For silicon (density 2330 kg/m³, specific heat 700 J/(kg·K)), take the room-temperature thermal conductivity as 150 W/(m·K) and the average phonon speed as 6000 m/s. Estimating the phonon mean free path l from κ = (1/3)Cvl, which has the same form as in the kinetic theory of gases (C is the heat capacity per unit volume), which value is closest?

Answer: C. About 46 nm

The heat capacity per unit volume is C = ρc = 2330×700 ≈ 1.63×10⁶ J/(m³·K). l = 3κ/(Cv) = 3×150/(1.63×10⁶×6000) ≈ 4.6×10⁻⁸ m, about 46 nm. About 15 nm drops the factor of 1/3 and uses κ/(Cv), and about 5.1 nm applies the 1/3 on the wrong side, giving κ/(3Cv). This estimate is an average that represents all phonons by a single speed and mean free path; in reality, low-frequency phonons with mean free paths of several hundred nm or more also carry much of the heat. For that reason, in silicon thin films and nanowires about 100 nm thick, surface scattering lowers the thermal conductivity well below the bulk value.