Wiedemann Franz Law

The Wiedemann Franz law states that the ratio of the thermal conductivity of a metal to its electrical conductivity is proportional to the temperature. This law was first proposed by Gustav Wiedemann and Rudolph Franz in 1853.

Mathematical Expression

The Wiedemann Franz law can be expressed mathematically as:

$$κ/σ = LT$$

where:

• κ is the thermal conductivity of the metal
• σ is the electrical conductivity of the metal
• L is the Lorenz number
• T is the temperature

The Lorenz number is a constant that is equal to 2.44 × 10^-8 WΩ/K².

Limitations

The Wiedemann Franz law is only valid for metals at temperatures above the Debye temperature. At temperatures below the Debye temperature, the law breaks down and the ratio of thermal conductivity to electrical conductivity decreases.

The Wiedemann Franz law is a fundamental law of physics that relates the thermal and electrical conductivity of metals. It has a number of applications, but it is only valid for metals at temperatures above the Debye temperature.

Factors Affecting Wiedemann Franz Law

The Wiedemann Franz law states that the ratio of the thermal conductivity to the electrical conductivity of a metal is proportional to the temperature. This law is only valid for metals at temperatures above the Debye temperature.

There are several factors that can affect the Wiedemann Franz law, including:

• Temperature: The Wiedemann Franz law is only valid for metals at temperatures above the Debye temperature. At lower temperatures, the thermal conductivity of metals decreases more rapidly than the electrical conductivity, so the ratio of the two decreases.
• Impurities: Impurities can scatter electrons and phonons, which can reduce the thermal and electrical conductivity of a metal. This can also affect the Wiedemann Franz law.
• Magnetic field: A magnetic field can affect the motion of electrons and phonons, which can also affect the thermal and electrical conductivity of a metal. This can also affect the Wiedemann Franz law.
• Crystal structure: The crystal structure of a metal can affect the thermal and electrical conductivity of a metal. This can also affect the Wiedemann Franz law.

The Wiedemann Franz law is a useful tool for understanding the thermal and electrical properties of metals. However, it is important to be aware of the factors that can affect this law, such as temperature, impurities, magnetic field, and crystal structure.

Wiedemann Franz Law Derivation

The Wiedemann-Franz law states that the ratio of the thermal conductivity of a metal to its electrical conductivity is proportional to the temperature. This law can be derived from the following considerations.

Assumptions

1. The metal is a good conductor of electricity and heat.
2. The electrons in the metal are free to move.
3. The mean free path of the electrons is much smaller than the size of the metal.
4. The temperature is constant.

Derivation

1. The thermal conductivity of a metal is given by:

$$k=\frac{1}{3}C_vl\bar{v}$$

where:

• $C_v$ is the specific heat capacity of the metal at constant volume
• $l$ is the mean free path of the electrons
• $\bar{v}$ is the average speed of the electrons

2. The electrical conductivity of a metal is given by:

$$\sigma=ne\mu$$

where:

• $n$ is the number of free electrons per unit volume
• $e$ is the charge of an electron
• $\mu$ is the electron mobility

3. The Wiedemann-Franz law states that:

$$\frac{k}{\sigma}=LT$$

where:

• $L$ is the Lorenz number
• $T$ is the temperature

4. Substituting the expressions for $k$ and $\sigma$ into the Wiedemann-Franz law, we get:

$$\frac{\frac{1}{3}C_vl\bar{v}}{ne\mu}=LT$$

5. Rearranging, we get:

$$L=\frac{1}{3n}\frac{C_vl\bar{v}}{e\mu T}$$

6. The Lorenz number is a constant for a given metal. Therefore, the Wiedemann-Franz law can be written as:

$$\frac{k}{\sigma}=LT$$

where $L$ is a constant.

The Wiedemann-Franz law is a fundamental law of metals that relates their thermal and electrical conductivities. This law can be derived from the basic principles of kinetic theory.

Wiedemann Franz Law Applications

The Wiedemann Franz law states that the thermal conductivity of a metal is proportional to its electrical conductivity. This law has a number of important applications in the field of materials science and engineering.

Applications of the Wiedemann Franz Law

• Thermal conductivity of metals: The Wiedemann Franz law can be used to measure the thermal conductivity of metals. This information is important for designing heat sinks and other thermal management devices.
• Electrical conductivity of metals: The Wiedemann Franz law can also be used to measure the electrical conductivity of metals. This information is important for designing electrical circuits and other electrical devices.
• Thermal properties of semiconductors: The Wiedemann Franz law can be used to study the thermal properties of semiconductors. This information is important for designing semiconductor devices, such as transistors and integrated circuits.
• Thermal properties of insulators: The Wiedemann Franz law can be used to study the thermal properties of insulators. This information is important for designing thermal insulation materials.
• Thermal properties of composite materials: The Wiedemann Franz law can be used to study the thermal properties of composite materials. This information is important for designing composite materials with specific thermal properties.

The Wiedemann Franz law is a fundamental law of physics that has a number of important applications in the field of materials science and engineering. This law can be used to measure the thermal conductivity and electrical conductivity of metals, semiconductors, insulators, and composite materials. This information is important for designing a wide variety of devices and materials.

Limitations of Wiedemann Franz Law

The Wiedemann Franz law states that the ratio of the thermal conductivity to the electrical conductivity of a metal is proportional to the temperature. This law is valid for most metals at room temperature, but there are some exceptions.

Deviations from the Wiedemann Franz Law

The Wiedemann Franz law does not hold for:

• At very low temperatures, the thermal conductivity of metals decreases more rapidly than the electrical conductivity, so the ratio of the two decreases. This is because the electrons that carry heat are scattered more strongly by impurities and defects at low temperatures.
• At very high temperatures, the thermal conductivity of metals increases more rapidly than the electrical conductivity, so the ratio of the two increases. This is because the electrons that carry heat are able to move more freely at high temperatures.
• In the presence of a magnetic field, the thermal conductivity of metals decreases, while the electrical conductivity is unaffected. This is because the magnetic field causes the electrons to move in a helical path, which reduces their ability to carry heat.
• In the presence of impurities or defects, the thermal conductivity of metals decreases, while the electrical conductivity may be unaffected. This is because the impurities or defects scatter the electrons that carry heat, reducing their ability to move freely.

The Wiedemann Franz law is a useful tool for understanding the thermal and electrical properties of metals. However, it is important to be aware of the limitations of the law, so that it can be used correctly.

Solved Examples on Wiedemann Franz Law

The Wiedemann-Franz law states that the ratio of the thermal conductivity to the electrical conductivity of a metal is proportional to the temperature. This law can be used to calculate the thermal conductivity of a metal if its electrical conductivity and temperature are known.

Example 1:

A copper wire has an electrical conductivity of 5.96 x 10$^{7}$ S/m and a thermal conductivity of 401 W/m-K at room temperature (293 K). Calculate the Wiedemann-Franz ratio for copper.

Solution:

The Wiedemann-Franz ratio is given by: $$ L = κ/σT $$

where:

• L is the Wiedemann-Franz ratio (in WΩ/K$^2$)
• κ is the thermal conductivity (in W/m-K)
• σ is the electrical conductivity (in S/m)
• T is the temperature (in K)

Substituting the given values into the equation, we get:

$$ L = (401 W/m-K) / (5.96 x 10^7 S/m * 293 K) = 2.23 x 10^{-8} WΩ/K^2 $$

Therefore, the Wiedemann-Franz ratio for copper is 2.23 x 10$^{-8}$ WΩ/$^2$.

Example 2:

A gold wire has an electrical conductivity of 4.11 x 10$^7$ S/m and a thermal conductivity of 318 W/m-K at room temperature (293 K). Calculate the thermal conductivity of gold using the Wiedemann-Franz law.

Solution:

We can use the Wiedemann-Franz law to calculate the thermal conductivity of gold as follows:

$$ κ = LσT $$

where:

• L is the Wiedemann-Franz ratio (in WΩ/K$^2$)
• σ is the electrical conductivity (in S/m)
• T is the temperature (in K)

The Wiedemann-Franz ratio for gold is 2.23 x 10$^{-8}$ WΩ/K$^2$ (as calculated in Example 1). Substituting the given values into the equation, we get:

$κ = (2.23 x ^{-8}$WΩ/K$^2$) * (4.11 x 10$^7$ S/m) * (293 K) = 242 W/m-K

Therefore, the thermal conductivity of gold is 242 W/m-K.

Wiedemann Franz Law FAQs

What is the Wiedemann Franz Law?

The Wiedemann Franz Law states that the ratio of the thermal conductivity of a metal to its electrical conductivity is a constant. This constant is known as the Lorenz number and is equal to 2.44 × 10$^{-8}$ WΩ/K2.

What is the physical significance of the Wiedemann Franz Law?

The Wiedemann Franz Law shows that the thermal and electrical conductivities of a metal are closely related. This is because both thermal and electrical conductivity are due to the movement of electrons in a metal.

What are the applications of the Wiedemann Franz Law?

The Wiedemann Franz Law is used to:

• Determine the thermal conductivity of a metal if its electrical conductivity is known.
• Determine the electrical conductivity of a metal if its thermal conductivity is known.
• Study the relationship between the thermal and electrical properties of metals.

What are the limitations of the Wiedemann Franz Law?

The Wiedemann Franz Law is only valid for metals at low temperatures. At high temperatures, the Lorenz number increases due to the increased scattering of electrons by phonons.

What are some other laws that relate the thermal and electrical properties of materials?

• The Wiedemann Franz Law is a special case of the more general Mott relation, which relates the thermal conductivity of a material to its electrical conductivity and Seebeck coefficient.
• The Nernst effect relates the thermal gradient in a material to its electrical potential.
• The Ettingshausen effect relates the magnetic field in a material to its thermal gradient.

Conclusion

The Wiedemann Franz Law is a fundamental law of physics that relates the thermal and electrical conductivities of metals. It has a number of applications in the study of the properties of metals.