Wiedemann Franz law: Thermal Conductivity & Derivation

Wiedemann Franz Law states that dissimilar metals at the same specific temperature, have nearly the same value. According to Gustav Wiedemann, Thermal Conductivity and Electrical Conductivity for dissimilar metals have about the same value at the same temperature. This empirical law is named after Gustav Wiedemann and Rudolph Franz, who discovered in 1853 that Ludvig Lorenz discovered the relation of/with temperature in 1872.

Key Takeaways: Thermal Conductivity, Electrical Conductivity, Heat, Wiedemann Franz Law.

Read More: Heat

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Wiedemann Franz Law

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The ratio of the thermal conductivity of the material and the electrical conductivity of the material is directly relative to the temperature.

The proportion of the electronic contribution of thermal conductivity (κ) to electrical conductivity (σ) of metal is slightly similar to the temperature(T), according to the law.

K/σ = LT

Where,

k = Thermal Conductivity. It is a degree of measurement of a material to conduct heat.

σ = Electrical Conductivity is noted as a degree of measurement of a material to conduct electricity (1/ρ).

L = Proportionality constant, and the number is known as the Lorenz number

L= k/σT

L = π²/3(kB/e)²

L= 2.44 × 10^-8 WΩK^-2

The relationship between thermal and electrical conductivity is based on the fact that heat and electrical movement involves freely moving electrons in the metal. The Wiedemann-Franz law is based on the fact that free electrons in the metal are involved in both heat and electrical transport. It also indicates that when particle velocity increases, electrical conductivity decreases but thermal conductivity increases.

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Derivation of the law

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To derive the law, we must assume that the material is homogeneous and isotropic. A temperature gradient is then applied to this material.

c_v → Specific heat

n → Number of particles per unit volume

λ → mean free path of collisions

v → velocity of electrons

Comparing the equations (1) and (2), we get

Equation- 3

We know that the energy of free electrons is

Equation- 4

Put the equation (4) in (3)

Now, for an ideal gas at constant volume, the specific heat,

Equation- 7 and 8

Put equation (8) in (6), we get

Equation- 9

Between collisions, there is a mean free path and a meantime.

Equation- 10

Equation- 11

e → Charge of the electron = 1.602 × 10^-9C

τ → Collision time or meantime: It is the average time for the electron to move or travel before scattering.

v_d → Drift Velocity

Equation- 12

Consider the electrons which move in metal without any application of the electrical field. Then the equipartition theorem is given by

Equation- 13

From equation (13) we get m as

Equation- 15 and 16

We assume v = vd, then equation (16) becomes

Equation- 17

This law is known as Wiedemann-Franz Lorenz Law.

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Factors Affecting Wiedemann Franz Law

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Wiedemann-Franz law is based on the fact that free electrons in the metal are involved in both heat and electrical transport. It also indicates that when particle velocity increases, electrical conductivity drops but thermal conductivity increases.

Read More: Thermal Expansion

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Thermal Conductivity of Wiedemann Franz law

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• Heat transfer by conduction includes the transmission of energy within a material without the material moving at all. The thermal conductivity and temperature gradient of a substance are determined by the amount of heat transfer.
• For most circumstances, arithmetic methods may be used to measure the conduction of heat transfer across smooth walls, but the heat transfer must be described in terms of the thermal gradient.
• Theoretically, thermal conductivity can be thought of as a container for medium-dependent things that define the amount of heat loss per unit range of temperature change.

Thermal Conductivity and Electrical Conductivity

• A direction-finding derivative is a function's measured gradient, which indicates where the function's highest rate-of-change is located.
• In the meantime, when the net energy transfer is from high to low temperature, the direction of heat transfer will be reversed to the temperature gradient.
• The heat transfer direction will be perpendicular to the equal-temperature surfaces adjacent to a heat source at its maximum value.

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Statement of Wiedemann and Franz law

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Wiedemann and Franz Law states that "The ratio of a metal's thermal conductivity to its electrical conductivity is relative to temperature". In terms of quality, this link is based on the fact that the heat and electrical transport in the metal contain unfettered electrons. The thermal conductivity increases as the average particle velocity rises, increasing the forward energy conveyance.

The electrical conductivity, on the other hand, decreases as particle velocity increases because the impacts divert electrons from furthering charge transit. The average velocity squared, which is proportional to the kinetic temperature, determines the ratio of thermal to electrical conductivity.

A classical mono-atomic gas's molar heat capacity is given as

cv = 3/2R 

cv = 3/2NAk

By monitoring electrons as a typical gas and comparing the thermal conductivity to the electrical conductivity, the Wiedemann-Franz law can be established.

The following are the expressions for thermal and electrical conductivity:

Thermal conductivity k = n(v)λk/ 2

Electrical conductivity σ = ne²λ/ m(v)

The mean particle speed can be calculated using kinetic theory as follows:

(v) = \(\sqrt{8}\)kT/ πm

The temperature can be used to express the ratio of these two quantities. The Wiedemann-Franz law is exemplified by the thermal to electrical conductivity ratio:

k/σ = 4k²T/ πe²

Read More: Thermal Conductivity

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Things to Remember

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• Experiments have revealed that, while the value of L is roughly constant, it is not identical for all materials.
• Kittel delivers some standards of L changing from L = 2.23×10^-2 W Ω K^-2  for copper at 0°C to L = 3.2×10^-8 W Ω K^-2  for tungsten at 100°C. 
• According to Rosenberg, the Wiedemann and Franz law is usually valid at high and low temperatures (i.e., a few Kelvins), but not for intermediate temperatures.
• The electrical and thermal conductivities of many high purity metals rise as the temperature drops.
• However, the value of L can change with temperature in some materials (such as aluminium or silver). L can decline by as much as ten times in the purest silver samples and at very low temperatures.

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