Question

Copper (cu) has crystal structure Atomic mass of Cu is 63.55 gmol^-1 and radius of a copper atom is 0.128 nm, mean speed of conduction electrons in Cu is 1.5 ×10^6 ms^-1 and frequency of vibration of the Cu atoms at 27 °C is 4×10^12 S^-1 Nordheim’s

Coefficient of Au in Cu is 5500

Ω −m

Caculate (a) the atomic density of copper (b) Estimate the drift mobility of electronics in cu (c) find the resistivity of pure copper cu (d) find the resistivity of copper if 1 at % of Au is mixed in Cu

Answer 1

a) Density by the definition $\rho=\frac{m}{V}$.

V is the volume of sphere $V=\frac{4}{3}\pi*R^3$

From the frequency of vibration $\nu=\frac{m}{M}->m=M*\nu$

$\rho=\frac{3M\nu}{4R^3\pi}$

$\rho=\frac{3*4*10^{12}*63.55*10^{-3}}{4*3.14*0.128^{3}*10^{-27}}=29650*10^{36}\frac{kg}{m^3}$

b) The drift velocity: $v=\frac{E\sigma}{ne}$ , where $\sigma$ - electrical conduction, $\sigma=\frac{1}{\rho}$ , $\rho$ - resistivity E - electrical intensity, $E=\frac{U}{l}$, U - voltage, $l$ - the length of conductor, n - concetration of electrons, $n=\frac{N}{V}$, N - number of electrons, V - volume, where electrons are concentrared, e - charge of electron, $e=-1.6*10^{-19}$ (Cl). Substite it to our formula. Then we get:

$v=\frac{VU}{Nle\rho}$.

The number of electrons $N=N_A*\nu$ and voltage $U=IR$ bu the Ohm's law. Then,

$v=\frac{VIR}{\nu*N_A*\rho*le}$ .

Now, write that $C=\frac{R}{l}$ - the Nordheim’s Coefficient.

$v=\frac{VIC}{\nu*N_A*\rho*e}$.

By the definition $I=\frac{e}{t}$ , t - time.

$v=\frac{VC}{\nu*N_A*\rho*t}$ .

Let's say that $t=\frac{1}{\nu}$ and we'll get:

$v=\frac{VC}{N_A*\rho}$

$V=\frac{4}{3}\pi*R^3$

$v=\frac{4\pi*R^3C}{3N_A*\rho}$

$v=\frac{3*3.14*(0.128)^3*10^{-27}*5500*10^{-9}}{3*6.02*10^{23}*1.7*10{-8}}=3.54*10^{-51} \frac{m}{s}$