Question

Calculate the surface area (in $sq\,m$ ) of a catalyst that adsorbs ${10^3}c{m^3}$ of nitrogen is reduced to STP in order to form a monolayer. The effective area in $(in\,\,{m^2})$ occupied by the ${N_2}$ molecules on the surface is: ( Surface area of ${N_2}$ = $1.62 \times {10^{ - 15}}c{m^2}$ )

Answer 1

There will be two formulas used in this question. One will be to find out the number of moles of ${N_2}$ gas which have been adsorbed then we know the area of one ${N_2}$ molecule, and if we simply multiply it with the total number of moles, we will have the total surface area occupied by all the ${N_2}$ molecules.
The formula to determine the number of molecules is: $Total\,number\,of\,molecules = \dfrac{{Avogadro's\,number}}{{22.414\,L}} \times Volume\,of\,gas$
Using the value of total number of moles, find out the value for total surface area.
$Total\,surface\,area\,of\,molecules = surface\,area\,of\,one\,molecule \times total\,number\,of\,molecules$
The information we have from the question is: $Surface\,area\,of\,1\,\,{N_2}\,molecule = \,1.62 \times {10^{ - 15}}c{m^2},\,Volume\,of\,gas = {10^3}c{m^3}$
Substitute these values to find the correct answer.

Formula Used:
$Total\,number\,of\,molecules = \dfrac{{Avogadro's\,number}}{{22.414\,L}} \times Volume\,of\,gas$

Complete step by step answer:
Let us start by understanding some terms in the questions and some terms which we will be needing to answer this question.
STP stands for Standard Temperature and Pressure conditions, this means that whatever reaction or process is taking place is happening when the Temperature is $273K$ and the pressure is $1atm$.
When the conditions are maintained at this temperature and pressure then the reaction process is said to be taking place at STP.
There is a rule which states that $1Mole$ of a gas will occupy $22.414L$ of volume in space and we know that $1\,mole = 6.022 \times {10^{23}}\,molecules$. This number is known as the Avogadro’s number.
The terms which we will need to know are adsorbent and adsorbate. Adsorbent refers to the catalyst on which the adsorption process takes place and Adsorbate refers to the gas which is getting adsorbed which in this case refers to the ${N_2}$ gas.
Now, we Know that $22.414L$ amount of space will be occupied by $6.022 \times {10^{23}}\,molecules$.
Hence, If we know the volume occupied by the adsorbate, we can find out the value of the total number of molecules.
The formula we will use by applying the above explanation is:
$Total\,number\,of\,molecules = \dfrac{{Avogadro's\,number}}{{22.414\,L}} \times Volume\,of\,gas$
We have been given:
$Volume\,of\,gas = {10^3}c{m^3},Avogadro's\,number = 6.022 \times {10^{23}}molecules$
Substituting this value in the above equation, we get
$Total\,number\,of\,molecules = \dfrac{{6.022 \times {{10}^{23}}molecules}}{{22.414L}} \times {10^3}c{m^3}$
Solving the equation we get:
$Total\,number\,of\,molecules = 2.69 \times {10^{29}}$
Now, If we know the value of the surface area of one ${N_2}$ molecules, we can multiply it by the total number of ${N_2}$ molecules to find out the value of total surface area.
The formula we will use by applying the above hypothesis is:
$Total\,surface\,area\,of\,molecules = surface\,area\,of\,one\,molecule \times total\,number\,of\,molecules$
From the question we know that:
$Surface\,area\,of\,1\,molecule = 1.62 \times {10^{15}}c{m^2}$
Substituting these values we get:
$Total\,surface\,area = 1.62 \times {10^{15}}c{m^2} \times 2.96 \times {10^{29}}molecule$
Solving the above equation we get:
$Total\,surface\,area = 4355{m^2}$

Note: These calculations are done by applying the Brunauer-Emmett-Teller theory of adsorption and this theory assumes that the surface of adsorbent is uniform and the adsorption at one site does not affect the adsorption at other sites.
However, We know from recent studies, that the surface of any material cannot be uniform and the rate of adsorption is dependent on adsorption taking place nearby.