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In an experiment, ${\rm{18}}{\rm{.04}}\;{\rm{g}}$ of mannitol were dissolved in ${\rm{100}}\;{\rm{g}}$ of water. The vapour pressure of water was lowered by ${\rm{0}}{\rm{.309}}\;{\rm{mmHg}}$ from ${\rm{17}}{\rm{.535}}\;{\rm{mmHg}}$. Calculate the molar mass of mannitol.
A. ${\rm{150}}{\rm{.0}}\;{\rm{g/mol}}$
B. $184.3\;{\rm{g/mol}}$
C. ${\rm{140}}{\rm{.4}}\;{\rm{g/mol}}$
D. ${\rm{123}}{\rm{.21}}\;{\rm{g/mol}}$

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Last updated date: 23rd Jun 2024
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Answer
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Hint: We can say that when the mass of two species are given and the difference of vapour pressure and original pressure also given, the Raoult’s law is utilized for determining the missing molar mass of the species. It generally is applied on the non-aqueous mixture.

Complete step by step answer:
We know that the molar mass is obtained through the mass of the sample which was given and is generally divided by the amount of substance which is measured in the moles.
Given, the mass of mannitol is ${\rm{18}}{\rm{.04}}\;{\rm{g}}$.
The mass of water is ${\rm{100}}\;{\rm{g}}$.
The vapour pressure of water was lowered by ${\rm{0}}{\rm{.309}}\;{\rm{mmHg}}$ from ${\rm{17}}{\rm{.535}}\;{\rm{mmHg}}$.
The molar mass of mannitol can be calculated by using the formula of Raoult’s law is given below.
$\dfrac{{{\rm{p}} - {{\rm{p}}^{\rm{0}}}}}{{{{\rm{p}}^{\rm{0}}}}} = \dfrac{{\dfrac{{{{\rm{W}}_{\rm{2}}}}}{{{{\rm{M}}_{\rm{2}}}}}}}{{\dfrac{{{{\rm{W}}_{\rm{1}}}}}{{{{\rm{M}}_{\rm{1}}}}} + \dfrac{{{{\rm{W}}_{\rm{2}}}}}{{{{\rm{M}}_{\rm{2}}}}}}}$
Where, ${\rm{p}} - {{\rm{p}}^{\rm{0}}}$ is the difference in pressure, ${{\rm{p}}^{\rm{0}}}$ is the main pressure, ${{\rm{W}}_{\rm{1}}}$ is the weight of mannitol, ${{\rm{M}}_{\rm{1}}}$ is the molar mass of mannitol, ${{\rm{W}}_{\rm{2}}}$ is the weight of water, ${{\rm{M}}_{\rm{2}}}$ is the molar mass.
The molar mass of water is ${\rm{18}}\;{\rm{g/mol}}$.
Substitute all the respective values in the above equation.
$\begin{array}{c}
\dfrac{{0.309}}{{{\rm{17}}{\rm{.535}}}} = \dfrac{{\dfrac{{{\rm{18}}{\rm{.04}}}}{{{{\rm{M}}_{\rm{2}}}}}}}{{\dfrac{{{\rm{100}}}}{{18}} + \dfrac{{{\rm{18}}{\rm{.04}}}}{{{{\rm{M}}_{\rm{2}}}}}}}\\
0.0176 = \dfrac{{\dfrac{{{\rm{18}}{\rm{.04}}}}{{{{\rm{M}}_{\rm{2}}}}}}}{{5.55 + \dfrac{{{\rm{18}}{\rm{.04}}}}{{{{\rm{M}}_{\rm{2}}}}}}}\\
 = 184.3\;{\rm{g/mol}}
\end{array}$
Hence, the calculated molar mass of mannitol is $184.3\;{\rm{g/mol}}$.

Therefore, the correct choice for this given question is B.

Note:
The Raoult’s law is one of the laws of physical chemistry and it is used in the area of thermodynamics. It is applicable on the same or different non aqueous mixture.