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Hi, I would like to know the closest meaning of modal "can". "Can" seems to take a general "be able" meaning.

For Zero conditional
When you are quiet, I can study.

Means:
When you are quiet, I am able to study.

How about First conditional?
When you are asleep (in the future) , I can study. (I can study when you are asleep)

Does "can" here mean:
When you are asleep, I am able to study. (I am able to study when you are asleep)
OR
When you are asleep, I will be able to study. (I will be able to study when you are asleep)

Thanks
Comments  
These are in the same tense:

When you are quiet, I can study.
When you are asleep, I can study.


If you wish to place it in the future, you must use this:

When you are asleep, I will be able to study.

There is no future form of 'can'.
I have been getting mixed answers on this.

Aren't these examples of "can" used for futurity?

I can do that tomorrow.
I can do that when you are free. (You are not free now)
Next week, I can't come to your party.

http://www.englishlanguageguide.com/english/grammar/conditionals.asp - under First conditional: A real possibility in the future, lists: If he gets good grades, he can go to university.
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When you add a specific future time adverbial, yes, but the verb form itself is simply 'timeless present' and without a denoted time reference can refer to any time, as your 2nd sentence does. (Note that you had to add 'you are not free now' to clarify, because the sentence itself does not reveal that fact.) There is no future form of 'can'.

If he gets good grades (at any time), he can go to university (at any time)-- This is a 'universal truth'.
May I ask then,
I can do that tomorrow = I am able to do that tomorrow OR I will be able to do that tomorrow?

I can do that when you are free = I am able to that when you are free OR I will be able to do that when you are free?
They seem synonymous to me, though the native speaker normally uses I'll be able, I think.
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Would you say that "can" unlike the other modals, is used for the zero conditional instead of the first?
It seems to me that they mostly work the same:

Water can/should/must freeze at -4 degrees.
If you put it under pressure first thing tomorrow morning, it can/should must freeze at -4 degrees.

But you'd better ask CJ– he's the master of zero conditional.