Question

# The density of a DNA sample is 1.1 g/ml and its molar mass determined by the cryoscopic method was found to be $\text{6 }\times \text{ 1}{{\text{0}}^{8}}\text{gmole}$. What is the volume occupied by one DNA molecule? $\text{(}{{\text{N}}_{\text{A}}}\text{ = 6 }\times \text{ 1}{{\text{0}}^{23}})$A. $\text{5}\text{.45 }\times \text{ 1}{{\text{0}}^{8}}\text{ ml}$B. $\text{1}\text{.83 }\times \text{ 1}{{\text{0}}^{-9}}\text{ ml}$C. $\text{9}\text{.06 }\times \text{ 1}{{\text{0}}^{-16}}\text{ ml}$D. $\text{1}\text{.09 }\times \text{ 1}{{\text{0}}^{-13}}\text{ ml}$

Hint: Density of a substance is defined as the ratio of mass by volume which has a unit of kilogram per cubic metre. Density is denoted by $\rho$. The formula to calculate no. of the molecule is $\dfrac{\text{No}\text{. of molecules}}{{{\text{N}}_{\text{A}}}}\text{ = }\dfrac{\text{Mass}}{\text{Molar mass}}$. Here, ${{\text{N}}_{\text{A}}}$ is the Avogadro's number and mass is the mass of the DNA which is to be calculated and Molar mass of the DNA is also given in the question.

-It is given in the question that the density of the DNA is 1.1 g/mol and we have to calculate the volume that is occupied by One DNA molecule.
-So, according to the formula of density, the mass of the DNA and volume should be needed.
-Now, we will calculate the mass of one DNA molecule first i.e.
$\dfrac{\text{No}\text{. of molecules}}{{{\text{N}}_{\text{A}}}}\text{ = }\dfrac{\text{Mass}}{\text{Molar mass}}$
$\text{Mass = }\dfrac{\text{No}\text{. of molecules }\times \text{ Molar mass}}{{{\text{N}}_{\text{A}}}}$
-Here, ${{\text{N}}_{\text{A}}}$ is called the Avogadro's number whose value is given in the question i.e. $\text{6 }\times \text{ 1}{{\text{0}}^{23}}$
$\text{Mass = }\dfrac{\text{1 }\times \text{ 6 }\times \text{ 1}{{\text{0}}^{8}}}{6\text{ }\times \text{ 1}{{\text{0}}^{23}}}$
$\text{Mass = 1}{{\text{0}}^{-15}}\text{gm}$
-Now, we know that the density of 1 molecule of DNA is 1.1 g/mol. So, the volume will be:
$\text{Volume = }\dfrac{\text{Mass}}{\text{Density}}$
$\text{Volume = }\dfrac{{{10}^{-15}}}{1.1}$
$\text{Volume = 0}\text{.906 }\times \text{ 1}{{\text{0}}^{-15}}\text{ml or 9}\text{.06 }\times \text{ 1}{{\text{0}}^{-16}}\text{ml}$

So, the correct answer is “Option C”.

Note: Cryoscopic method is the method which is used to determine the process of unknown molecular weight of a substance with the help of cryoscopic constant that is denoted by ${{\text{K}}_{\text{f}}}$. Moreover, it gives the relationship between the freezing point and molality (Ratio of no. of a mole of solute to the volume of solvent in Kg).