Question

According to Weidemann-Franz-Lorentz Law, the theoretical value of Lorentz number (L) for metals is:

• A. \(2.45 \times 10^{-10}\) Watt ohm/deg\(^2\)
• B. \(2.45 \times 10^{-18}\) Watt ohm/deg\(^2\)
• C. \(2.45 \times 10^{-8}\) Watt ohm/deg\(^2\)
• D. \(2.45 \times 10^{-14}\) Watt ohm/deg\(^2\)

Hint:

This is a famous result in solid-state physics. Memorizing the value \(L \approx 2.45 \times 10^{-8}\) W\(\Omega\)/K\(^2\) is very useful, as it is often asked as a direct factual question.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Wiedemann-Franz law states that the ratio of the thermal conductivity (\(\kappa\)) to the electrical conductivity (\(\sigma\)) of a metal is proportional to the absolute temperature (T). The constant of proportionality is called the Lorentz number (L).
Step 2: Key Formula or Approach:
The law is expressed as:
\[ \frac{\kappa}{\sigma} = LT \]
Or, the Lorentz number is defined as:
\[ L = \frac{\kappa}{\sigma T} \]
The theoretical value of the Lorentz number, as derived from the free electron model (Sommerfeld model), is a combination of fundamental constants:
\[ L = \frac{\pi^2}{3} \left(\frac{k_B}{e}\right)^2 \]
where \(k_B\) is the Boltzmann constant and \(e\) is the elementary charge.
Step 3: Detailed Explanation:
Let's calculate 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 \( (J/C)^2/K^2 = V^2/K^2 \). Since \(W = V \cdot A\) and \( \Omega = V/A \), we have \( W\Omega = V^2 \). So the units are \( W\Omega/K^2 \) or Watt ohm/deg\(^2\).
Step 4: Final Answer:
The theoretical value of the Lorentz number is approximately \(2.45 \times 10^{-8}\) W\(\Omega\)/K\(^2\).