What is the mean free path of conduction electrons in Na at T = 273 K?

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SUMMARY

The mean free path (λ) of conduction electrons in sodium (Na) at a temperature of 273 K is calculated to be 3.56 nm using the resistivity (4.2 µΩ·cm) and the average Fermi speed (1.07e6 m/s). The relevant equation applied is ρ = (Me)(Vav)/(Ne)(e)^2λ, where Vav is defined as sqrt(3kT/Me). The discussion highlights the importance of correctly interpreting the variables and units involved in the calculation, particularly the relationship between resistivity and Fermi speed.

PREREQUISITES
  • Understanding of electrical resistivity and its units
  • Familiarity with Fermi speed and its significance in solid-state physics
  • Knowledge of the equation relating resistivity, electron density, and mean free path
  • Basic grasp of thermodynamic principles, specifically the Boltzmann constant (k)
NEXT STEPS
  • Study the derivation and application of the equation ρ = (Me)(Vav)/(Ne)(e)^2λ
  • Learn about Fermi energy and its calculation in metals
  • Explore the concept of mean free path in different materials and its dependence on temperature
  • Investigate the relationship between resistivity and electron mobility in conductors
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Physics students, materials scientists, and electrical engineers interested in solid-state physics and the behavior of conduction electrons in metals.

w3390
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Homework Statement



The resistivities and Fermi speeds of Na, Au, and Sn at T = 273 K are 4.2 µ\Omega·cm, 2.04 µ\Omega·cm, and 10.6 µ\Omega·cm, and 1.07e6 m/s, 1.39e6 m/s, and 1.89e6 m/s respectively. Use these values to find the mean free paths λ for the conduction electrons in these elements.

Homework Equations


The Attempt at a Solution



I think the equation I should be using is \rho=(Me)(Vav)/(Ne)(e)^2\lambda. The main issue with this problem is that the units are all over the place. Using this equation, I get \lambda= 3.56nm for Na. All I'm concerned with right now is Na. I have a feeling though that since I am given the Fermi speed, I need to use a formula that incorporates the Fermi speed, but I cannot find any that seem relevant.
 
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w3390 said:
I think the equation I should be using is \rho=(Me)(Vav)/(Ne)(e)^2\lambda.

What is V_av in this equation?
The units of this equation are correct. The unit of Ne is m^-3, The units of e is As.
 
Vav in this equation is sqrt(3kT/Me), where k= 1.38e-23 J/K. This is where I think the problem lies. Using the equation I stated in my first post, there is no place to substitute the Fermi speeds. I cannot find any equation that fits the Fermi speed to the resistivity. Since I am given the Fermi speed, I know how to find the Fermi energy, but that gets me nowhere.
 
Nevermind...figured it out
 

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