Question

relationship between molarity molality and density

Answer 1

m = \dfrac{1000 \times M}{1000 \times d - M \times M_{solute}}

Answer 2

Let M be the molarity of the solution (mol/L), m be the molality (mol/kg), d be the density of the solution (g/mL or g/cm³), and M_solute be the molar mass of the solute (g/mol).

Consider 1 L of the solution.
Number of moles of solute = M * 1 L = M moles.
Mass of solute = (Number of moles of solute) * M_solute = M * M_solute (g).

Volume of solution = 1 L = 1000 mL.
Mass of solution = Density * Volume = d (g/mL) * 1000 (mL) = 1000 * d (g).

Mass of solution = Mass of solute + Mass of solvent.
Mass of solvent = Mass of solution - Mass of solute = (1000 * d - M * M_solute) (g).

Molality (m) is defined as the number of moles of solute per kilogram of solvent.
Mass of solvent in kg = (1000 * d - M * M_solute) (g) / 1000 (g/kg) = (1000 * d - M * M_solute) / 1000 (kg).

Molality m = (Number of moles of solute) / (Mass of solvent in kg)
m = M moles / [(1000 * d - M * M_solute) / 1000 kg]
m = (M * 1000) / (1000 * d - M * M_solute)

This is the relationship between molarity (M), molality (m), density of the solution (d), and molar mass of the solute (M_solute).

Rearranging the formula, we can also express molarity in terms of molality:
1000 * d - M * M_solute = (1000 * M) / m
1000 * d = M * M_solute + (1000 * M) / m
1000 * d = M * (M_solute + 1000 / m)
M = (1000 * d) / (M_solute + 1000 / m)

Alternatively, we can express density:
1000 * d = M * M_solute + 1000 * M / m
d = (M * M_solute / 1000) + (M / m)

The most common form relating molality and molarity is:

$m = \dfrac{1000 \times M}{1000 \times d - M \times M_{solute}}$

where:
m = molality (mol/kg)
M = molarity (mol/L)
d = density of solution (g/mL or g/cm³)
M_solute = molar mass of solute (g/mol)