Questions: Thermal Conductivity and Material Properties

Question 1

Why do metals have far higher thermal conductivity than insulators like aerogel or wood?

• A. Metals have higher specific heat capacity, allowing them to store and release more thermal energy
• B. Metals contain free electrons that move at Fermi velocities (~10⁶ m/s) with long mean free paths, carrying heat far more efficiently than phonons
• C. Metal atoms are more tightly packed, letting phonon vibrations propagate without gaps
• D. Metals have lower density, so thermal energy waves encounter less resistance

Answer: (B)

The key is the microscopic heat carrier. In metals, free electrons carry heat; they move at Fermi velocities (~10⁶ m/s) with mean free paths of hundreds of nanometers. In insulators, only phonons (lattice vibrations) carry heat; phonons are slower (~10³ m/s) and scatter more frequently. This difference in carrier speed and mean free path — captured by k = (1/3)C_v·v_avg·λ — explains the ~20,000× difference in k between copper and aerogel.

Question 2

A material has very high electrical conductivity. What does the Wiedemann-Franz law predict about its thermal conductivity?

• A. Low thermal conductivity — good electrical conductors store charge rather than transmitting heat
• B. High thermal conductivity — the same free electrons carry both charge and heat, so k and σ scale together
• C. Moderate thermal conductivity regardless of electrical properties, because heat is carried by phonons not electrons
• D. The relationship depends on density, not on the type of electrical carrier

Answer: (B)

The Wiedemann-Franz law (k/σ = L₀T) states that for metals, the ratio of thermal to electrical conductivity is proportional to temperature via the Lorenz number. This holds because the same free electrons carry both. Measuring one gives you information about the other. This law does NOT apply to insulators, where phonons carry heat but electrons carry no charge — so the connection between k and σ breaks down for non-metals.

Question 3

Increasing the temperature of a pure metal usually increases its thermal conductivity.

• T. True
• F. False

Answer: False

In metals at room temperature, rising temperature increases phonon population, which intensifies electron-phonon scattering. More scattering shortens the electron mean free path (λ), which decreases k. So for metals, k typically decreases with increasing temperature at room temperature and above. The relationship reverses near absolute zero, where phonon scattering is minimal and mean free paths become very long — this non-monotonic behavior is characteristic of phonon and electron conductors alike.

Question 4

A material's thermal conductivity depends on both how much energy its heat carriers can store (heat capacity) and how far they travel before being scattered (mean free path).

• T. True
• F. False

Answer: True

This is exactly what the kinetic expression k = (1/3)·C_v·v_avg·λ encodes. C_v is the heat capacity per unit volume (energy stored per degree), v_avg is the carrier speed, and λ is the mean free path (distance before scattering). Large k requires all three to be high. Aerogel has low k not just because it has few carriers, but because its structure creates extremely short mean free paths for whatever phonons exist.

Question 5

Explain why aerogel (k ≈ 0.02 W/m·K) has roughly 20,000 times lower thermal conductivity than copper (k ≈ 400 W/m·K), even though both are solid materials.

Model answer: Copper has free electrons as heat carriers — they move at ~10⁶ m/s with mean free paths of hundreds of nanometers, making electron-mediated heat transport extremely efficient. Aerogel has no free electrons; heat is carried only by phonons through its sparse silica network. Aerogel's nanoporous structure creates extremely short mean free paths (phonons scatter constantly at the silica-air interfaces), and the material has very low heat capacity per unit volume due to its mostly-air composition. All three factors in k = (1/3)C_v·v_avg·λ favor copper and disfavor aerogel simultaneously.

The comparison illustrates that the right question is not 'what is this material?' but 'what carries heat through it, and how far can each carrier travel?' The carrier type (electron vs phonon), carrier speed, and mean free path together determine k. Aerogel's engineered nanostructure is specifically designed to minimize all three, making it one of the best thermal insulators known.