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**Hint:**The terms like molarity and molality are used in consideration with the concentration of a solution. Molarity is related to the volume, whereas molality is relatable with the mass. We can define the relation by relating them based on density in consideration with their formulae.

**Complete Solution :**

-Now, first, we will differentiate between the molarity and the molality.

Molarity | Molality |

As mentioned it relates to the volume of the solution, so we can say the total number of moles per litre of solution is termed as molarity. | As we know it is related to the mass, so it defined as the total moles of a solute present in a kilogram of a solvent is termed to be molality. |

From the definition; it depends on the volume of the solution. | It is dependable on the mass of the solution, from the definition. |

The unit of molarity is expressed as M. | The unit of molality is expressed as m. |

Molarity can be determined by dividing moles of the solute to that of the volume of solution in the litres. | Molality can be determined by dividing the moles of solute to that of the mass of solvent in kilograms. |

The unit of molarity can be derived i.e. moles/litre. | The unit of molality can be derived i.e. moles/kg. |

-Thus, the above-shown table, we can say that it differentiates between the molarity and the molality.

-Now, we have to tell the relation between the formula of molarity, and the molality.

-First, we will write the formula of molarity i.e.

$Molarity=\dfrac{\text{number of moles of solute}}{\text{volume of solution in litres}}$

- So, next, we have molality; it can be written as:

$Molality=\dfrac{\text{moles of solute}}{\text{kilograms of solvent}}$

-As we know, molarity is represented by M, and molality is represented by m.

-So, now we will derive the relation.

Let us assume the mass of the given solute be W.

Let the volume of the solution be V.

Let the molality be m.

Let the molar mass of solute be $M'$ .

Let the molarity be M.

Let the weight of the solvent be $W'$.

Therefore, we can write molarity, as

$M=\dfrac{W}{M'}\times \dfrac{1000}{V}...(1)$

Molality can be written as, $m=\dfrac{W}{M'}\times \dfrac{1000}{W'}...(2)$

We know that, $Density(d)=\dfrac{Mass}{Volume}=\dfrac{M}{V}=\dfrac{W+W'}{V}$

From equation (1),

$V=\dfrac{W\times 1000}{MM'}...(3)$

From equation (2),

$W'=\frac{W\times 1000}{mM'}$

Therefore, from rearranging the above two equations, we get,

$W+W'=W+\dfrac{W\times 1000}{mM'}+\dfrac{1000}{M'}\left[ \dfrac{mM'}{1000m}+\dfrac{1}{m} \right]W'...(4)$

Now dividing equation (4) by equation (3), we get

$\begin{align}

& \dfrac{d}{M}=\dfrac{1}{m}+\dfrac{M'}{1000} \\

& \dfrac{1}{m}=\dfrac{d}{M}-\dfrac{M'}{1000} \\

\end{align}$

$\Rightarrow m=\dfrac{M\times 1000}{(d\times 1000)-MM'}$

-In the last, we can conclude that the value of molality can be changed into the molarity by the above-mentioned formula. We also get to know that molality and molarity are two different terms.

**Note:**Don’t get confused while defining the relation between the molarity, and the molality. Molality relates to the mass of the solute i.e. given mass used in the solution but to relate the terms we use the molar mass of solute too in consideration with the volume.

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