Question

How can you tell what a metal is or its atomic weight just by knowing the specific heat.

Answer 1

The specific heat is approximately
\[C = 3Nk_B = 3 n R\]
so you can find the number of moles in a sample:
\[ n = \frac{C}{3R} \]
If you know the mass of the sample (say, M) then the molar mass is just
\[ \frac{M}{n} = \frac{3MR}{C} \]