Specific heat of solids

Picture a solid as N atoms, each oscillating about a fixed point in its lattice.

A one-dimensional oscillation has average energy 2 × ½k_BT = k_BT; in three dimensions an atom has 3k_BT.

For one mole (N = N_A) the energy is U = 3k_BT × N_A = 3RT.

At constant pressure ΔQ = ΔU + PΔV ≈ ΔU, because a solid's volume change is negligible. So C = ΔQ/ΔT = ΔU/ΔT = 3R.

Table 12.3 (molar specific heats, J mol⁻¹ K⁻¹, at room temperature and atmospheric pressure): aluminium 24.4, carbon 6.1, copper 24.5, lead 26.5, silver 25.5, tungsten 24.9. The specific heats in J kg⁻¹ K⁻¹ are 900.0, 506.5, 386.4, 127.7, 236.1 and 134.4.

3R ≈ 24.9 J mol⁻¹ K⁻¹ matches these values at ordinary temperatures; carbon is the exception.