Worked example: Estimate the intrinsic carrier density of silicon at 300 K (E_g = 1.1 eV, effective density of states N_c ≈ N_v ≈ 10¹⁹ cm⁻³). Step 1: Intrinsic density n_i = √(N_c·N_v)·e^{−E_g/2kT}. Step 2: kT at 300 K = 0.0259 eV; E_g/2kT = 1.1/(2×0.0259) = 21.2. Step 3: e^{−21.2} ≈ 6.1×10⁻¹⁰; n_i = 10¹⁹ × 6.1×10⁻¹⁰ ≈ 6×10⁹ cm⁻³. Step 4: Compare with atomic density ~5×10²² cm⁻³: only ~1 atom in 10¹³ contributes a carrier — pure silicon is nearly insulating! Step 5: Doping with 10¹⁵ donors/cm³ swamps this completely — conductivity becomes engineerable. Conclusion: the e^{−E_g/2kT} factor makes intrinsic carriers exquisitely temperature-sensitive — the physics behind thermistors and why chips need controlled doping.