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the most common units that we'll always be working with is J, g, K, L, and atm. It's good practice to convert to these units because a lot of the given units of, for examples, constants will be given in J/mol or kJ/mol. the 8.314 constant is also in K or kelvin units so leaving things in celsius can be troublesome if you do forget to convert units.
When you are calculating the specific heat capacity this will be in grams, but when you are calculating the molar heat capacity this will be in moles. As far as degrees C goes, what every your constant is given in (kelvin or celcius) is what you will have to work with. It does not seem to matter that much, just whatever is given in the problem is what you should use for temperature units.
I think it really depends on what the problem asks/provides. Like what other people said, the common units are J, K, g, L, etc., since they match with the other units when calculating. As long as you know how to convert them, then I think all should be good. Hope that helps!
The biggest concern with units is just making sure that everything you are calculating is in consistent units. A given question may ask for different units, so just pay attention to that and make sure that you use the correct constants.
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