Weighing Electrons

Don Herbison-Evans

(donherbisonevans@outlook.com)

**1. Gravimetric Chemistry**

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.
Bearing in mind that, generally, atomic masses are determined by Mass Spectrometry,
and so are inertial masses, then if electrons do have gravitational weight:
1 gram of S will be produce T/S grams of T.
But if electrons have no gravitational weight:
(S-se) grams of S will produce (T-te) grams of T.
So then 1 gram of S will produce
(T-te)/(S-se) grams of T if electrons have no weight

If x is the difference in weight of product T from 1 gram of S

= [S(T-te) - T(S-se)]/[S.(S-se)]

= (sTe - tSe)/[S.(S-se)]

≈ e.(sT - tS)/(S

≈ e.(s/S - t/T).(T/S) : equation 2

So let us consider the reaction:

The principle is to find out what weight of BeF_{2}
is produced from a weighed amount of Be.

Electrons only contribute about 548 micrograms per Gram Dalton, but balances are now commercially available that can measure 2.5 grams to an accuracy of 1 microgram, such as the Sartorius Cubis Ultramicro Balance MSA

With this reaction: we have

S = 9.012182 ≈ 9

s = 4

T = 2 x 18.998403 + 9.012182 = 47.009042 ≈ 47

t = 22

sT ≈ 4 x 47 = 188

tS ≈ 22 x 9 = 198

S

x ≈ e.[188 - 198)]/81 = -e.[10/81] = -0.123e = -67.7 micrograms

One would of course work with arbitrary
but accurately weighed amounts of the reactants
and scale the results accordingly.
So if the values are scaled to consider
the production of 1 gram of BeF_{2},
the electrons would make a difference of -12.9 micrograms.
Limiting the effects of impurities and interactions with the atmosphere
and reaction vessels to values well below this would be a challenge,
particularly as metallic Be in air becomes covered in a thin oxide layer,
and F_{2} attacks nearly everything.

From equation 2, we appear to need to maximise the difference s/S - t/T,
while minimising T/S.
Hydrogen has s/S ≈ 1, but gases are hard to work with and weigh accurately.
Other lighter elements have s/S ratios near 0.5,
and the heavier elements have values approaching 92/238 ≈ 0.39.
So we might try to drop the t/T ratio by using a reaction
that converts between heavy elements to light elements,
for example: reacting BeI_{2} with F_{2},
and evaporating off the iodine and iodine fluorides produced,
would give

sT ≈ 110 x 47 = 5,170

tS ≈ 22 x 263 = 5,786

S

x ≈ e.(5170 - 5786)/69169 = -e.616/69169 ≈ -0.008906e

≈ -4.88 micrograms

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

Number | Atomic Mass | |

As can be seen, only four of these are non-metals, which severely limits our options. However, it is also possible to use compounds with multi-isotopic elements although the weights would be complicated by the isotopic ratios, which would need to be measured and allowed for.

**2. Beam sag**

Consider a vacuum tube L metres long with an electron gun at one end
accelerating an electron beam by some voltage V
striking a phosphorescent screen at the other end.
If electrons have a gravitational weight proportional to their inertial mass:
the beam will sag due to the electrons falling by gravity
as they fly from one end of the tube to the other.

The amount of sag can be calculated by noting that the electrons will have an energy of

e is the electron charge ≈ 2 x 10

m is the electron mass ≈ 10

v = (2Ve/m)

So the time of flight travelling L metres will be

The result varies as the square of the tube length and inversely with the accelerating voltage. The minimum voltage will be determined by the thermal distribution of electron velocities as they leave the cathode of the electron gun.

If the cathode temperature is T Kelvin, the electrons will have a range of velocities
about a median which is the equivalent of a voltage V_{c} where

These considerations suggest that this way of determining the weight of electrons may just be on the limit of feasibility, but would require the building of extensive and expensive apparatus.

**Conclusion**

All the above considerations suggest that gravimetric chemistry is a better way of determining the weight of electrons than using beam sag. The gravimetric chemistry method is probably feasible with available equipment and materials, but would require very pure reactants, ideally 99.9999% pure, very careful handling, and possibly the use of inert atmospheres.

written 26 August 2014, updated 17 November 2018