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.
FAQs on Wiedemann Franz Law
1. What is the Wiedemann-Franz law?
The Wiedemann-Franz law is a fundamental principle in physics that states that the ratio of the electronic contribution to the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is directly proportional to its absolute temperature (T). Essentially, it explains why materials that are good conductors of electricity are typically also good conductors of heat.
2. What is the formula for the Wiedemann-Franz law, and what does each term signify?
The mathematical expression for the Wiedemann-Franz law is:
κ / σ = L * T
The terms in this formula represent:
- κ (kappa): The thermal conductivity of the material, which measures its ability to conduct heat.
- σ (sigma): The electrical conductivity of the material, which measures its ability to conduct electric current.
- L: The Lorenz number, a constant of proportionality.
- T: The absolute temperature in Kelvin (K).
3. What is the Lorenz number and why is it significant in the Wiedemann-Franz law?
The Lorenz number (L) is the constant of proportionality in the Wiedemann-Franz law. Its theoretical value is approximately 2.44 × 10-8 W·Ω·K-2. Its significance lies in the fact that it is nearly constant for a wide range of metals at high temperatures. This uniformity suggests that the underlying mechanism for both electrical and thermal conduction in these materials is the same—the movement of free electrons.
4. Why are good electrical conductors also good thermal conductors, according to the Wiedemann-Franz law?
The direct relationship between electrical and thermal conductivity exists because both processes in metals are dominated by the same charge carriers: free electrons. When an electric field is applied, these electrons move to create an electric current (high electrical conductivity). Similarly, when a temperature gradient is applied, these same electrons transport kinetic energy from hotter regions to colder regions, resulting in efficient heat transfer (high thermal conductivity). The law quantifies this shared transport mechanism.
5. Under what conditions does the Wiedemann-Franz law fail or show significant deviations?
The Wiedemann-Franz law is a powerful approximation but fails under certain conditions, including:
- Intermediate Temperatures: At temperatures that are neither very high nor very low, inelastic scattering processes become significant, causing the Lorenz number to deviate from its constant value.
- Very Low Temperatures: At temperatures near absolute zero, electron-electron scattering can become more dominant than electron-impurity scattering, leading to a breakdown of the law.
- Semiconductors and Insulators: The law does not apply to semiconductors or insulators. In these materials, heat is primarily transported by lattice vibrations (phonons) rather than electrons, decoupling thermal conductivity from the very low electrical conductivity.
6. What are some practical applications or implications of the Wiedemann-Franz law?
The Wiedemann-Franz law has important practical implications in material science and engineering. Its primary uses include:
- Material Characterisation: It allows for the estimation of a metal's thermal conductivity, which can be difficult to measure directly, by measuring its electrical conductivity, which is much simpler.
- Design of Heat Sinks: In electronics, materials like copper and aluminium are chosen for heat sinks because their high electrical conductivity implies high thermal conductivity, allowing them to efficiently dissipate heat from components like CPUs.
- Thermoelectric Devices: Understanding deviations from the law is crucial for developing efficient thermoelectric materials, which aim to have high electrical conductivity but low thermal conductivity to maintain a temperature gradient.
7. How does the Wiedemann-Franz law distinguish metals from insulators?
The Wiedemann-Franz law serves as a clear distinction between the conductive properties of metals and insulators.
In metals, there is a vast sea of free electrons that act as the primary carriers for both charge and heat, so the law generally holds true.
In insulators, there are virtually no free electrons. Consequently, their electrical conductivity (σ) is extremely low. Heat is conducted primarily through lattice vibrations (phonons), a mechanism completely independent of charge transport. Therefore, the direct relationship between κ and σ described by the law does not exist for insulators.
