Question

A plant virus is found to consist of uniform cylindrical particles of \[150{A^o}\] in diameter \[1500{A^o}\] long. The specific volume of the virus is \[0.75\text{ }mL{{g}^{-1}}\]. If the virus is considered to be a single particle, find its molar mass.

Answer 1

Hint: Try to make use of the specific volume to proceed with your answer. Manipulating the values, you can find the mass using the specific volume. Also, you can find out the volume of the substance. Using the normal mass, you can find the mass of 1 mole, which is the molar mass of the substance.

Complete answer:
As it is given that there is a specific volume of the system, we can calculate the mass of the virus. So,
\[Sp.\text{ }Volume=\text{ }Volume\div mass\text{ }of\text{ }particle\]
Which means that
\[Mass\text{ }=\text{ }Volume\div sp.\text{ }Volume\]
Let us calculate the volume for the particle. Now,
\[V\text{ }=\text{ }3.14\times radiu{{s}^{2}}\times height\], so
\[V=\text{ }\left( 3.14\times 150\times {{10}^{-8}}\times 5000\times {{10}^{-8}} \right)\div 2\]
\[V=\text{ }8.839\times {{10}^{-17}}c{{m}^{3}}^{{}}\]
Now, we have the volume with us. Let us calculate the mass of the particle. We have
\[Mass=\text{ }\left( 8.839\times {{10}^{-17}} \right)\div 0.75\]
\[Mass=\text{ }11.78\times {{10}^{-17}}g\]
So we find out the mass to be 11.78×10-17g.
This means that each particle has a mass of 11.78×10-17g. Molar mass means the mass of the substance which has a number of particles equal to the Avogadro number. If we multiply the mass of each particle with the total number of particles present in 1 mole of the substance, we get our final answer i.e the molar mass of the whole substance.
Therefore, the molar mass of the substance is \[(11.78\times {{10}^{-17}})\times (6.022\times {{10}^{23}})\]
Which comes out to be as \[7.098\times {{10}^{7}}g/mol\], which is the required answer to our question. That means Avogadro number of particles which constitute 1 mole of the substance has a mass of \[7.098\times {{10}^{7}}g/mol\] in it.

Note:
Specific volume of a substance is defined as the ratio as the real volume of the substance and its mass. Or, it can be also called the reciprocal of the density. Hence, a more dense substance will have less specific volume. It is an intrinsic property.