Concept explainers
(a) Derive the equation relating the molality (m) of a solution to its molarity (M),
where d is the density of the solution (g/mL) and ℳ is the molar mass of the solute (g/mol). (Hint: Start by expressing the solvent in kilograms in terms of the difference between the mass of the solution and the mass of the solute.) (b) Show that, for dilute aqueous solutions, m is approximately equal to M.
(a)

Interpretation:
The given equation relating molality (m) of the solution to its molarity (M) has to be derived.
Concept introduction:
Molality (m): Molality is the number of moles of solute present in one kilogram of solvent.
Molarity (M): Molarity is number of moles of the solute present in the one liter of the solution.
Molarity is estimation of moles in the total volume of the solution while molality is estimation of moles in relationship with solvent in the solution
Explanation of Solution
Given data:
Where,
Relate molality (m) to molarity (M):
The equations relating the molality (m) of the solution to its molarity (M) are as follows,
If we assume 1L of the solution then we can determine the mass of the solution from its volume and density and the mass of the solute from molar mass and molarity.
Substituting these expressions into equation (1) we get,
From molality definition we know that,
We assume that 1L of the solution, we also know that mol solute (n) = Molarity (M), then the equation (3) becomes,
Substitute this expression into equation (2) we get,
Take inverse of both sides of the equations,
(b)

Interpretation:
For dilute aqueous solutions, m is approximately equal to M has to be shown.
Concept introduction:
Molality (m): Molality is the number of moles of solute present in one kilogram of solvent.
Molarity (M): Molarity is number of moles of the solute present in the one liter of the solution.
Molarity is estimation of moles in the total volume of the solution while molality is estimation of moles in relationship with solvent in the solution
Explanation of Solution
m is approximately equal to M For dilute aqueous solutions is shown as follows,
For dilute aqueous solution the density is approximately
In dilute solutions,
For example, 0.010M
The derived equation reduces to
Because
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Chapter 12 Solutions
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