From Molarity to Molality and Vice Versa

From molarity to molality and vice versa

Last updated: November 27, 2018

Summary The concentration of a solution is often expressed either as molarity or molality. Conversion from one to the other requires knowledge of the density at the temperature of a speciﬁc solution. The molar mass of the solute is also needed. Here, we derive the well-known formulas for molarity-to-molality and molality-to-molarity conversion. Further, we refer to online calculators for binary systems that include molarity- molality conversions via estimated solution densities. This document is available at: www.axeleratio.com/chem/conc/molarity molality conversion.pdf

Copyright c 2018 by author. This work is licensed under the Creative Commons Attribution Inter- national Licence (CC BY). http://creativecommons.org/licence/by/4.0/

Keywords. Solution chemistry, concentration, molarity, molality, vol- umetric property, density, binary system.

1 Introduction

Herein, we consider the molarity-molality relation for solutions that consist of a single solute dissolved in a solvent. Subscripts 1, 2 and sln are used to identify the solvent, the solute, and the solution, respectively. The molarity in mol/L is deﬁned as c2 = n2/Vsln, where n2 is the amount of solute in moles (mol) and Vsln is the volume of the solution in liter (L). The molality is deﬁned in mol/kg as b2 = 1000 · n2/m1, where m1 is the mass of the solvent in gram. We take m1 in g, since we want stay with gram-based units

1 −1 for other mass-containing variables: molar mass in g · mol and density in −3 − g · dm (g · L 1).

2 Molarity to molality

Task. Our task is to calculate the molality, b2, from the molar concentration, c2, given for a binary system (solvent-solute system):

1000 · n2 n2 b2 = ← c2 = (1) m1 Vsln

The calculation requires known values for the density of the solution, ρsln, and the molar mass of the solute, M2. Approach. Assuming a solution amount with a volume Vsln, the corre- sponding mass of the solution is

msln = Vsln · ρsln (2)

The mass of the solute is

m2 = n2 · M2 = c2 · Vsln · M2 (3)

Then, the mass of the solvent is

m1 = msln − m2 = Vsln(ρsln − c2 · M2) (4) and the molality formula is derived as follows:

1000 · n2 1000 · n2 1000 · c2 b2 = = = (5) m1 Vsln(ρsln − c2 · M2) ρsln − c2 · M2 Since the solute concentration is the same for any taken amount of the solution, we could have assumed any other volume instead of Vsln. But the above assumption allowed for a derivation without introducing any extra variables.

3 Molality to molarity

Task. Our task is to calculate the molarity (molar concentration, c2, from the molality, b2, given for a binary system:

n2 1000 · n2 c2 = ← b2 = (6) Vsln m2

2 The calculation requires known values for the density of the solution, ρsln, and the molar mass of the solute, M2. Approach. The molar concentration is derived by solving equation (5) for c2: b2 · ρsln c2 = (7) b2 · M2 + 1000

4 Applications

Online Calculators. Online calculators to estimate solution densities at user-speciﬁed molarities are currently tested and will soon be uploaded. Basaed on the estimated solution density, the solution molarity will be con- verted to molality.