Question #206323

) State and prove Wiedemannâ€“Franz law (or) Derive expressions for electrical and thermal conductivities on the basis of classical free electron theory and deduce the value of Lorentz number

Expert's answer

The ratio of a metal's thermal conductivity (K) to its electrical conductivity () is proportional to the metal's absolute temperature.

"\\frac{K}{\\sigma} \u221d T \\implies \\frac{K}{\\sigma} =L T"

Using the formulas for electrical and thermal conductivity of metals, we can drive Widemann-Franz's law using classical theory.

The expression for thermal conductivity

"K= \\frac{K_B nv \\lambda}{2}"

The expression for elecetricl conductivity

"\\sigma = \\frac{ne^2 \\tau}{m}"

"\\frac{K}{\\sigma}= \\frac{1\/2 K_B nv \\lambda}{ne^2 \\tau \/ m}"

"\\frac{K}{\\sigma}= \\frac{1}{2} \\frac{mK_B v^2}{e^2 }"

"\\frac{K}{\\sigma}= \\frac{1}{2} mv^2\\frac{K_B}{e^2 }"

We know that kinetic energy s given by

"\\frac{1}{2}mV^2= \\frac{3}{2} K_BT"

"\\frac{K}{\\sigma}= \\frac{3}{2} K_BT\\frac{K_B}{e^2 }"

"\\frac{K}{\\sigma T}= \\frac{3}{2}\\frac{K^2_B}{e^2 }"

"\\frac{K}{\\sigma T}= L"

L is the Lorentz number

As a result, it is demonstrated that the ratio of a metal's thermal and electrical conductivity is proportional to the metal's absolute temperature.

"\\implies L = \\frac{3}{2} \\frac{K_B^2}{e^2}"

"L = \\frac{3}{2} \\frac{(1.38*10^{-23})^2}{2(1.6*10^{-19})^2} = 1.12 * 10^{-8} W \\Omega K^{-2}"

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