Q:

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...

