(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is a problem in numerical estimation.

Estimate a typical maximum momentum pmax for the free electrons in a metal where

pmax = [tex] \left(\frac{3nh^{3}}{8\pi}\right)^{1/3}[/tex]

(In estimating n, the number density of free electrons, assume there is one such electron associated with every atom in the metal, and either look up or guestimate how dense

typical metals are compared with water – for which you should know the density.)

2. Relevant equations

Density of water = 1000 kg/m^{3}

Molar mass of water = 18 g/mol

Molar mass of iron = 55 g/mol (Iron, I suppose, exhibits the typical properties of metals.)

3. The attempt at a solution

Molar mass of iron/Molar mass of water = 3. Therefore, one iron atom is 3 times heavier than a water molecule.

But I do not know anything about the packing of atoms/molecules in iron or water.

SO, how do I guesstimate the number density of free electrons in a free metal?

Anyway, if I proceed in my calculation with n = 10^{28}, then

pmax = [tex]\left(\frac{3nh^{3}}{8\pi}\right)^{1/3}[/tex]

= [tex]\left(\frac{3*10^{28}*(6.63*10^{-34})^{3}}{8\pi}\right)^{1/3}[/tex]

= [tex](6.63*10^{-34})\left(\frac{3*10^{28}}{8*3}\right)^{1/3}[/tex]

= [tex] (6.63*10^{-34})\left(\frac{10^{28}}{8}\right)^{1/3}[/tex]

= [tex]\left(\frac{(6.63*10^{-34})*10^{9}}{2}\right)[/tex]

= [tex]3.3*10^{-25}[/tex]

Does the answer look reasonable. How can the estimation working be improved?

Any help would be greatly appreciated.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Numerical Estimation of the momentum of free electrons

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