Question

In gold at room temperature, the mobility of mobile electrons is about $\mathbf{4}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{\left(}\frac{{\displaystyle \raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.}}{{\displaystyle \raisebox{1ex}{${\displaystyle \text{V}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{m}}$}\right.}}\mathbf{\right)}$, and there are about $\mathbf{5}\mathbf{.}\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}$mobile electrons per cubic meter. Calculate the conductivity of gold, including correct units.

Short answer

The conductivity of gold is $4.1\times {10}^7\frac{\text{C}·\text{V}}{{\text{m}}^{\text{3}}·\text{s}}$.

Step-by-step solution

A conductivity:

Conductivity is proportional to the product of carrier mobility and concentration.

A given data:

Mobility of mobile electrons,$n=5.9\times {10}^{28}\raisebox{1ex}{$\text{mobiles electrons}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^3$}\right.$

Mobility of the charge carriers, $u=4.3\times {10}^{-3}\left(\frac{{\displaystyle \raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.}}{{\displaystyle \raisebox{1ex}{${\displaystyle \text{V}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{m}}$}\right.}}\right)$

Charge of the electron, $q=-1.6\times {10}^{-19}\text{C}$

The conductivity of gold, including correct units:

Conducting of gold is given by,

$\sigma =\left|q\right|nu$

Substitute known values in the above equation.

$\begin{array}{rcl}\sigma & =& \left|-1.6\times {10}^{-19}\text{C}\right|\times \left(5.9\times {10}^{28}\raisebox{1ex}{$\text{mobiles electrons}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^3$}\right.\right)\times \left(4.3\times {10}^{-3}\frac{{\displaystyle \raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}}{{\displaystyle \raisebox{1ex}{$\text{V}$}\!\left/ \!\raisebox{-1ex}{$\text{m}$}\right.}}\right)\\ & =& 4.1\times {10}^7\frac{\text{C}·\text{V}}{{\text{m}}^{\text{3}}·\text{s}}\end{array}$

Hence, the conductivity of gold is$\begin{array}{rcl}& & 4.1\times {10}^7\frac{\text{C}·\text{V}}{{\text{m}}^{\text{3}}·\text{s}}\end{array}$.