Question

The critical volume of a gas is 0.036$ lit.mo{{l}^{-1}}$. The radius of the molecule will be( in cm):
(Avogadro Number = $\text{6 x 1}{{\text{0}}^{23}}$)
(A) ${{(\dfrac{9}{4\pi }\text{ x 1}{{\text{0}}^{-23}})}^{\dfrac{1}{3}}}$
(B) ${{(\dfrac{8\pi }{3}\text{ x 1}{{\text{0}}^{-23}})}^{\dfrac{1}{3}}}$
(C) ${{(\dfrac{3}{8\pi }\text{ x 1}{{\text{0}}^{-23}})}^{\dfrac{1}{3}}}$
(D) none of these

Answer 1

Hint: Write down the formula to calculate volume of 1 molecule of gas. We assume the gas to be spherical in nature. Now in 1 mole of the gas there will be Avogadro number of molecules of gas. So multiply the volume of 1 gas molecule with Avogadro number to obtain the total volume and thus find radius of the gas molecule.

Complete step by step answer:
-Critical temperature is defined as the temperature of a substance in its critical state beyond which it cannot be liquefied.
-Critical pressure of a fluid is defined as the vapor pressure of the fluid at the critical temperature.
-Critical volume is the volume occupied by the fluid or substance in its critical state (Critical pressure and temperature).
-Under critical conditions we will define the value of pressure, volume and temperature considering n=1.
$\begin{align}
  & {{P}_{c}}=\dfrac{a}{27{{b}^{2}}} \\
 & {{T}_{c}}=\dfrac{8a}{27Rb} \\
 & {{V}_{c}}=3b \\
\end{align}$
Critical volume for 1 molecule of gas = 3b
For 1 mole, critical volume is given = 0.036 L

$\Rightarrow \text{ 0}\text{.036 x 1}{{\text{0}}^{3}}\text{c}{{\text{m}}^{3}}\text{ = 3}\left( \dfrac{4}{3}\pi {{\text{r}}^{3}} \right)\text{ x }{{\text{N}}_{\text{A}}}$
$\Rightarrow \text{ r = }{{\left( \dfrac{36}{24\pi }\text{ x 1}{{\text{0}}^{-23}} \right)}^{\dfrac{1}{3}}}\text{ = }{{\left( \dfrac{3}{8\pi }\text{ x 1}{{\text{0}}^{-23}} \right)}^{\dfrac{1}{3}}}$
So, the correct answer is “Option C”.

Note: -Compressibility factor(Z) is a correction factor which describes the deviation of a real gas from its behavior of an ideal gas. It is simply the ratio of molar volume of the gas to the molar volume of an ideal gas subjected to the same identical temperature and pressure.
- For ideal gases, Z = 1 as PV = nRT.
  However for real gases we follow the Van der Waals equation i.e.
  $[P+\dfrac{a{{n}^{2}}}{{{V}^{2}}}][V-nb]=nRT$
Where,
P stands for pressure,
V stands for volume,
n stands for number of moles,
a is pressure correction constant
b is volume correction constant
T stands for temperature
R stands for universal gas constant