We used it on the chain as well as the other metal parts, and it worked well; however, after discussing with some bike experts and receiving comments about it, it has been determined that using it on the chain is a little risky. If you properly clean off the acid (by rinsing and drying thoroughly), it should be fine, but remnant acid could weaken the chain. So be careful!
Step 1: Materials
Citric acid: We used some lime juice we had leftover in the fridge, but any citrus-y juice will work--lemon, lime, etc. According to Urawaza, the citric acid reacts with the iron oxide (rust) so it can be removed.
Abrasive: We used steel wool, but anything gritty to help remove the rust will do.
Paper towels or an old rag. To clean.
A bike. Trying to remove the rust from a non-existent bicycle could prove too much for this Instructable.
Step 2: Pour Acid Onto Abrasive
Step 3: Attack the Bike!
Anything with rust that is accessible to clean and dry. Tire rims, supports, and gears (carefully). As for the chain, it is possible if you clean it thoroughly but you apply the acid at your own risk.
I find that the easiest way to clean the bike without taking it apart is to flip the bicycle upside down so that it is resting on the seat and the handlebars.
Your hands will probably get dirty--this is part of the fun. Take lots of dirty-hands pictures.
Step 4: Rinse
Step 5: Dry Thoroughly
Step 6: Lubricate
This cleaning process is just one of many variations on a theme--you can try any combination of citrus and grittiness. For example, the Urawaza calls for a specific mixture of salt and lemon juice. Let us know the results if you try any other way!
Your bicycle should be ready to ride smoothly through the streets and/or mountains. Onward, Joshmobile III, onward!