) and may be generalized. If
=
p(x)
then it may be transformed to
360) = 1/6 X 10- sew Volume 49, Number 12, December 1972
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