Question

If $M$ is the molarity, $d$ is the density (in$g/l$), ${M_B}$​ is the molar mass of solute and $m$ is the molality, then choose the correct relation ?
(A) $\dfrac{{1000M}}{{1000d\, + \,M.{M_B}}}\, = \,m$
(B) $\dfrac{{1000M}}{{1000d\, - \,m.{M_B}}}\, = \,M$
(C)$\dfrac{{1000M}}{{1000d\, - M.{M_B}}}\, = \,\,m$
(D)$\dfrac{{1000M}}{{1000d\, + m.{M_B}}}\, = \,\,M$

Answer 1

Hint: Molarity and Molality both are different terms that are used to express the concentration of solutions as there are various ways to express the concentration of solutions. Both are calculated using different formulas.

Complete step by step answer:
Molarity or molar concentration is defined as the number of moles of solute that are present in $1$ liter of solution. It has a unit of $\dfrac{{mol}}{{lit.}}$ abbreviated as $M$ , while a $mol$ is a term of measurement which denotes $6.022 \times {10^{23}}$ particles. The Molarity of a solution depends on the temperature. As you increase the temperature the molarity of the solution decreases. Molarity is given by the formula:
$Molarity\, = \,\dfrac{{W \times 1000}}{{{M_B} \times V}}$
Where $W$ is the given mass of solute and ${M_B}$ is the molar mass.
Molality is defined as the total number of moles of solute present in one kilogram of solvent. Its unit is expressed as $m$. The Molality of a solution does not depend on the temperature. It is represented by the formula;
$Molality\, = \,\dfrac{{W \times 1000}}{{{M_B} \times W'}}$
Where $W'$ is the given mass of solvent and ${M_B}$ is the molar mass
 The relation between both the terms can be deduced by their respective formulas.
Let’s consider a solution having $1000\,ml$ of Volume. As we know
$density(d)\, = \,\dfrac{{mass(M)}}{{volume(V)}}$
So, the mass of solution will be,
$mass = 1000\, \times d$
The mass of given solute is,
$ = no.\,of\,moles\, \times \,molar\,mass$
$ = M \times {M_B}$
Mass of the solvent will be
$ = mass\,\,of\,solution\, - \,mass\,\,of\,solute\,$
$(1000\, \times d)\, - \,(M \times {M_B})$
Now, as we discussed,
$molality(m)\, = \,\dfrac{{moles\,of\,solute}}{{Kg\,of\,solvent}}$
$ \Rightarrow \,m\, = \,\dfrac{M}{{(1000\, \times d)\, - \,(M \times {M_B}) \times \dfrac{1}{{1000}}}}$
$ \Rightarrow \,\,\,m\, = \,\dfrac{{1000M}}{{1000.d - M.{M_B}}}$
Hence we can say that option (C) is correct.

Note:
The concentration of a compound can be expressed in different ways in chemistry. Concentration is defined as the amount of solute present in a unit volume of solution. Different terms used for expressing the concentration are Molarity, Molality, Normality, Mole fraction, Parts per million(ppm), etc.