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[LordBucket]

Scientists have proposed an experiment anticipated to be conducted next year, which will test, in demonstrable fashion, whether it's possible to turn light into physical matter.

So now that we're all thinking about Star Trek replicators:

The question:
Assuming 100% efficiency, how much light (in practical everyday intuitive units of your choice) would it take to make a a cup of tea, Early Gray, hot?

[Baffler]

This is a fun silly problem. The first step, I think, is to find out how much tea is in a cup of Earl Gray, (we could just call a cup 1 cup in customary units, but I have a feeling liters will be easier to work with.) According to Google, 1 cup = .236588L. Not an inspiring number. We'll say that the Captain's got the big mug with the big chair, and that his cup holds exactly 0.25L of tea.

Next is to consider the components of the tea. Earl Gray is, essentially, black tea flavored with bergamot. There are other ingredients, of course, but even just "bergamot" poses serious problems. I'm having some trouble finding out just what exactly those other ingredients are, though no doubt the mix contains dozens, maybe even hundreds of different compounds. Each of which we'd have to determine the exact quantity present, the individual atoms that those compounds are composed of, the energy required to get those atoms to form the right bonds and therefore the compounds needed, and the energy required to produce each and every neutron, proton, and electron those atoms are themselves composed of from the photon feedstock.

And that's ignoring the problems posed by getting those compounds to form properly. Each component would probably have to be synthesized one at a time, or in groups of sufficiently different components to prevent them from interfering with each other. Not to mention the problem of ensuring that every single synthesis goes to 100% completion 100% of the time with all of the byproducts being separated and recycled to prevent random other stuff from ending up in the tea.

I'm guessing a lot of energy.

[LordBucket]

Next is to consider the components of the tea. Earl Gray is, essentially, black tea

 I'm having some trouble finding out just what exactly those other ingredients are

From Chemical composition of tea:

Table 11. Composition of a black tea beverage

% dry weight -- substance

4.6 Epi-galloctechin gallate
1.1 Epi-galloctechin
3.9 Epi-atechin gallate
1.2Epi-atechin
trace Flavonol glycoides
trace Bisflavanols
2.6 Theaflavins
trace Theaflavic acid
35.9 Thearbigins
7.6 Cafeine
0.7 Theobromine
0.3 Theophyllne
1.2 Gallc acid
0.2 Chlorogenic acid
1.5 Oxalic acid
0.02 Malonie acid
0.1 Succinie aeid
0.3 Malie acid
0.0 1 Acnitic aeid
0.8 Citrie acid
4.8 Lipids
6.9 Monosacchardes
0.2 Peetin
4.2 Polysacchardes
6.0 Peptides
3.6 Theanine
3.0 Other amino acids
4.8 Potasium
4.7 Other mineraIs
0.01 Volatiles

"it is convenient and realistic to assume usage of 13.5 g black tea per litre of hot water"

[Frumple]

... not really mathy enough to take a stab at it, but I just have to ask for clarification: Does that "volatiles" at the bottom mean black tea is technically (very) mildly explosive?

[BurnedToast]

In star trek cannon, they didn't actually replicate everything directly from energy, they had a big mass tank that recycled the matter over and over again, and the replicator just formed it into whatever was ordered. But oh well, you want light directly into matter so....

90 megajoules of energy adds about 1 microgram of mass to a system.

Let's call it 6 ounces of tea, which is about 170 grams. One porcelain teacup is about 120 grams (a surprisingly difficult bit of information to find). So the total mass is about 290 grams

As I'm sure we all know, there's 1,000,000 micrograms in a gram. So that means we've got about 290 million micrograms, times 90 megajoules per microgram, which gives us 26.1 petajoules of energy required.

You ask for that in common, everyday units. I'm not sure that's practical. So I'll give you another unit instead - that's very roughly the energy released by 390 "little boy" atomic bombs.

Edit: or another way of looking at it, the largest nuclear power plant in the world (kashiwazaki-kariwa, located in japan) produces 60 TWH per year. That's about 216 petajoules. So, it would take about 8% of the yearly power output of the largest nuclear power plant on earth to make your cup of tea.

[Baffler]

... not really mathy enough to take a stab at it, but I just have to ask for clarification: Does that "volatiles" at the bottom mean black tea is technically (very) mildly explosive?

All it actually means is that it has a tendency to form a gas.

[Baffler]

Next is to consider the components of the tea. Earl Gray is, essentially, black tea

 I'm having some trouble finding out just what exactly those other ingredients are

From Chemical composition of tea:

Table 11. Composition of a black tea beverage

% dry weight -- substance

4.6 Epi-galloctechin gallate
1.1 Epi-galloctechin
3.9 Epi-atechin gallate
1.2Epi-atechin
trace Flavonol glycoides
trace Bisflavanols
2.6 Theaflavins
trace Theaflavic acid
35.9 Thearbigins
7.6 Cafeine
0.7 Theobromine
0.3 Theophyllne
1.2 Gallc acid
0.2 Chlorogenic acid
1.5 Oxalic acid
0.02 Malonie acid
0.1 Succinie aeid
0.3 Malie acid
0.0 1 Acnitic aeid
0.8 Citrie acid
4.8 Lipids
6.9 Monosacchardes
0.2 Peetin
4.2 Polysacchardes
6.0 Peptides
3.6 Theanine
3.0 Other amino acids
4.8 Potasium
4.7 Other mineraIs
0.01 Volatiles

"it is convenient and realistic to assume usage of 13.5 g black tea per litre of hot water"

Now, I think, we just need to find the bond energies for all of these, add them together, and adjust those numbers based on the % mass of each. Add it to the number BurnedToast got, and if I'm right we've got a fair approximation.

Doing the above probably won't have much of an effect on the number, but it's still enough energy to be worth considering. We'll assume that the replicator is able to just stick atoms onto the structure right where they need to be. Though some of these categories, like lipids and peptides, are pretty broad.

[BurnedToast]

Now, I think, we just need to find the bond energies for all of these, add them together, and adjust those numbers based on the % mass of each. Add it to the number BurnedToast got, and if I'm right we've got a fair approximation.

The bond energies would literally just be rounded off and not change the answer at all.

For water, it's 458.9 kj per mol. Pretending the cup is made of water too (because lazy), and saying about 18g of water per mol gives us about 16 mol or ~7330 kj.

That's 7.33 × 10-9 petajoules, or (if I'm counting my zeros right) 0.000000007330 petajoules

[LordBucket]

...wait. I just realized there's a much easier way to do this. We're creating matter. Energy and matter are equivalent. It doesn't matter what we're making, it only matter how much mass we're creating.

According to http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence  25 kilowatt-hours corresponds to 1 microgram.

Six ounces of water corresponds to 177.44 grams. That's not accounting for temperature, but it happens that the formal definition of a gram is weight of specified volume of water at a specified temperature...so that conversion should be easy when we do it. Proceeding for casual estimate...

There are 1000000 micrograms per gram, so 1000000 * 177.44 =  177,440,000 micrograms of water in 6 ounces.

177,440,000 * 25 = 4,436,000,000 kw/hours

A teabag is 2 grams, but the vast majority of that doesn't make it into the tea. I suspect that if we weighed 6oz of water before and after soaking a teabag in it, our instruments would probably not be sensitive enough to detect the difference. So we don't even need to worry about the mass contributed by the tea.

So now we just need to identify the mass of the teacup, convert it into kilowatt hours, add it to the above and find out how much light of our chosen wavelength that much energy is.


EDIT for fun:

Kerosene has an energy density of 33 megajoules per liter.

http://en.wikipedia.org/wiki/Joule

1 kw/h = 3.6 MJ

4,436,000,000 kw/hours = 15,969,600,000 megajoules

15,969,600,000 megajoules / 33 = 5,323,200,000

So, assuming 100% conversion efficiency, if we burned 5,323,200,000 liters of kerosene, that would provide enough energy to make a cup of tea, not including the cup.

[Baffler]

Now, I think, we just need to find the bond energies for all of these, add them together, and adjust those numbers based on the % mass of each. Add it to the number BurnedToast got, and if I'm right we've got a fair approximation.

The bond energies would literally just be rounded off and not change the answer at all.

For water, it's 458.9 kj per mol. Pretending the cup is made of water too (because lazy), and saying about 18g of water per mol gives us about 16 mol or ~7330 kj.

That's 7.33 × 10-9 petajoules, or (if I'm counting my zeros right) 0.000000007330 petajoules

That's not right, but it does just round off either way in the face of numbers like we're dealing with here. I'm mostly just considering it for curiosity's sake.

[BurnedToast]

Now, I think, we just need to find the bond energies for all of these, add them together, and adjust those numbers based on the % mass of each. Add it to the number BurnedToast got, and if I'm right we've got a fair approximation.

The bond energies would literally just be rounded off and not change the answer at all.

For water, it's 458.9 kj per mol. Pretending the cup is made of water too (because lazy), and saying about 18g of water per mol gives us about 16 mol or ~7330 kj.

That's 7.33 × 10-9 petajoules, or (if I'm counting my zeros right) 0.000000007330 petajoules

That's not right, but it does just round off either way in the face of numbers like we're dealing with here. I'm mostly just considering it for curiosity's sake.

Where is it wrong?

Genuine question, I'm not a chemist so it was based on about 10 minutes of google research and I'm curious how badly wrong I got it.

[Baffler]

The 458.9 number is the energy required to break one of water's O-H bonds. The number needed here, enthalpy of formation, is -285.8 kJ*mol^-1 (the negative means that the process is exothermic.) It's not really my strong suit, but I wasn't really right either calling it bond energy.

[LordBucket]

15,969,600,000 megajoules

Here we go: total energy from the sun that hits the earth per second: 1.74 x 10^17 joules

15,969,600,000 megajoules is ~1.597 x 10^16 joules

1.597 x 10^16 / 1.74 x 10^17 = .0917

So, to make six ounces of tea would take approximately 9% of the total energy of all wavelengths of light that reach the earth from the sun in one second.