The resistivity of pure copper due to lattice vibrations is 1.70 µΩ·cm at 300 K and 0.20 µΩ·cm at 77 K. For a copper alloy in which added impurities give a temperature-independent residual resistivity of 0.40 µΩ·cm, which value is closest to the resistivity ratio ρ(300 K)/ρ(77 K) according to Matthiessen's rule?

Answer: C. About 3.5

By Matthiessen's rule, the total resistivity is the sum of a temperature-dependent lattice-vibration component and a temperature-independent impurity and defect component, ρ = ρ_th(T) + ρ_res. At 300 K it is 1.70 + 0.40 = 2.10 µΩ·cm and at 77 K 0.20 + 0.40 = 0.60 µΩ·cm, so the ratio is 2.10/0.60 = 3.5. About 8.5 is the ratio of the lattice-vibration components of pure copper alone (1.70/0.20), ignoring the residual resistivity. About 10.5 adds the residual resistivity only to the 300 K value, (1.70 + 0.40)/0.20, overlooking that it adds equally at both temperatures. Because the residual component makes up a larger share at lower temperatures, the ratio of resistance at room temperature to that at cryogenic temperature (the residual resistivity ratio) is used to evaluate the purity and defect content of metals.