Resistivity mean free path and scattering time of copper

[SMC]

so this is the question I'm having a little trouble with:

Assume that the ratio of copper resistivity at room and absolute zero temperatures (so called "residual resistance ratio") is 1000. Estimate the electron mean free path in copper at low temperatures.

we also know this:

Let us assume that at room temperature copper has the electrical conductivity of 10⁵ Ohm-1 cm-1 , and that its electron density is 10²³ cm-3 and its Fermi velocity is 10⁶ m/s.

so i know: mean free path = scattering time x velocity of electron
and velocity of electron can be approximated to fermi velocity

the question tells us: resistivity(room temp)/resistivity(T=0)=1000

resistivity in inversely proportional to scattering time so: scattering time(T=0)/scattering time(room temp)=1000

now my lecture notes say the typical value for scattering time in a metal (i assume at room temp.) is 10^-14 seconds
so scattering time(T=0) is 1000 x 10^-14=10^-11 seconds right?

i also know from lecture notes that v_f = \sqrt{\frac{2kT_f}{m}} and typical value for fremi velocity in metals is 10⁶ m/s. (k is Boltzmann constant and T_f is fermi temp. )

but how do i find the value of v_f for low temp?

ps sorry i haven't used latex in a while so I'm not really sure why the expression for fermi velocity isn't showing correctly

 

[mfb]

The fermi velocity is far above thermal velocities, it won't change significantly between absolute zero and room temperature.

now my lecture notes say the typical value for scattering time in a metal (i assume at room temp.) is 10^-14 seconds

I'm not sure if you are supposed to use this. That would ignore the copper-specific values for conductivity and electron density.