Weighing Electrons

Don Herbison-Evans
donherbisonevans@outlook.com
26 August 2014, revised 3 June 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. 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 use of H, He, Li, B, C, O, N, can be complicated by this fractionation, but Be and F have only one stable isope so considered below as possible choices for this experiment.

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):
H,D 1
Be 4
F 9

and

Atomic (inertial) masses derived from mass spectrometry:
H 1.00782505 Daltons
D 2.01410178
Be 9.01218315
F 18.99840316
e 0.00054858

Be to BeF2 with Weighty Electrons

The simplest suitable reaction seems to be that of the 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

BeH2 to BeF2 with weighty Electrons

N molecules of BeH2 will weigh
= 9.01218315 + 2 x 1.00782505 = 11.02783325 grams

So reducing to unit weight of BeH2, 1 gram of BeH2 will produce
47.00898947 / 11.02783325 = 4.26275846 grams BeF2

BeH2 to BeF2 with weightless Electrons

1 gm molecule of BeH2 has 6 electrons/molecule, with electron inertial mass
= 6 x 0.00054858 = 0.00329148 grams

N molecules of BeH2 (nuclei) will weigh
= 11.02783325 - 0.00329148 = 11.02454177 gms

So reducing to unit weights:-
1 gram of BeH2 will produce
46.99692071 / 11.02454177 = 4.26293643 grams BeF2

BeD2 to BeF2 with weighty Electrons

N molecules of BeD2 will weigh
= 9.01218315 + 2 x 2.01410178 = 13.04038671 gramss

1 gram of BeD2 will produce
47.00898947 / 13.04038671 = 3.60487695 grams BeF2

BeD2 to BeF2 with weightless Electrons

1 gm molecule of BeD2 has 6 electrons/molecule, with electron inertial mass
= 6 x 0.00054858 = 0.00329148 grams

N molecules of BeD2 (nuclei)
= 13.04038671 - 0.00329148 = 13.03709523 grams

So reducing to unit weights:-
1 gram of BeD2 will produce
46.99692071 / 13.03709523 = 3.60486135 grams BeF2

Discussion

The results are shown in table 1:

grams BeF2
produced
from
1 gram Be
from
1 gram BeH2
from
1 gram BeD2
with
electrons
5.21616002
4.26275846
3.60487695
without
electrons
5.21609090
4.26293643
3.60486135
difference
micrograms
69.12
-177.97
15.60

weights in grams of of BeF2 produced by 1 gram each of Be, BeH2, and BeD2

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, BeH2, and BeD2, and scale the results accordingly. These calculations suggest that taking the above BeH2 reactions, the difference between heavy and weightless electrons in producing BeF2 would be about 178 micrograms per gram of BeH2.

The H in the BeH2 would not have to be isotopically pure to get a measurable result, but the purer it is isotopically, the greater the difference will be in the answers. 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 monoisotopic elements:

Na, Al, P, Sc, Mn, Co, As, Y, Nb, Rh, I, Cs, Pr, Tb, Ho, Tm, Au,

although the neutron imbalance of the heavier elements would make the result harder to measure.

For the general case, consider the transformation of molecule A with molecular mass 'A' Daltons into a molecule B with molecular mass 'B' Daltons. Let 'a' be the number of electrons in A, and 'b' be the number electrons in B. Let 'e' be the mass of an electron in Daltons. Then if electrons have gravitational weight: then 1 gram of A will produce B/A grams of B. If electrons have no gravitational mass: (A-ae) grams of A will produce (B-be)grams of B. So 1 gram of A will produce (B-be)/(A-ae) grams of B. The difference 'x' will be

x = B/A - (B-be)/(A-ae) = [B(A-ae) - A(B-be)]/[A.(A-ae)] = (Abe - Bae)/[A.(A-ae)] = = (b/B - a/A).e/(A-ae) This formula can be useful in seeking more convenient and less poisonous alternatives to the Beryllium and Fluorine considered above The formula suggests that 'A' and 'a' be chosen to be small, Hence the choice of Be above. The formula suggests that B and b chosen to be as different from each other as possible, as in the heaviest elements which have a high neutron imbalance.

So for BeH2 -> BeF2

x ≈ e x (13/23 - 6/6)/6 = -0.43e but for BeH2 -> BeI2 x ≈ e x (159/385 - 6/6)/6 = -0.58e Although this has a better result, the spontaneous inflammability of BeI2 might make the experiment even more hazardous than BeH2 -> 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 noone done this?

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