Appendix 3: Selected Physical Constants.

Appendix 3 Physical Constants

Physical Constants
(taken from CODATA internationally recommended values:
description letter variable Value (uncertainty)* relative uncertainty**
Speed of light in vacuum c 299,792,458 m s1 exact
Planck Constant h 6.62607015 × 1034 J s exact
Bohr radius a0 5.29177210903(80) × 1011 m 8.0×1022
electron mass me 9.10938291(40) × 1031 kg 4.4×108
elementary (electron) charge e 1.602176634 × 1019 C exact
Avagadro Constant NA 6.02214076 × 1023 mol1 exact
Boltzman Constant k 1.380649 × 1023 J K1 exact
Faraday Constant F 96485.3312 C mol1 exact
Molar Ideal Gas Constant R 8.314 462618 J mol1 K1 exact
Rydberg Constant RH 2.179872171 × 1018 J 4.4 × 108

Note that these values are quoted to a very large number of sig figs since the values have been measured quite accurately.  These values are periodically updated as better and better measurement techniques are developed.  The values quoted here are current as of 2018.  Any updates to these numbers will likely only occur in the last couple of decimal places so you can feel safe in using these numbers.  Note that some of the numbers have now been deemed to be defined numbers. Those have no uncertainty in them because they are defined numbers.

* The uncertainties quoted here are generally determined statistically as a result of multiple measurements by several researchers.  They are the statistical standard error. The values are given as a number with an uncertainty expressed in parenthesis.

Consider the Bohr Radius: The value is quoted as a_0 = 5.29177210903(80) \times 10^{-11}\textrm{ m}.  In this formulism, we see that the last two digits of the number, i.e., 03, are in error by + 80 so that means the value of the Bohr Radius can be as high as a_0 = 5.29177210983 \times 10^{-11}\textrm{ m} and as low as a_0 = 5.29177210823 \times 10^{-11}\textrm{ m}.  As far as our accuracy goes, somewhere in that range is the real value of the Bohr radius.

** The relative uncertainty is simply the standard error divided by the actual value.  For example, the relative uncertainty for the bohr radius is calculated as follows

\frac{0.00000000080 \times 10^{-11}\mathrm{ J s}}{5.29177210903 \times 10^{-11}\mathrm{ J s}} = 1.5\times 10^{-10}.

A good way to visualize the meaning of relative uncertainty is to look at the power of 10 exponent.  It gives 1 less than the number of decimal places in the numeric value.  Since there is one more sig fig in front of the decimal, the Bohr radius is known to about 12 sig figs.

In first year chemistry, we generally use only 3 or 4 sig figs in our problem solutions.  So most chemistry texts don’t even bother quoting the uncertainty in these physical constants.  These texts merely quote the physical constant to less digits (5 or 6) and the students are left to take these as being perfectly accurate.  Doing this simplification will not induce much error as long as the number of digits used in the physical constant is at least two more than the number of sig figs in the problem values.  Note that the textbooks that do this are introducing a round-off error into any calculations used.  They assume that error is negligible for most cases a first-year student might need.


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