# In $_6^{14}\mathrm C$ decay, how is mass-energy conserved?

The mass of a carbon 14 atom is $14.003\,241\,988\,7\:\mathrm{u}$, nitrogen 14 has a mass of $14.003\,074\,004\,78\:\mathrm{u}$, and the rest-mass of an electron is $0.000\,548\,579\,9\:\mathrm{u}$

In $\beta^-$ decay,

$$_6^{14}\mathrm C \ \longrightarrow \ _7^{14}\mathrm N + e^- + \nu_e^+.$$

The mass of nitrogen and an electron is substantially larger than that of the carbon atom, so where did the extra mass-energy come from?

• Check your figures (and math) again. Jan 30, 2017 at 16:25
• 14.003242 - 14.003074 < 0.000548 Jan 30, 2017 at 16:38
• @Sam51 - clue - which elementary particles make up the atomic mass? Jan 30, 2017 at 16:42
• Off-topic to your question, but the neutrino is neutral and shouldn't have a superscript plus sign. Jan 31, 2017 at 10:28
• @Joren It's not completely off-topic. With the superscript plus sign it looks like the charge is conserved at first glance. That would explain OP's confusion. Jan 31, 2017 at 11:13

One needs to be very careful in doing mass-energy balances in nuclear decay reactions, especially in beta-decay (electron or positron)

The reaction as written in the OP is correct, and is exactly as it is normally written but is slightly misleading (not the fault of the OP!).

Consider an individual carbon-14 atom on the LHS of the reaction. It consists of 6 protons, 8 neutrons, and six orbital electrons. The orbital electrons are not involved in the nuclear reaction, and are usually ignored.

The major product of the decay of this carbon-14 atom is an atom of nitrogen-14. But this atom still has the same six orbital electrons as the parent carbon -14.

However, the atomic masses are tabulated for the whole neutral atom. So the mass used for the carbon-14 atom is correct, but the mass used for the daughter nitrogen-14 atom is actually too large by the mass of the seventh orbital electron found in a neutral atom. To emphasize, the actual reaction product has six orbital electrons, while the mass used is for a seven-orbital-electron nitrogen-14 atom.

So, one needs to subtract an electron from the tabulated nitrogen-14 atom, and then add back in the mass of the actual beta-particle produced in the reaction.

This is the same as just using the tabulated atomic mass values, and not including the beta-particle mass.

• TL; DR: The OP is counting the electron produced by Beta decay twice, as it's already counted in the N14 mass. Jan 30, 2017 at 18:16
• OP might have been able figure out their own mistake by also checking for charge conservation. "Hey, where did that extra negative charge come from? Could it somehow be related to that mysterious mass abundance I just discovered?"
– jkej
Jan 30, 2017 at 22:15

The atomic masses include both the nuclei and the electrons for a neutral atom. Nitrogen has one more electron than Carbon.