Question

A plant virus consists of uniform cylindrical particles of $150\mathop {\text{A}}\limits^ \circ$ in diameter and $5000\mathop {\text{A}}\limits^ \circ$ long. The specific volume of the virus is $0.75c{m^3}{g^{ - 1}}$. If the virus is considered to be a single particle, its molecular mass is:
A: $7.09 \times {10^7}gmo{l^{ - 1}}$
B: $8.09 \times {10^7}gmo{l^{ - 1}}$
C: $9.07 \times {10^7}gmo{l^{ - 1}}$
D: $9.70 \times {10^7}gmo{l^{ - 1}}$

Answer 1

Specific volume of a substance is defined as the ratio of the volume of substance to the mass of substance. Basically it is reciprocal of density. Density of a substance is defined as mass per unit volume and specific volume is defined as volume per unit mass of substance.
Formula used: Molecular mass$= \dfrac{{{\text{volume}}}}{{{\text{specific volume}}}} \times {N_A}$
${N_A} = 6.022 \times {10^{23}}$
Volume of cylinder$= \pi {\left( {\dfrac{d}{2}} \right)^2}l$
Where ${N_A}$ is Avogadro number, $d$ is diameter and $l$ is length

Complete step by step answer:
Specific volume of a substance is defined as the volume per unit mass of substance. This means to calculate mass from specific volume we need to divide volume of the substance with specific volume. As formula to calculate specific volume is:
Specific volume$= \dfrac{{{\text{volume}}}}{{{\text{mass}}}}$
We know that the particle is cylindrical. To find the volume of cylinder radius and length of cylinder is required and these quantities are given in question. This means we can calculate the volume easily and a specific volume is given. This means with the help of specific volume and volume we can calculate molecular mass of the virus.
Volume of cylinder $= \pi {\left( {\dfrac{d}{2}} \right)^2}l$
Given diameter is $150\mathop {\text{A}}\limits^ \circ$, which in $cm$ is equal to $150 \times {10^{ - 8}}cm$
Given length is $5000\mathop {\text{A}}\limits^ \circ$, which in $cm$ is equal to $5000 \times {10^{ - 8}}cm$
So, volume$= \pi {\left( {\dfrac{{150 \times {{10}^{ - 8}}}}{2}} \right)^2}5000 \times {10^{ - 8}}$
Solving this we get volume$= 8.83 \times {10^{ - 17}}$
Molecular mass is the mass of the Avogadro number of particles. Avogadro number is equal to $6.022 \times {10^{23}}$. This means molecular mass will be equal to mass times Avogadro number. So, formula to calculate molecular mass is,
molecular mass$= \dfrac{{{\text{volume}}}}{{{\text{specific volume}}}} \times {N_A}$
Substituting the values,
molecular mass$= \dfrac{{8.83 \times {{10}^{ - 17}}}}{{0.75}} \times 2.023 \times {10^{23}}$
Solving this we get molecular mass equal to $7.09 \times {10^7}gmo{l^{ - 1}}$.
So, the correct answer is option A.

Note:
Specific gravity of a substance is defined as the density of the substance with density of reference material. For liquids generally water is taken as reference material and for gases air is taken as reference material.