Question

The Critical density of the gas $C{O_2}$ is $0.44gc{m^{ - 3}}$ at a certain temperature. If $r$ is the radius of molecule, ${r^3}$ in $c{m^3}$ is approximately: (NA is Avogadro number)
(A) $\dfrac{{25}}{{\pi N}}$
(B) $\dfrac{{100}}{{\pi N}}$
(C) $\dfrac{{6.25}}{{\pi N}}$
(D) $\dfrac{{25}}{\pi }$

Answer 1

Hint : We know that the critical density is the average density of matter which is required for the Universe to just halt its expansion, but only after an infinite time. Therefore, a Universe with the critical density will be said to be flat.

Complete Step By Step Answer:
We know that the formula for the density is:
$Density = \dfrac{{mass}}{{volume}}$
Also, the atomic mass of $C{O_2}$ is 44 amu.
Mass of $6.022 \times {10^{23}}$ molecules $= 44g$
So, Mass of one molecule $= \dfrac{{44}}{{{N_A}}} \times 1$
Here volume that we will take will be the critical volume,
We know that $C{O_2}$ is consist of two atoms of carbon and one atom of oxygen so there are overall three atoms. Therefore Volume of gas-liquid equilibrium at critical pressure and temperature given by
${V_c} = 3 \times \dfrac{4}{3}\pi {r^3}$
On substituting the values, we get
$0.44 = \dfrac{{\dfrac{{44}}{{{N_A}}}}}{{3 \times \dfrac{4}{3} \times \pi \times {r^3}}}$
On further solving we get
${r^3} = \dfrac{{25}}{{\pi N}}$
Therefore, the correct answer is Option A.

Additional Information:
The temperature at and above which vapor of the substance cannot be liquefied, no matter how much pressure is applied is known as the critical temperature of that substance.
Avogadro's number is the number of units in one mole of any substance which is also defined as its molecular weight in grams and is equal to $6.02214076 \times {10^{23}}$ . The units may be in the form of electrons, atoms, ions, or molecules which depend on the nature of the substance and the character of the reaction.

Note :
Critical temperatures (the maximum temperature at which a gas can be liquefied by pressure) for helium range from $5.2K$ to high to measure. Critical pressures (the vapor pressure at the critical temperature) are generally in range of about 40–100 bars.