Question

Derive the relation between molarity and molality. Calculate the molality of 1M HCl solution having density $1.5365{gmol}^{-1}$.

Answer 1

Hint: Molarity is also known as molar concentration. It is the concentration of the solution. It is the number of moles of solute per liter of solution.
Molality: It is defined as the number of moles of solute in one kilogram of solvent.

Complete step by step solution:
The formulas of molarity and molality are given below:
$Molarity={\displaystyle \frac{molesofsolute}{litersofsolution}}$
and $Molality={\displaystyle \frac{molesofsolute}{kilogramsofsolvent}}$

Now, the relationship between Molarity and Molarity:
Let,
The mass of given solute = W
The volume of the solution = V
Molality = m
The molar mass of solute = M
Molarity = M
Therefore, molarity, M will be = ${\displaystyle \frac{W}{M^{{}^{\prime }}}}\times {\displaystyle \frac{1000}{V}}$.........(1)
And molality, m will be = ${\displaystyle \frac{W}{M^{{}^{\prime }}}}\times {\displaystyle \frac{1000}{W^{{}^{\prime }}}}$………..(2)
As we know, Density = mass/volume
$d={\displaystyle \frac{M}{V}}$
From here, $d={\displaystyle \frac{W+W^{{}^{\prime }}}{V}}$
Now, from equation (1), we get
$V={\displaystyle \frac{W\times 1000}{m\times M^{{}^{\prime }}}}$ .........(3)
And from equation (2), we get
$W^{{}^{\prime }}={\displaystyle \frac{W\times 1000}{m\times M^{{}^{\prime }}}}$
Therefore,
$W+W^{{}^{\prime }}=W+{\displaystyle \frac{W\times 1000}{m\times M^{{}^{\prime }}}}\times {\displaystyle \frac{1000}{M^{{}^{\prime }}}}[{\displaystyle \frac{{mM}^{{}^{\prime }}}{1000m}}+{\displaystyle \frac{1}{m}}]W^{{}^{\prime }}$.........(4)
Now, by dividing equation (4) by (3), we get
${\displaystyle \frac{d}{m}}={\displaystyle \frac{1}{m}}+{\displaystyle \frac{M^{{}^{\prime }}}{1000}}{\displaystyle \frac{1}{m}}$
${\displaystyle \frac{d}{m}}-{\displaystyle \frac{M^{{}^{\prime }}}{1000}}\times m$
${\displaystyle \frac{M\times 1000}{(d\times 1000)-M\left(M^{{}^{\prime }}\right)}}$
Hence, this is the relation between molality and molarity.

NUMERICAL:
It is given that,
Density = $1.5365{gmol}^{-1}$.
Molecular weight of solute (HCl) = $36.46{gmol}^{-1}$.
Molarity = 1M
Molality, m =?
Using the formula;
$Molality={\displaystyle \frac{molarity}{(density-molarity)\times molecularweightofsolute}}$
Now, by putting the values in the above formula, we get
Molality = $Molality={\displaystyle \frac{1}{\left({1.5365gmol}^{-1}-1\right)\times 36.46}}$
Molality = $Molality={\displaystyle \frac{1}{0.5365\times 36.46}}$
               = $Molality={\displaystyle \frac{1}{19.560}}=0.05112$
Hence, the molality of 1M HCl solution = $0.05112M$.

Note: The possibility to make a mistake is that don’t make mistakes in the values given in the question or any error using formulas or analysis. Don’t confuse between molality and molarity also convert the values from kg to g in molality.