Weighing Electrons

Don Herbison-Evans
donherbisonevans@outlook.com
26 August 2014, revised 6 October 2017

Introduction

No-one has actually weighed electrons, and maybe their gravitational mass is not the same as their inertial mass as Einstein assumed in his Equivalence Principle. The Eotvos Experiment indirectly implies that electrons are subject to gravity, but it would be nice actually to weigh them.

Electrons constitute the highest proportion of the mass of molecules that are composed of atoms of the lowest atomic number, so it is tempting to use the lightest elements for this experiment. However measurements based on chemical reactions can be complicated by changes in isotopic ratios for molecules containing elements with more than one naturally ocurring stable isotope. Thus the only elements in the lightest dozen that have only one stable isope and so avoid this fractionation are Be, F and Na.

Electrons only contribute about 549 micrograms per Grm Dalton, but balances are now commercially available that can measure several grams to an accuracy of 1 microgram, such as the Sartorius Cubis MSE6.6S-0CE-DM balance.

In the following considerations, we shall use

N = Avogadro's number

and

Atomic numbers (number of electrons per atom):
Be 4
F 9

and

Atomic (inertial) masses derived from mass spectrometry:
Be 9.01218315 Daltons
F 18.99840316 Daltons
e 0.00054858 = 548 micro-Daltons

Be to BeF2 with Weighty Electrons

Consider the simple conversion of metallic Beryllium to Beryllium Fluoride:

1 gm atom of Be containing N atoms:
= 9.01218315 grams

1 gm molecule of BeF2
= 9.01218315 + 2 x 18.99840316 = 47.00898947 grams

Normalising : 1 gram of Beryllium would produce

47.00898947 / 9.01218315 = 5.21616002 grams BeF2

Be to BeF2 with Weightless Electrons

1 gm atom of Be containing N atoms minus the mass of 4.N electrons:
= 9.01218315 - 4 x 0.00054858 = 9.01218315 - 0.00219432 = 9.00998883 grams

1 gm molecule of BeF2 minus the 4.N electrons in the Be, and 2 x 9.N electrons in the 2F:
= 47.00898947 - 22 x 0.00054858 = 47.00898947 - 0.01206876 = 46.99692071 grams

Normalising : 1 gram of Beryllium would produce
46.99692071 / 9.00998883 = 5.21609090 grams BeF2

The difference in the weight of BeF2 produced by by 1 gram of Beryllium is

5.21616002 - 5.21609090 = 69.12 micrograms BeF2

Discussion

Using inertial masses, 1gm Be with HF would produce 5.21616002 gms BeF2 if electrons are subject to gravity, and 5.21616002 if not, a difference of 69.12 micrograms easily measured by good balances. Has anyone done this?

One would of course work with arbitrary but accurately weighed amounts of Be and scale the results accordingly. Be and F are monoisotopic so there are no fractionation problems to be considered in these reactions, but spray, impurity levels, and corrosion are problems that would need solving.

One might alternatively consider other reactions involving the other mono-isotopic elements:

  Element  
  Atomic  
Number
  Approx  
Atomic
Mass
Be
4
9
F
9
19
Na
11
23
Al
13
27
P
15
31
Sc
21
45
Mn
25
55
Co
27
58
As
33
75
Y
39
89
Nb
41
93
Rh
45
103
I
53
127
Cs
55
133
Pr
59
141
Tb
65
159
Ho
67
165
Tm
69
169
Au
79
197

For the general case, consider the conversion of molecule S with molecular mass 'S' Daltons into a molecule T with molecular mass 'T' Daltons. Let 's' be the number of electrons in S, and 't' be the number electrons in T. Let 'e' be the mass of an electron in Daltons. Then if electrons have gravitational weight: then 1 gram of S will produce T/S grams of T. If electrons have no gravitational mass: (S-se) grams of S will produce (T-te)grams of T. So 1 gram of S will produce (T-te)/(S-se) grams of T. The difference 'x' will be

x = (T-te)/(S-se) - T/S = [S(T-te) - T(S-se)]/[S.(S-se)] =
   = e.(Ts - St)/[S.(S-se)] ≈ e.(T/S2).(s/T - t/S)
This formula can be useful in seeking more convenient and less poisonous alternatives to the Beryllium and Fluorine considered above The formula suggests that 'S' be chosen to be small, and T large, and that S and T should then be chosen to make s/S and t/T as different from each other as possible.

So for the example above: Be -> BeF2: S = 9, s = 4, T = 47, t = 22:

x ≈ e.(47x4 - 9x22)/81 = e.(188 - 198)/81= -0.123e and for say Be -> BeI2: S = 9, s = 4, T = 263, t = 110: x ≈ e.(263x4 - 9x110)/81 = e.(1052 - 990)/81 = 0.765e Although this has a better result, the spontaneous inflammability of BeI2 might make the experiment even more hazardous than Be -> BeF2.

Question To Quora: 6 June 2017 1gm Be with HF produces 5.216160 gm BeF2 if gravity affects electrons, 5.216090 if not, difference = 70 mugm, easily measured, so why no-one done this?

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