Answer:
The fraction of the volume of the atom that is taken up by the nucleus is
.
The density of a proton is
.
Explanation:
Diameter of the atom ,d = 2.50 Å
Radius of the atom ,r = 0.5 d=0.5 × 2.50 Å = 1.25Å
Volume of the sphere= ![\frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
Volume of atom = V
..[1]
Diameter of the nucleus ,d' =
Radius of the nucleus ,r' = 0.5 d'=
Volume of nucleus = V'
..[2]
Dividing [2] by [1]
![\frac{V'}{V}=\frac{\frac{4}{3}\pi r'^3}{\frac{4}{3}\pi r^3}](https://tex.z-dn.net/?f=%5Cfrac%7BV%27%7D%7BV%7D%3D%5Cfrac%7B%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%27%5E3%7D%7B%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%7D)
![=\frac{r'^3}{r^3}=\frac{(4.5\times 10^{-5}\AA)^3}{(1.25 \AA)^3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7Br%27%5E3%7D%7Br%5E3%7D%3D%5Cfrac%7B%284.5%5Ctimes%2010%5E%7B-5%7D%5CAA%29%5E3%7D%7B%281.25%20%5CAA%29%5E3%7D)
![\frac{V'}{V}=4.6656\times 10^{-14}](https://tex.z-dn.net/?f=%5Cfrac%7BV%27%7D%7BV%7D%3D4.6656%5Ctimes%2010%5E%7B-14%7D)
The fraction of the volume of the atom that is taken up by the nucleus is
.
Diameter of the proton ,d = ![1.72\times 10^{-15} m = 1.72\times 10^{-13} cm](https://tex.z-dn.net/?f=1.72%5Ctimes%2010%5E%7B-15%7D%20m%20%3D%201.72%5Ctimes%2010%5E%7B-13%7D%20cm)
1 m = 100 cm
Radius of the proton,r = 0.5 d=![0.5\times 1.72\times 10^{-13} cm=8.6\times 10^{-14} cm](https://tex.z-dn.net/?f=0.5%5Ctimes%201.72%5Ctimes%2010%5E%7B-13%7D%20cm%3D8.6%5Ctimes%2010%5E%7B-14%7D%20cm)
Volume of the sphere= ![\frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
Volume of atom = V
![V=\frac{4}{3}\times 3.14\times (8.6\times 10^{-14} cm)^3=2.664\times 10^{-39}cm^3](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B4%7D%7B3%7D%5Ctimes%203.14%5Ctimes%20%288.6%5Ctimes%2010%5E%7B-14%7D%20cm%29%5E3%3D2.664%5Ctimes%2010%5E%7B-39%7Dcm%5E3)
Mass of proton, m = 1.0073 amu = ![1.0073\times 1.66054\times 10^{-24} g](https://tex.z-dn.net/?f=1.0073%5Ctimes%201.66054%5Ctimes%2010%5E%7B-24%7D%20g)
![1 amu = 1.66054\times 10^{-24} g](https://tex.z-dn.net/?f=1%20amu%20%3D%201.66054%5Ctimes%2010%5E%7B-24%7D%20g)
Density of the proton : d
![d=\frac{m}{V}=\frac{1.0073\times 1.66054\times 10^{-24} g}{2.664\times 10^{-39}cm^3}=6.278\times 10^{14} g/cm^3](https://tex.z-dn.net/?f=d%3D%5Cfrac%7Bm%7D%7BV%7D%3D%5Cfrac%7B1.0073%5Ctimes%201.66054%5Ctimes%2010%5E%7B-24%7D%20g%7D%7B2.664%5Ctimes%2010%5E%7B-39%7Dcm%5E3%7D%3D6.278%5Ctimes%2010%5E%7B14%7D%20g%2Fcm%5E3)
The density of a proton is
.