Question

Molarity and molality of the given solution are $4$ molar and $5$molal respectively. The molecular weight of the solute is$100$.
The density of the given solution is
A. $1$ gm/ml
B. $1.2$ gm/ml
C. $10/12$gm/ml
D. $0.80$ gm/ml

Answer 1

To determine the answer we should use the formulas of molarity, molality and density. First we will determine the relation among all three. Then by substituting the given values we can determine the density.

Complete solution: The formulas of molarity, molality and density are as follows:
${\text{molarity}}\,\,{\text{ = }}\,\dfrac{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}}{{{\text{volume}}\,{\text{of}}\,{\text{solution}}}}$......$(1)$
${\text{molality}}\,\,{\text{ = }}\,\dfrac{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}}{{{\text{kg}}\,{\text{of}}\,{\text{solvent}}}}$....$(2)$
${\text{d = }}\dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solution (kg)}}}}{{{\text{volume of}}\,{\text{solution(L)}}}}$....$(3)$
On rearranging the equation $(1)$for volume of solution,
${\text{volume}}\,{\text{of}}\,{\text{solution}}\,\,{\text{ = }}\,\dfrac{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}}{{{\text{molarity}}}}$.....$(4)$
On substituting the volume from equation $(4)$into $(3)$,
${\text{d = }}\dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solution(kg)}}}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}} \times {\text{molarity}}$
Mass of solution is the sum of mass of solvent and mass of solute so,
${\text{d = }}\dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solute(kg)}}\,{\text{ + }}\,{\text{mass}}\,\,{\text{of}}\,{\text{solvent(kg)}}}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}} \times {\text{molarity}}$
On opening the equation we get,
${\text{d = }}\dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solute(kg)}}\,}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}} + \dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solvent(kg)}}}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}} \times {\text{molarity}}$
As we know that we take molar mass of a compound (solute) in g/mole, so, we have to replace the mass of solute from kilogram to gram.
We know, ${\text{1}}\,{\text{kg}}\,{\text{ = }}\,{\text{1000}}\,{\text{g}}$
So,
${\text{d = }}\dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solute(g)}}\,}}{{{\text{1000}} \times {\text{moles}}\,{\text{of}}\,{\text{solute}}}} + \dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solvent(kg)}}}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}} \times {\text{molarity}}$.....$(5)$
We know the mole is determined by dividing the mass (g) by molar mass (g/mol).
${\text{mole}}\,{\text{ = }}\,\dfrac{{{\text{mass}}}}{{{\text{molar}}\,{\text{mass}}}}$
On rearranging the above formula for molar mass we get,
${\text{molar}}\,{\text{mass}}\,{\text{ = }}\,\dfrac{{{\text{mass}}}}{{{\text{mole}}}}$…..$(6)$
On substituting the molar mass from equation $(6)$ to$(5)$ we get,
${\text{d = }}\dfrac{{{\text{molar}}\,{\text{mass}}\,{\text{of}}\,{\text{solute}}\,}}{{{\text{1000}}}} + \dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solvent(kg)}}}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}} \times {\text{molarity}}$.....$(7)$
On inverting the equation $(2)$ we get,
$\dfrac{{\text{1}}}{{{\text{molality}}}}\,\,{\text{ = }}\,\dfrac{{{\text{mass}}\,\,{\text{of}}\,{\text{solvent(kg)}}}}{{{\text{moles}}\,{\text{of}}\,{\text{solute}}}}$.....${\text{(8)}}$
On substituting from equation ${\text{(8)}}$into equation${\text{(7)}}$,
${\text{d = }}\,\left( {\dfrac{{{\text{molar}}\,{\text{mass}}\,{\text{of}}\,{\text{solute}}\,}}{{{\text{1000}}}} + \dfrac{{\text{1}}}{{{\text{molality}}\,}}} \right) \times {\text{molarity}}$....${\text{(9)}}$
So, the equation ${\text{(9)}}$ gives the relation in molarity, molality and density.
On substituting $4$for molarity , $5$ for molality and $100$for molecular mass,
${\text{d = }}\,\left( {\dfrac{{100\,}}{{{\text{1000}}}} + \dfrac{{\text{1}}}{{5\,}}} \right) \times {\text{4}}$
${\text{d = }}\,\left( {0.1 + 0.2} \right) \times {\text{4}}$
${\text{d = }}\,1.2$
So, the density of the given solution is $1.2$gm/ml.

Therefore, option (B) is correct.

Note: Molarity is defined as the mole of solute dissolved in per litter of the solution and molality is defined as the mole of solute dissolved in per kg of the solvent. Density is defined as the mass of solution present in a volume of solution. Units are very important to determine the correct answer. Unit of molarity is mol/L and the unit of molality is mol/kg. The unit of density is g/ml or kg/L.