Step 1: Concept Overview:
The Wiedemann-Franz law posits that the ratio of thermal conductivity (\(\kappa\)) to electrical conductivity (\(\sigma\)) in a metal is directly proportional to absolute temperature (T). This proportionality constant is known as the Lorentz number (L).
Step 2: Core Formula:
The law is mathematically represented as:
\[ \frac{\kappa}{\sigma} = LT \]
Alternatively, the Lorentz number can be expressed as:
\[ L = \frac{\kappa}{\sigma T} \]
Based on the free electron model (Sommerfeld model), the theoretical Lorentz number is defined using fundamental constants:
\[ L = \frac{\pi^2}{3} \left(\frac{k_B}{e}\right)^2 \]
Here, \(k_B\) represents the Boltzmann constant and \(e\) signifies the elementary charge.
Step 3: Detailed Calculation:
Let's compute the theoretical value:
Boltzmann constant, \( k_B \approx 1.38 \times 10^{-23} \) J/K
Elementary charge, \( e \approx 1.60 \times 10^{-19} \) C
\[ L = \frac{\pi^2}{3} \left(\frac{1.38 \times 10^{-23}}{1.60 \times 10^{-19}}\right)^2 \]\[ L \approx 3.29 \times (0.8625 \times 10^{-4})^2 \]\[ L \approx 3.29 \times (7.44 \times 10^{-9}) \]\[ L \approx 2.448 \times 10^{-8} \text{ (J/C/K)}^2 \]
The units are equivalent to \( (J/C)^2/K^2 = V^2/K^2 \). Given that \(W = V \cdot A\) and \( \Omega = V/A \), it follows that \( W\Omega = V^2 \). Therefore, the units can also be expressed as \( W\Omega/K^2 \) or Watt ohm per squared Kelvin.
Step 4: Conclusion:
The approximate theoretical value of the Lorentz number is \(2.45 \times 10^{-8}\) W\(\Omega\)/K\(^2\).