---
title: "Conductivity of sodium from electron density"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:sodium-conductivity-electron-density"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/sodium-conductivity-electron-density
site: BCT Engineering
---

# Conductivity of sodium from electron density

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2073 Chaitra, 5 marks
- Video: [Electrical Conductivity | Problem | Free electron theory](https://www.youtube.com/watch?v=fQsIMEQ5ei0) — Sasthra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/sodium-conductivity-electron-density
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

What will be the conductivity of sodium metal having electron density 2.5×10²⁸ m⁻³ and relaxation time 3×10⁻¹⁴ sec?

## Short answer

Using σ = ne²τ/m, with n = 2.5×10²⁸ m⁻³, τ = 3×10⁻¹⁴ s, e and m constants, conductivity σ ≈ 2.1×10⁷ S/m.

## Answer

Conductivity from the free-electron model is σ = ne²τ/m.

**Given:** n = 2.5×10²⁸ m⁻³, τ = 3×10⁻¹⁴ s, e = 1.602×10⁻¹⁹ C, m = 9.11×10⁻³¹ kg.

```text
σ = n e² τ / m
  = (2.5×10²⁸)(1.602×10⁻¹⁹)²(3×10⁻¹⁴) / (9.11×10⁻³¹)
  = (2.5×10²⁸)(2.566×10⁻³⁸)(3×10⁻¹⁴) / (9.11×10⁻³¹)
  = 1.925×10⁻²³ / 9.11×10⁻³¹
  = 2.11×10⁷ (Ω·m)⁻¹
```

**Answer: σ ≈ 2.1×10⁷ Ω⁻¹m⁻¹ (S/m)**, so the resistivity is ρ = 1/σ ≈ 4.7×10⁻⁸ Ω·m, which is the right order for a metal.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The conductivity formula is
   A. σ = n e^2 τ/m
   B. σ = n e τ/m^2
   C. σ = n τ/(e^2 m)
   D. σ = e^2 τ/(n m)
   Answer: A. Free-electron model. (Key point: Formula)

2. σ is about
   A. 2.1 × 10^7 S/m
   B. 2.1 × 10^5 S/m
   C. 2.1 × 10^9 S/m
   D. 4.7 × 10^−8 S/m
   Answer: A. 1.925e−23/9.11e−31 = 2.11e7. (Key point: Result)

3. The resistivity is about
   A. 4.7 × 10^−8 Ω·m
   B. 2.1 × 10^7 Ω·m
   C. 4.7 × 10^−6 Ω·m
   D. 2.1 × 10^−7 Ω·m
   Answer: A. ρ = 1/σ. (Key point: Resistivity)

## Close variants set in other exams

- [Conductivity of sodium metal](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asodium-conductivity-atomic-weight) — Engineering Physics, chapter 8, asked 2×

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