Question

It has been estimated that is all the $O_3$ in the atmosphere were brought to sea level at $STP$, the gas would form a layer 0.3 cm thick. Estimate the number of $O_3$ molecules in the earth's atmosphere. (radius of the earth = 4000 mile; 1 km = 0.6214 mile).

• A. $3.9\times {10}^{37}$
• B. $4.2\times {10}^{37}$
• C. $4.5\times {10}^{37}$
• D. $6.79\times {10}^{37}$

Answer 1

The correct option is B $4.2\times {10}^{37}$

Radius = 4000 miles
$4000miles=\frac{4000}{0.6214}km=\frac{4\times {10}^6}{0.6214}m$
Thickness $=0.3cm=0.003m$
Volume of $O_3$ molecules at the surface = $4\pi r^2$ thickness
$=4\times \frac{22}{7}\times {\left(\frac{4\times {10}^6}{0.6214}\right)}^2\times 0.003m^3$
$V=1.56273\times {10}^{12}{m}^3$
$P=1atm=1.01325\times {10}^{5}Pascal$
$Moles=n=\frac{pV}{RT}$
$=\frac{1.01325\times {10}^5\times 1.56273\times {10}^{12}}{8.3143\times 273}$
$=6.976\times {10}^{13}$
$\therefore Molecules=n{N}_0=6.976\times {10}^{13}\times 6.02\times {10}^{23}=4.2\times {10}^{37}$.
Hence option B is correct.