How long will a cube of Plutonium-238 keep my tea warm? Plutonium-238 gives off heat, which reduces over time due to its half-life. Suppose I have a mug of tea in a room temperature environment, and instead of dropping in a sugar cube, I dropped in a plutonium cube. How long will the mug of tea remain at a reasonable temperature?
If the cube either fails to prevent the tea cooling down, or is so hot it causes the water to boil, please adjust the size of the cube.
 A: Somebody on Wikipedia helpfully did the calculation. I will repeat some of the highlights here:
Given the half life of Pu-238 (87.7 years), the molar mass (238!), one gram of Pu-238 has a specific activity of 634 GBq.
We know the mean energy of the emitted alpha particle, and conclude that the energy generated by a gram of Pu-238 (assuming the alpha particles don’t escape which is reasonable when it’s submerged in a cup of tea) is about 0.57 W.
Now a typical cup of tea might start out with a temperature of 50 ˚C, and gets “cold” in 15 minutes - let’s say it cooled down by 20 ˚C (this is just spitballing - your actual values may change but the principle of the next bit should be sound). If you had 250 ml of tea with a heat capacity of about 4.2 J/g/K or around 1000 J/˚C for the cup, then you would need 20000 J in 15 minutes to keep the tea “at temperature”. That would require about 20 W of heat - that seems like a lot. A USB powered “cup heater” draws about 1A at 5V, so that would be 5 W.
So we are looking at somewhere between 5 and 20 W of power needed (this depends on how hot you like your tea, and the motion and relative humidity of the air: when dry air is moving over the surface, the tea will lose heat much more rapidly.
To get 5 W you would need about 10 g of Pu-238; its density is almost 20 g/cc, so you would need just half a cubic centimeter; at the high end, it would be 2 cc. The volume of a “standard sugar cube” is about 3 cc (they come in many sizes so it’s not possible to give a definitive answer - but most sugar cubes seem to be bigger than you’d need your radioactive sugar cube to be).
That suggests that a sugar-cube-sized block of pure Pu-238 would be sufficient to keep your tea hot. Of course your tea will evaporate at a reasonable rate with this much heat being added - so you still have to drink it quickly. Alternatively, if you cover the tea with a cover (even a saucer) it will keep warm by itself much better.
