Assertion–Reason: intrinsic resistivity of Si versus Ge Assertion (A): The intrinsic resistivity of silicon is lower than that of germanium. Reason (R): The free-electron concentration in intrinsic germanium is greater than that in intrinsic silicon.

Introduction / Context:
Intrinsic carrier concentration n_i and mobility determine the intrinsic resistivity ρ_i of semiconductors. Comparing silicon and germanium at 300 K highlights how bandgap size and mobilities affect ρ_i.

Given Data / Assumptions:

• Intrinsic material (no intentional doping) at room temperature.
• Resistivity relation: ρ_i = 1 / (q * n_i * (μ_n + μ_p)).
• Typical values: n_i(Ge) ≫ n_i(Si) at 300 K due to Ge's smaller bandgap.

Concept / Approach:
Germanium has a smaller bandgap (~0.66 eV) than silicon (~1.12 eV), so its intrinsic carrier concentration is much higher at a given temperature. Since ρ_i is inversely proportional to n_i (and mobilities are of the same order), germanium has a much lower intrinsic resistivity than silicon. Therefore the assertion that silicon has lower intrinsic resistivity is false. The reason correctly states that intrinsic germanium has greater free-electron concentration than silicon.

Step-by-Step Solution:
Note n_i(Ge) ≫ n_i(Si) at 300 K.Compute trend: ρ_i ∝ 1 / n_i → ρ_i(Ge) < ρ_i(Si).Conclude: A is false; R is true.

Verification / Alternative check:
Handbook values: ρ_i(Si) ~ 2.3 × 10^5 Ω·cm, ρ_i(Ge) ~ 46 Ω·cm (order-of-magnitude figures), confirming the trend.

Why Other Options Are Wrong:
(a) and (b) treat A as true; it is not. (c) rejects the true reason. (e) denies both statements, which is inaccurate.

Common Pitfalls:

• Confusing doped (extrinsic) with intrinsic behaviour.
• Assuming mobility differences outweigh n_i differences; the exponential n_i dependence dominates.

Final Answer:
A is false but R is true