Question

When stearic acid \[\left( {{C_{18}}{H_{36}}{O_2}} \right)\] is added to water, its molecules collect at the surface and form a monolayer. The cross-sectional area of each stearic acid molecule is \[0.21n{m^2}\]. If \[1.4 \times {10^{ - 4}}g\] of stearic acid is needed to form a monolayer over water in a dish of diameter\[20cm\]. (The area of the circle of the radius \[r\] is \[\left( {\pi {r^2}} \right)\], then the value of Avogadro’s number is:
A. \[3 \times {10^{23}}\]
B. \[6 \times {10^{23}}\]
C. \[4 \times {10^{23}}\]
D. \[1 \times {10^{23}}\]

Answer 1

Hint: To answer this problem first we need to be clear about the basic concepts first, one mole of a substance is equivalent to \[6.022 \times {10^{23}}\] units of that substance, particles or atoms. The number \[6.022 \times {10^{23}}\] is known as Avogadro's number or Avogadro's constant. The Avogadro number is the proportionality factor that relates the number of constituent particles in a sample with the amount of substance in that sample.

Complete step-by-step answer:The area of the dish of \[20cm\] in diameter which is given in the question is given by the formula \[\pi {r^2}\]
Hence the radius is given by
$r = \dfrac{{20}}{2} = 10cm = 0.01{m^2}$
By substituting the values, the formula for area is given as,
$\pi \times 0.01{m^2}$
As given in the question \[0.21 \times {10^{ - 18}}{m^2}\]is the area of one molecule
Hence, \[\pi \times 0.01{m^2}\]is the area of \[ = \dfrac{{\pi \times 0.01}}{{0.21 \times {{10}^{ - 18}}}}\]molecules
Since the molecular weight of stearic acid is $284g/mol$.
The number of moles of stearic acid present in \[1.4 \times {10^{ - 4}}g\] of stearic acid is given as,
\[1.4 \times {10^{ - 4}}g\]stearic acid \[ = \dfrac{{1.4 \times {{10}^{ - 4}}}}{{284}}mol\]
\[ = \dfrac{{1.4 \times {{10}^{ - 4}}}}{{284}}{N_o}\]molecules
By equating the number of molecules present in \[1.4 \times {10^{ - 4}}g\] of stearic acid and \[\dfrac{{\pi \times 0.01}}{{0.21 \times {{10}^{ - 18}}}}\] molecules we get,
\[\
   = \dfrac{{1.4 \times {{10}^{ - 4}}}}{{284}}{N_o} = \dfrac{{\pi \times 0.01}}{{0.21 \times {{10}^{ - 18}}}} \\
  {N_o} = 3.30 \times {10^{23}} \\
  3.30 \times {10^{23}} \\
\ \]
Therefore, Avogadro’s number is \[3.30 \times {10^{23}}\]

Additional Information:
Self-assembled monolayers (SAM) of natural molecules are molecular congregations shaped unexpectedly on surfaces by adsorption and are sorted out into pretty much enormous arranged domains. The units may be electrons, atoms, ions, or molecules, depending on the nature of the substance and the character of the reaction.

So, the correct option is A.

Note: Area is fairly that is occupied by an object when it is laying on a surface. That area is the space which is given by the object. Though cross-sectional area is an area which we acquire when a similar object is cut into two pieces. The area of that specific cross segment is known as cross sectional area.