Question

When the size of a spherical nanoparticle decreases from 30 nm to 10 nm, The ratio of surface area/ volume becomes:
A.$\dfrac{1}{3}$ of the original
B.3 times the original
C.$\dfrac{1}{9}$ of the original
D.9 times the original

Answer 1

Hint: Here, first we have to find out the ratio of surface area to volume of a sphere. Formulas to be used are, surface area of sphere, that is, $4\pi {r^2}$ and volume of sphere, that is, $\dfrac{4}{3}\pi {r^3}$. Then we have to compare the values of the two given sizes using the obtained ratio.

Complete step by step answer:
Here, the sizes of the nanoparticle are given in diameter, so we have to convert the formulas of surface area and volume to diameter instead of radius.
The formula of the surface area of the sphere is $4\pi {r^2}$, where, r is radius.
We know that, diameter is two times of r, that is, $d = 2r$ . So, the formula of surface area of sphere is,
Surface area = $2 \times 2 \times \pi \times r \times r = \pi {d^2}$
So, the formula of the surface area of the sphere is $\pi {d^2}$.
Similarly, we have to convert the formula of volume of sphere in diameter.
The formula of volume of the sphere is $\dfrac{4}{3}\pi {r^3}$, where, r is radius.
We can also write, volume=$\dfrac{4}{3}\pi {r^3} = \dfrac{{2 \times 2}}{3}\pi \times r \times r \times \dfrac{d}{2} = \dfrac{{\pi {d^3}}}{6}$
Now, we have to find the ratio of surface area to volume.
Ratio=$\dfrac{{\pi {d^2}}}{{\dfrac{{\pi {d^3}}}{6}}} = \dfrac{6}{d}$
Now, we have to compare the ratio of two diameters, ${d_1} = 30$ and ${d_2} = 10$ using the obtained ratio.
$\dfrac{{{{\left( {\dfrac{\rm{SA}}{\rm{V}}} \right)}_2}}}{{{{\left( {\dfrac{\rm{SA}}{\rm{V}}} \right)}_1}}} = \dfrac{{\left(\dfrac{6}{d_2}\right)}}{{\left(\dfrac{6}{d_1}\right)}}$
$ \Rightarrow \dfrac{{{{\left( {\dfrac{\rm{SA}}{\rm{V}}} \right)}_2}}}{{{{\left( {\dfrac{\rm{SA}}{\rm{V}}} \right)}_1}}} = \dfrac{6}{{10}} \times \dfrac{{30}}{6} = 3$
So, the ratio of the final surface area to volume is 3 times the original ratio.

Hence the option B is the correct one.

Note: Always remember that we can calculate surface and volume for any 3 dimensional geometrical shapes. The term ‘surface area’ indicates the region or area occupied by the surface of any object and the term ‘volume’ indicates the available area in an object.