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Wiedemann Franz Law

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Franz Law is one of the important laws in physics. This law was discovered and termed after the German physicists, Gustav Wiedemann and Rudolph Franz, in 1835. Gustav Wiedemann revealed that thermal Conductivity (κ) and electrical Conductivity (σ) are roughly having an identical value at the same temperature for dissimilar metals.

This empirical law is named after Gustav Wiedemann and Rudolph Franz, who, in 1853, described that κ / σ has about the identical value for dissimilar metals at the precise temperature. In 1872, the proportionality of κ / σ with temperature was revealed by Ludvig Lorenz.


Wiedemann Franz law

In Wiedemann Franz law

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).

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

\[\frac{K}{σ}\]= LT

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Here,

L = proportionality constant, and it is named as the Lorenz number. 

L = \[\frac{K}{σT}\]=\[\frac{π^{2}}{3}(\frac{K_B}{e})^{2}\]WΩK-2    

  • The connection in the middle of the thermal and electrical conductivity is centered on the point that heat and the electrical movement contain freely roaming electrons in the metal. 

  • The thermal conductivity raises the velocity of the average particle and also surges in the frontward energy movement. Alternatively, electrical conductivity reduces the velocity of the particle.


Thermal Conductivity of Wiedemann Franz law

Heat transfer by conduction includes the transmission of energy inside a material deprived of any movement of the material altogether. The amount of heat transfer establishes the thermal conductivity and the temperature gradient of the material. 

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Arithmetical methods can be utilized for the measurement of the conduction of heat transfer across smooth walls, but the heat transfer must be stated concerning the thermal gradient for most cases.

Theoretically, thermal conductivity can be assumed as the vessel for the medium-dependent things which describe the level of heat loss per unit range to the rate of change of temperature.

The measured gradient of a function is a direction-finding derivative, which indicates in the track of the maximum rate-of-change of the function. 

The heat transfer's direction will be reversed to the temperature gradient in the meantime when the net energy transfer will be from high to low temperature. 

The maximum value of the heat transfer direction will be perpendicular to the equal-temperature surfaces adjacent to a heat source.


State Wiedemann and Franz law

This law depicts that "the ratio of the thermal conductivity to the electrical conductivity of a metal is relative to the temperature." Qualitatively, this connection is centered on the detail that the heat and electrical transport contain the unrestricted electrons in the metal. 

The thermal conductivity upsurges with the average particle velocity, that rises the forward carriage of energy. 

However, the electrical conductivity cuts with particle velocity rise because the impacts distract the electrons from forwarding transportation of charge. This revenue that the ratio of thermal to electrical conductivity hangs upon the average velocity squared, which is relational to the kinetic temperature. 

The molar heat capacity of a classical mono-atomic gas is indicated as

cv = \[\frac{3}{2}R=\frac{3}{2}N_Ak\]

The Wiedemann-Franz law can be assumed by observing the electrons like a conventional gas & comparing the resultant thermal conductivity to the Electrical Conductivity. 

The thermal and electrical conductivity's expressions are outlined here below:

Thermal conductivity k = \[\frac{n(v)λk}{2}\]

Electrical conductivity σ = \[\frac{ne^{2}λ}{m(v)}\]

The mean particle speed from kinetic theory can be expressed as;

(v)=\[\sqrt{\frac{8kT}{πm}}\]

The ratio of these quantities can be stated in terms of the temperature. The ratio of thermal to electrical conductivity exemplifies the Wiedemann-Franz Law as:

\[\frac{k}{σ}\] = \[\frac{4k^{2}T}{πe^{2}}\]

This is in the form of the Wiedemann Franz Law.

The value of the constant has an error in this conventional calculation. When the quantum mechanical conduct is done, the rate of the constant is initiated as:

L = \[\frac{k}{σT}\] = \[\frac{π^{2}k^{2}}{3e^{2}}\]= 2.45 × \[10^{-8}WΩ/K^{2}\]

The point that the ratio of thermal to electrical conductivity times the temperature is constant forms the core of the Wiedemann-Franz Law. 

Notably, it is also free of the number density of the particles and the particle mass.

Wiedemann Franz law limitations

Experimentations have uncovered that the value of L, while approximately constant, is not precisely identical for all materials. 

Kittel delivers some standards of L changing from L = 2.23×10−8 W Ω K−2 for copper at 0 °C to L = 3.2×10−8 W Ω K−2 for tungsten at 100°C. 

Rosenberg archives that the Wiedemann and Franz law is usually functional for high temperatures and low temperatures (i.e., a few Kelvins), but may not hold at in-between temperatures.

  • In many high purity metals, both the electrical and thermal conductivities increase as temperature declines. 

  • In some materials (such as aluminum or silver), however, the value of L also may fall with temperature. In the cleanest silver samples and at very little temperatures, L can drop by as much as an amount of 10.

FAQ (Frequently Asked Questions)

Q1. What are the vital factors that must be available in Wiedemann Franz's law?

Ans- K is noted as thermal conductivity, and σ is termed as electrical conductivity. This law formulates that the ratio of electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of metal is somewhat similar to the temperature (T).

k/σ = LT

Q2. Which factors are affecting the Wiedemann Franz law?

Ans- The Lorenz number L in dissipated semiconductors has durable enslavement on some system limits, the strong point of interatomic interactions, dimensionality, and Fermi level. This law is not worthy, or the value of the Lorenz number can be bridged, at least in some cases.

Q3. Describe the main difference between electrical Conductivity and Thermal Conductivity?

Ans- Electrical conductivity originates from the motion of electrons, or holes under the impact of pressure, known as voltage. Thermal conductivity originates from the transmission of kinetic energy under the effect of a pressure termed as heat.

Q4. If the thermal conductivity is 45 W/K-m and the electrical conductivity is 4.7 * 106 at a given Lorentz number i.e., of metal, then what will be the temperature?

Ans- As we know; k/σ = LT

Here, k = 78 and σ= 4.7 * 106 and L= 2.45 × 10-8WΩ/K2

By putting all them all in the equation we will get;

T = 45/4.7 × 106  × 2.45  × 10-8 = 390.78 K