Numerical Estimation of the momentum of free electrons

[hasan_researc]

Homework Statement

This is a problem in numerical estimation.
Estimate a typical maximum momentum pmax for the free electrons in a metal where
pmax = \left(\frac{3nh^{3}}{8\pi}\right)^{1/3}

(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.)

Homework Equations

Density of water = 1000 kg/m³
Molar mass of water = 18 g/mol
Molar mass of iron = 55 g/mol (Iron, I suppose, exhibits the typical properties of metals.)

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 = 10²⁸, then

pmax = \left(\frac{3nh^{3}}{8\pi}\right)^{1/3}
= \left(\frac{3*10^{28}*(6.63*10^{-34})^{3}}{8\pi}\right)^{1/3}
= (6.63*10^{-34})\left(\frac{3*10^{28}}{8*3}\right)^{1/3}
= (6.63*10^{-34})\left(\frac{10^{28}}{8}\right)^{1/3}
= \left(\frac{(6.63*10^{-34})*10^{9}}{2}\right)
= 3.3*10^{-25}

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

Any help would be greatly appreciated.

 

[diazona]

You could easily look up the actual density (mass/volume) of iron and calculate the electron density from that...

 

[hasan_researc]

I surely can, but I want to find the density using my own reasoning (w/o help from external sources).

 

[Redbelly98]

The problem statement says to either look up or guesstimate the density of a typical metal. Can you estimate a metal's density, relative to water, from your own personal experience? I.e., is it heavier or lighter than water? By what factor, roughly?

If you can't make an estimate you are reasonably confident about, my advice is to look it up.

 

[hasan_researc]

Thank you for your help!

" Can you estimate a metal's density, relative to water, from your own personal experience?" :

I'm not just sure whether using personal experience is a good method to estimate the density of a typical metal !? Otherwise, I could easily quote a factor (eg, 3 or 4).

What I want to be able to do is work out the metal density relative to water density by comparing the atomic structure and properties of metals and water. Unfortunately, my knowledge is not adequate to answer that. So, I am hoping someone could help me with that.

[I can easily look up the value in a book, but then I will not have learned how to guesstimate ths quantity, so...]

 

[diazona]

I'm not just sure whether using personal experience is a good method to estimate the density of a typical metal !? Otherwise, I could easily quote a factor (eg, 3 or 4).

Yes. That's exactly what estimation is. How accurate do you think you need to be with this?

What I want to be able to do is work out the metal density relative to water density by comparing the atomic structure and properties of metals and water.

This would be a detailed calculation, not an estimation. If you want to know how to do it, I'd suggest you head over to the General Physics forum and ask how you can calculate the density of iron from its crystal structure, or the density of water from its molecular bonding properties. But I can't imagine that anyone would actually calculate the density of iron this way if all they need is a numerical value for that density.