Wiedemann–Franz Law
The Wiedemann–Franz law is one of the important laws in solid state physics and thermal physics. It explains the relationship between electrical conductivity and thermal conductivity in metals. This law was proposed by the German scientists Gustav Wiedemann and Rudolph Franz in 1853. They observed that metals which conduct electricity well also conduct heat well. This important relation helped scientists understand the behavior of free electrons inside metals.In simple words, the same electrons that carry electric current in a metal are also responsible for carrying heat energy. Therefore, if a metal has high electrical conductivity, it usually has high thermal conductivity too.
Statement of Wiedemann–Franz Law
The law states that:
The ratio of thermal conductivity to electrical conductivity of a metal is directly proportional to the absolute temperature of the metal.
Mathematically:
K/σ=LT
Where:
K = Thermal conductivity of the metal
σ = Electrical conductivity
T = Absolute temperature in Kelvin
L = Lorenz number (constant)
This equation means that when temperature increases, the ratio of thermal conductivity to electrical conductivity also increases proportionally.
Meaning of Terms
1. Thermal Conductivity (K)
Thermal conductivity is the ability of a substance to conduct heat. If one end of a metal rod is heated, heat flows to the colder end. The ease with which heat flows is measured by thermal conductivity.
High thermal conductivity materials:
Low thermal conductivity materials:
2. Electrical Conductivity (σ)
Electrical conductivity is the ability of a material to allow electric current to pass through it.
Metals like copper and silver have very high electrical conductivity because they contain many free electrons.
3. Absolute Temperature (T)
Temperature measured in Kelvin scale is called absolute temperature.
T (K) = t( C°) + 273
4. Lorenz Number (L)
The proportionality constant in Wiedemann–Franz law is called Lorenz number.
L=2.44 × 10-8 W ΩK-2
This value is approximately constant for many metals at ordinary temperatures.
Physical Explanation of the Law
To understand the Wiedemann–Franz law, we must know the role of free electrons in metals.
In a metal:
Outer electrons are loosely bound.
These electrons move freely throughout the metal.
They are called conduction electrons or free electrons.
These electrons perform two important functions:
1. Electrical Conduction
When a potential difference is applied, free electrons drift from one end to another, producing electric current.
2. Thermal Conduction
When one end of a metal is heated:
Electrons near the hot end gain kinetic energy.
They move rapidly toward cooler regions.
During collisions with atoms and other electrons, they transfer energy.
Thus heat is conducted through the metal.
Since the same electrons are responsible for both heat conduction and electrical conduction, the two properties are related. This is the basis of Wiedemann–Franz law.
Simple Example
Take a copper spoon and place one end in hot tea.
After some time:
The handle becomes hot.
Heat is transferred through the spoon.
Now connect copper wire in an electric circuit:
Current flows easily.
This shows that copper is a good conductor of both heat and electricity.
Mathematical Form of the Law
According to classical electron theory:
K = 1/3Cvvλ
Where:
(Cv) = specific heat per unit volume
(v) = average electron speed
(λ) = mean free path
Electrical conductivity:
σ = ne2τ/m
Where:
(n) = number of electrons per unit volume
(e) = electron charge
(m) = electron mass
(τ) = relaxation time
By combining these equations, scientists obtained:
K/ σ T
This constant is the Lorenz number.
Importance of Wiedemann–Franz Law
The law was very useful in the development of electron theory of metals. It showed that electrons carry both charge and heat.
It also gave evidence for:
Free electron model
Drude theory
Sommerfeld theory
Later quantum mechanics improved the theoretical explanation of the law.
Applications of Wiedemann–Franz Law
1. Estimation of Thermal Conductivity
If electrical conductivity of a metal is known, thermal conductivity can be estimated.
2. Engineering Design
Used in selecting materials for:
Heat exchangers
Electrical wiring
Cooling systems
3. Electronics Industry
Materials with high electrical and thermal conductivity are used in:
- CPU heat sinks
- Power devices
- Connectors
4. Scientific Research
Used in studying:
- Metals
- Alloys
- Superconductors
- Nanomaterials
Examples of Good Conductors
These metals approximately obey Wiedemann–Franz law.
Limitations of Wiedemann–Franz Law
Although important, the law is not perfectly valid in all cases.
1. Low Temperatures
At very low temperatures, electron scattering changes, so deviations occur.
2. Alloys
Alloys contain impurities which disturb electron motion.
3. Semiconductors
In semiconductors, heat may be carried by lattice vibrations (phonons), not only electrons.
4. Insulators
Insulators have no free electrons, so the law does not apply.
5. Superconductors
Electrical resistance becomes zero, so the law fails in normal form.
Modern View
According to quantum theory:
Electrons obey Fermi-Dirac statistics.
Only electrons near Fermi level participate in conduction.
This gives more accurate Lorenz number and better agreement with experiment.
Experimental Verification
Scientists measure:
Electrical conductivity using circuits
Thermal conductivity using heat flow methods
Then calculate:
L= K/ σ T
For many pure metals, this value remains nearly constant
Advantages of the Law
Simple relation between two properties
Useful in material science
Supports electron theory
Helpful in engineering applications
Disadvantages
Not universal
Fails for non-metals
Deviations at low temperature
Requires pure metals for best accuracy
Conclusion
The Wiedemann–Franz law is an important law in solid state physics that connects thermal and electrical properties of metals. It clearly shows the role of free electrons in conduction phenomena. Though it has limitations, it remains a fundamental concept in material science and condensed matter physics. It laid the foundation for modern theories of metallic conduction and continues to be useful in practical engineering and scientific applications.
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