Question

The density of gold is 19 \[g/c{m^3}\]. If \[1.9\] \[ \times {10^{ - 4}}\]g of gold is dispersed in one litre of water to give a sol having spherical gold particles of radius 10nm, then the number of gold particles per \[m{m^3}\]of the solution will be:
(A) \[1.9 \times {10^{12}}\]
(B) \[6.3 \times {10^{14}}\]
(C) \[6.3 \times {10^{10}}\]
(D) \[2.4 \times {10^6}\]

Answer 1

Hint: We are provided the density of gold which is mass by volume. Using the radius of the sphere, the volume of the sphere can be calculated. With these values of two volumes, the number of particles can easily be calculated by dividing them.

Complete step-by-step answer:
Density is a physical property of a liquid or sol determined by ratio of mass to volume. When a large number of molecules combine to form the colloidal particles, it is called sol (multimolecular colloid). Gold sol is one such colloid, which is negatively charged due to the adsorption of negatively charged gold oxide ions on dispersed phase.
These are lyophobic sol in which dispersed phase has no attraction for the dispersion medium., where dispersed phase is solute and dispersed medium is solvent. These are irreversible sols.
Volume of gold that is dispersed in one litre water = \[\dfrac{{mass}}{{density}}\]
Volume = \[\dfrac{{1.9 \times {{10}^{ - 4}}g}}{{19g/c{m^3}}}\]\[ = 1 \times {10^{ - 5}}c{m^3}\]
Provided that, Radius of spherical gold sol particle (r) = \[10nm = 10 \times {10^{ - 9}}m = {10^{ - 6}}cm\]
So, Volume of the spherical gold sol particle = \[\dfrac{4}{3}\pi {r^3}\]
Putting the value of radius in it, we get
\[ = \dfrac{4}{3} \times 3.14 \times ({10^{ - 6}}) = 4.19 \times {10^{ - 18}}c{m^3}\]
Number of gold sol particles in \[1 \times {10^{ - 5}}c{m^3}\]= \[\dfrac{{1 \times {{10}^{ - 5}}c{m^3}}}{{4.19 \times {{10}^{ - 18}}c{m^3}}}\]
And number of gold sol particles in 1 \[m{m^3}\]= \[\dfrac{{2.38 \times {{10}^{12}}}}{{{{10}^6}}} = 2.38 \times {10^6}\]
\[ \approx 2.4 \times {10^6}\]

Hence, the correct option is (D).

Note: Gold is a metal and so gold sol are metal sol. It is prepared by making metal electrodes and then immersed in a dispersion medium when an electric arc is struck. Dispersion and condensation are involved in this process. It has colloidal suspension of nanoparticles of gold.