Weighing Electrons
Don Herbison-Evans
(donherbisonevans@outlook.com)
26 August 2014

I have this silly notion that no-one has actually weighed electrons, and maybe their gravitational mass is not the same as their inertial mass. The Eotvos experiment certainly implies that electrons are subject to gravity, but I thought it would be nice to actually weigh them.

The reactions I had in mind are
BeH2 + 2HF -> BeF2 + 2H2
vs
BeD2 + 2HF -> BeF2 + (some irrelevant combination of D2 + H2 + HD)

Basically : find out what weights of BeF2 are produced by weighed amounts of BeH2 and BeD2. Electrons of course may only contribute about 585 microgms per Grm Dalton, but balances are now commercially available that can measure 2.5 gms to an accuracy of 1 microgm, like

the Sartorius CubisŪ Ultramicro Balance MSA2.7S-000-DM

Atomic numbers
H,D 1
Be 4
F 9

Atomic (inertial) weights assuming they are derived from mass spectrometry
H 1.00782505 Daltons
D 2.01410178
Be 9.01218315
F 18.99840316
e 0.00054858

Assuming electrons have gravitational mass:

1 gm molecule of BeH2 containing N molecules: (N = Avogadro's number)
= 9.0121831(5) + 2 x 1.00782505
= 9.01218315 + 2.01565010
= 11.02783325 gms

1 gm molecule of BeD2
= 9.0121831(5) + 2 x 2.01410178
= 9.01218315 + 4.02820356
= 13.04038671 gms

React each with excess aqueous HF in an inert atmosphere and evaporate to dryness, being careful to inhibit spray from bubbling H2/D2 and boiling (tricky) assuming each sample has less than 0.01ppm impurity (tricky) :-

Each will produce 1 gm molecule of BeF2
= 9.0121831(5) + 2 x 18.99840316
= 9.01218315 + 37.99680632
= 47.00898947 gms

So reducing to unit weights, 1 gm of BeH2 will produce
47.00898947 / 11.02783325 gms BeF2
= 4.26275846 gms BeF2

1 gm of BeD2 will produce
47.00898947 / 13.04038671 gms BeF2
= 3.60487695 gms BeF2

If electrons have no gravitational mass -

1 gm molecule of BeH2 and BeD2 has 6 electrons/molecule, with electron inertial mass = 6 x 0.00054858
= 0.00329148 gms

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

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

N molecules of BeF2 have 22N electrons with mass
= 22 x 0.00054858
= 0.01206876 gms

Each reaction will produce N molecules of BeF2 (nuclei)
= 47.00898947 - 0.01206876
= 46.99692071 gms

So reducing to unit weights:-
1 gm of BeH2 will produce
46.99692071 / 11.02454177 gms BeF2
= 4.26293642 gms BeF2

1 gm of BeD2 will produce
46.99692071 / 13.03709523 gms BeF2
= 3.60486135 gms BeF2

BeH2 -> BeF2 (gravitational - inertial) difference
= 4.26293642 - 4.26275846
= 0.00017796 gms

BeD2 vs BeF2 difference
= 3.60486135 - 3.60487695
= -0.00001560 gms

Net difference between BeF2 amounts
= 0.00017796 - (-0.00001560)
= 0.00019356 gms
= 194 micrograms

One would of course work with smaller and arbitrary but accurately weighed amounts of BeH2 and BeD2, and scale the results accordingly. These calculations suggest that taking the above two reactions, the difference between heavy and weightless electrons would be about 4 microgm per gm of BeF2.

The H in the BeH2 and the D in the BeD2 would not have to be isotopically pure to get a result, but the purer they are 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 between the other monoisotopic elements

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

in compounds with other multi-isotopic elements, although the neutron imbalance of the heavier elements makes the result harder to measure.