How much energy would be required to make one tea cup full of Earl Gray tea at 100F? On the TV show "Star Trek: The Next Generation", Captain Picard is often pictured using a replicator to materialize a cup of "Earl Gray tea, hot". Besides wondering what they do with all the empty teacups, I've often wondered just how much pure energy is contained in that cup of tea.
 A: If "materializing" means creating matter from energy, then a 250 ml cup of water will contain approximately 250 g of water with a smidge of other molecules. Add another 100 g for the cup (light weight... this is space). From $E=mc^2$, you find an energy content of roughly $3\cdot 10^{15}~\rm{ J}$.
Note that the thermal energy content (difference between "cold" and "hot") is many orders of magnitude smaller than this - completely negligible.
If we assume the mass of the Enterprise to be about $3\cdot10^9~\rm{kg}$ (source), this corresponds to the kinetic energy of the entire starship moving at 3000 m/s (Mach 10).
I'm thinking that it would be cheaper to get a Keurig...
A: Well, the first thing to do is to discard the tea itself, because it not only accounts for a truly miniscule overall volume within the beverage, but also it is composed of molecules enormously more complex than those of water and hence we would be here all day. The second thing to do is discard the cup - it is too much of an unknown quantity in its many possible sizes and compositions. Just water, then. He'll have to fetch a cup from somewhere.
The energy required to heat a 250ml cup of water from absolute zero to 100F, using relevant figures of specific heat of water from engineering toolbox, is approximately 325,306 Joules, or 325kJ.
Heat = mass * specific heat * temperature change. 
However, if you mean the absolute amount of energy, or relativistic energy, necessarily manifest in the mass itself of that water, it becomes a bigger calculation. We'll assume it's totally pure and that 27.78 grams of that water is hydrogen, with the remaining 223.22 grams being oxygen. These figures follow from their respective masses.
Using e = mc^2 for the energy of all that hydrogen, we get e = 8,328,234,483.24 J, or about 8.328 GJ. Again for the oxygen, we get 66,919,672,474.8 J, or 66.919 GJ.
Combining the relativistic energies we get 75,247,906,958 or 75.247 GJ. Adding the now paltry amount of heat energy in a 100F cup of water we get 75,248,232,264 J or 75.248 GJ.
Finally, we ought to involve the bond energy of molecular water, which I find to be 458.9 kJ/mol, for our 250ml of water would be roughly 6,373,611 J or 6.373 MJ. Our total energy is now 75,248,232,264 + 6,373,611 or 75,254,605,875. That's 75.254 GJ.
