(a) Calculate the number of electrons in a small, electrically neutral silver pin that has...

Step 1
a) It's necessary to determine the number of electrons in a small, electrically neutral silver pin of mass 10g. Molar mass of silver is known ${\displaystyle M=107.87\frac{g}{m}ol}$. It is possible to determine the number of atoms inside the pin
${\displaystyle N_{Ag}=\frac{m_{\pi n}}{M}\cdot 6\cdot {10}^{23}}$ atoms
${\displaystyle N_{Ag}=5.56225\cdot {10}^{22}}$ atoms
Inside every atom there a 47 electrons, it's necesarry to calculate the total amount of charge.
${\displaystyle N_e=47N_{Ag}}$
${\displaystyle N_e=2.61\cdot {10}^{24}}$ elektrons
${\displaystyle q_e=47\cdot N_e}$
${\displaystyle q_e=435087C}$
Step 2
It's necessary to calculate the number of electrons in 1 mC.
${\displaystyle N=\frac{1mC}{1.67\cdot {10}^{-19}C}=5.99\cdot {10}^{15}}$ electrons
Now, we can determine the number of electrons for even ${\displaystyle {10}^9}$.
${\displaystyle n=\frac{N}{\frac{N_e}{{10}^9}}=2.29}$ for ever ${\displaystyle {10}^9}$ electons

Part A
Moles of silver ${\displaystyle =10.0\frac{g}{107.8}\frac{g}{m}ol=0.093moles}$
calculate number silver atom using the Avogadro law
${\displaystyle =0.093\times 6.023\times {10}^{23}=5.58\times {10}^{22}atoms}$
number of electron is therefore number of atom x atomic number that is ${\displaystyle 5.58\times {10}^{22}\times 47=2.62\times {10}^{24}electrons}$
Part B
we well know electron charge is ${\displaystyle 1.60\times {10}^{-19}c}$
convert 1.00mc to coulombs ${\displaystyle =\frac{1}{1000}=1.0\times {10}^{-3}c}$
Number of electron in ${\displaystyle 1.0\times {10}^{-3}}$ is therefore
${\displaystyle \left\{(1.0\times \frac{{10}^{-3}}{1.60}\times {10}^{-19})\right\}=6.25\times {10}^{15}}$
number of ${\displaystyle \left({10}^9\right)}$ electrons ${\displaystyle \left\{(2.62\times \frac{{10}^{24}}{{10}^9})\right\}=2.62\times {10}^{15}}$
number of electron added per ${\displaystyle {10}^9}$ is therefore ${\displaystyle =\left\{(6.25\times \frac{{10}^{25}}{2.62}\times {10}^{15})\right\}=2.3per\left({10}^9\right)}$