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You want to use either value depending on the information present. The problem would typically provide you with one or the other, and you may have to convert a value in moles to grams or the other way around. These are two different values; specific heat is joules per gram per degree Kelvin and Molar heat capacity is joules per mole per degree Kelvin.
The units of specific heat capacity and molar heat capacity can vary in terms of temperature units, but they are distinct based on whether there are grams or moles in the denominator. Joules or kilojoules can be in either numerator, and units of kelvin or celsius are also interchangeable in both. I believe the standard or general term that you would use more often is specific heat capacity, which uses grams over moles.
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