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When we talk about some mode for a device like "hyperactive mode" we use the preposition "in" on some occasions, for example "The microwave is in hyperactive mode", "The spacecraft is in the recovery mode" etc.

But in some cases we use "on" as in "The computer is on standby/Put the computer on standby" and
"The airplane was on autopilot". Is there any reason behind this or is it just idiomatic ?
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I can think of no specific reason; you can think of it as idiomatic or as a fixed phrase (in...mode).
Is it idiomatic to say "Put the cellphone on charge" and "Put the file on download" ?
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Yes, but I don't know about the second because I don't understand the process you are referring to. If you are simply downloading a file, I don't think so. Presumably, with this phrase ('on X'), 'X' is a designated position on a switch or control apparatus of some kind -- it is essentially a label, like 'ON' and 'OFF'.

But it may be too easy just to call these all idioms. There are many, and they are prepositions that have collocated so highly that they are fixed to their set of nouns:

The cellphone on charge, the phone on hold, the blender on mince
At home, at church, at school
In trouble, in danger, in peril
On fire, on location, on camera

etc
I get it MM. But, since I am not a native, it's not easy for me to tell which phrase is idiomatic and which is not. I have got one final question related to this idiomatic use of "on".

"Why have you put on this channel ?"
Is this an idiomatic thing to say to someone who is watching some channel on TV or probably not watching it but has just put it on ? Do you think "turning on a channel" is more idiomatic than "putting on a channel" ?
Why have you put on / turned on this channel ? -- These both sound common and natural to me.
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I think you would use in when referencing something intangible, like mode,thezone,control,the process of etc whereas using on at least refers to something more concrete such on the look out, on board, on autopilot, on drugs.