Using Maxwell-Boltzman Statistics

[says]

Homework Statement

Determine if the classical approximation (Maxwell-Boltzman statistics) could be employed for the following case: a) Electron gas in a metal at 2.7K (cubic metal lattice of spacing 2Å)

Homework Equations

Maxwell-Boltzman statistics are acceptable to use if the de broglie wavelength, λ << d

Relation - Particle spacing, d, and density, (N/V) = 1/d³
d = (V/N)^1/3

Kinetic energy, KE = p²/2m = 3/2K_bT
λ = h/p = h/(3mK_bT)^1/2

The Attempt at a Solution

2Å = 2*10^-10m
(N / V) = 1 / (2Å)³
(N / V) = 1 / 8*10^-30m

λ << (V/N)^1/3

h³/(3mK_bT)^3/2 << (V/N)

λ = h³/(3mK_bT)^3/2

h³/(3mK_bT)^3/2 << (V/N)

(N/V)(h³/(3mK_bT)^3/2) << 1

(1 / 8*10^-30)(h³/(3mK_bT)^3/2)

(1 / 8*10^-30)(6.62*10^-34)³) / (3)(9.109*10^-31)(1.38*10^-23)(2.7)

4.10*10^-124 << 1

So the classical approximation could be used. I'm pretty sure my derivation is correct, but I'm not sure if the calculation is.

 

[mjc123]

Far too complicated, and the answer's very wrong. You have a formula for λ, why not use it and compare the result with the given value of d?

 

[says]

So I should just evaluate:
h/(3mK_bT)^1/2 << 2Å

λ=6.62*10^-34/((3)(9.109*10^-31)(1.38*10^-23)(2.7))λ= 6.5*10¹⁸, which is not << 2Å, so we can't use Maxwell-Boltzman statistics

 

[mjc123]

I got 6.5*10^-8m, which is still > 2Å. Did you forget to square-root the denominator?

 

[says]

Yes, I must have forgot to square-root the denominator. so 6.5*10^-8 m> 2Å so Maxwell-Boltzman statistics would not be a good approximation to use.