Question

Is there any relationship berween thermal conductivity and heat capacity?

Answer 1

For metals there is a connection between the thermal conductivity and electric conductivity (Wiedemann–Franz law).

However specific heat is not directly related. This is because electric and thermal conductivity are due to the electrons, however the specific heat is mostly due to the ion vibrations (phonons).

Despite "classical" intuition electrons contribute almost nothing for specific heat in metals. Electrons in a typical metal behave close to an ideal fermion gas, in a very deep quantum range (typical Fermi temperature is about 40K Kelvins

hope this help you