Why does my mass calculation give higher mass after the beta-decay from Thorium-234 to Protactinium-234? I'm trying to do a mass calculation on the beta-decay of Thorium-234, which I believe goes like this:
Th-234 -> Pa-234 + e + energy
Masses


*

*Thorium-234: 234.043601 u [1].

*Protactinium-234: 234.043308 u [2].

*Electron: 0.000548 u [3].


This gives mass on right hand side of reaction 234.043856 u, which is more then we had in the first place.
What am I doing wrong? Aren't the fission beta-decay suppose to release energy, given by the mass difference?
 A: In $\beta^-$ decay, you don't actually need to take the mass of the electron into account.  The reason for this is that after the decay, the daughter nucleus still has the same number of electrons orbiting it as the parent did;  but that's one less than it "should have" since the daughter's proton number has increased by one.  The masses given in nuclide mass tables are the masses for the neutral atoms.
In other words, you have
$$
\underbrace{{}^{234}_{90} \mathrm{Th}}_{\text{90 protons, 144 neutrons, 90 electrons}} \to \underbrace{{}^{234}_{91} \mathrm{Pa}^+}_{\text{91 protons, 143 neutrons, 90 electrons}} + \underbrace{\beta^-}_{\text{1 electron}} + \bar{\nu}.
$$
If we say that $m({}^Z_A \mathrm{X})$ is the mass of a neutral atom, then the mass before is
$$
m_i = m({}^{234}_{90} \mathrm{Th})
$$
while the mass after is
$$
m_f = \left[ m({}^{234}_{91} \mathrm{Pa} ) - m_e \right] + m_e + m_\nu \approx m({}^{234}_{91} \mathrm{Pa} )
$$
since $m_\nu$ is, of course, negligible.  Thus, the mass defect in the reaction is just the difference in the masses of the neutral atoms.
Note that when you do $\beta^+$ decay, the logic actually works the other way:  the final mass is actually the mass of the neutral atom plus two electron masses.  Try going through the logic in a similar way to see why this is so.
