@dot_Sp0T So here goes, first take a look at the chart of nuclides
here specifically the decay information linked
here.
What the second link tells you is three things: Ra-226 has a 100% chance to decay into an $\alpha$ particle (that is in the 'Decay Mode' column of the chart)
It also tells you that the half-life is 1600 years, in the 'Parent T_{1/2}' column
It also tells you that there are five different alpha decay quantizations possible, listed in the section 'Alphas'
The five quanitzations have an energy in keV listed in the first column, and an intensity, or percentage chance of happening in the second.
In this case, 93% of the alpha particles decay releasing 4784 keV of energy.
Now, to determine the energy released by one atom every second, you will convert the half-life to a decay rate. 1600 years = 5.05e10 seconds. Decay rate is ln(2)/half-life = 1.37e-11 with units 1/s.
That represents the chance that a single atom with decay each second
So, now multiply that rate by the energy released (lets round it to 4700 keV since there are five quantizations that we need to average together)
to get 6.6e-5 eV/s released.
Now, next thing is to covert from a per-atom rate to a per-kilogram rate. The atomic mass of Ra-226 is .... 226 grams per mole, and a mole is Avogadro's number (6.022e23) of particles. So 6.6e-5 * 6.022e23 /0.226 = 1.75e20 eV or 28 J
But wait! I got 168 J last time, not 28 what happened? Well what I calculated there was just the decay of Radium itself. But once radium decays, it becomes Radon-222, which has a half-life of just 3 days; and when Radon decays it becomes Polonium-218 with a halflife of just 3 minutes. So, as soon as the very long time span Radium atom pops, the rest of the decay change happens relatively quickly. Therefore, we can approximate the total energy released as the total energy of all decays
all the way down to the next decay that will take a timescale of years.
Well, that next decay product will be Lead-210, six more steps down the chain. Complicating this further, there are multiple ways to get there. Radium-226 only decayed one way, but Bismuth-212,
for example can decay six different ways!
I'm not entirely sure what energy level I used for that calculation, as a quick substitute I estimated 27620 keV from that wikipedia chart
And plugging that in for 2700 in our original equation, it works out to 162 J/s/kg, which is pretty close to the 168 that I got in that answer.
I actually remember building a spreadsheet to do the decay math out for that answer, but I can't find it now, so I'm reasonably confident that 168 W/kg is a good estimate.
@dot_Sp0T Ok thats it, you get all that?