Residual resistivity of binary alloys at absolute zero At T = 0 K, the residual resistivity of a binary alloy primarily is:

Introduction / Context:Even at 0 K, real metals and alloys exhibit a finite “residual resistivity” due to static disorder such as impurity atoms and lattice imperfections. In alloys, this disorder is intrinsic to the mixed lattice.

Given Data / Assumptions:

• Binary substitutional alloy with a host metal and a solute element.
• Low-temperature limit (phonon scattering negligible).
• Random distribution of solute atoms (no long-range order).

Concept / Approach:Residual resistivity in alloys follows the Nordheim rule: scattering from compositional disorder depends on the probability of encountering unlike neighbors, typically scaling with c(1 − c), where c is the atomic fraction of the solute. Thus, ρres is composition-dependent rather than a simple arithmetic operation on the pure-metal residual resistivities.

Step-by-Step Solution:Set T → 0 K ⇒ phonon contribution ρph → 0.Total ρ ≈ ρres determined by static scattering centers.In a random alloy, scattering rate increases with compositional disorder ⇒ ρres = f[c(1 − c)].Therefore, ρres varies with solute concentration and is not simply additive or multiplicative.

Verification / Alternative check:Experimental plots of resistivity versus composition for systems like Cu–Ni or Ag–Au show maxima near intermediate compositions, consistent with c(1 − c) behavior.

Why Other Options Are Wrong:Sum/difference/product of pure-metal residual resistivities ignores the dominant alloy-disorder scattering. Composition independence is contradicted by abundant data.

Common Pitfalls:Assuming Matthiessen’s rule implies simple addition of host and solute “resistivities” without considering how disorder scales with composition.

Final Answer:dependent on the concentration of the minor (solute) component in the alloy