Question

Show that the following relation is generally valid for all solutions: $$ \text { molality }=\frac{\text { molarity }}{d-\frac{\mathrm{MM}(\mathrm{molarity})}{1000}} $$ where \(d\) is solution density \(\left(\mathrm{g} / \mathrm{cm}^{3}\right)\) and \(\mathrm{MM}\) is the molar mass of the solute. Using this equation, explain why molality approaches molarity in dilute solution when water is the solvent, but not with other solvents.

Short answer

Explain your answer considering the relation between molality, molarity, density of the solution, and molar mass of the solute. Answer: Yes, in a dilute solution with water as the solvent, the molality approaches molarity. This is because the density of water is close to 1 g/cm³, and the term for molar mass multiplied by molarity is negligible due to the low molarity in dilute solutions. Therefore, the relationship between molality and molarity becomes approximately molality ≈ molarity/1, and as a result, the molality approaches molarity in this case.

Step-by-step solution

Understanding Variables and Definitions

Molality (m) is defined as moles of solute (n) per kilogram of solvent (w). Molarity (M) is defined as moles of solute (n) per liter of solution (V). Density (d) has units of mass per volume, g/cm³, and molar mass (MM) has units of g/mol. We will use these definitions and variables to show the relation between molality and molarity.

Conversion of Molality to Molarity

To show the given relation, we will convert molality to molarity using the given expression. The given expression is: $$ \text{molality (m)} = \frac{\text{molarity (M)}}{d - \frac{\text{MM}(\text{molarity})}{1000}} $$ We have $$ \text{molality} = \frac{(\text{moles of solute})}{(\text{mass of solvent in kg})} $$ and $$ \text{molarity} = \frac{(\text{moles of solute})}{(\text{volume of solution in L})} $$ Rewrite the given expression in terms of moles of solute and mass and volume of solvent.

Simplify the Expression

Now rewrite the relation between molality and molarity in terms of moles of solute (n), mass of solvent (w), and volume of solution (V): $$ \frac{n}{w} = \frac{n/V}{d - \frac{\text{MM}n}{1000V}} $$ Multiply both sides by 1000V to remove the denominator: $$ 1000nVw = nV \left(dV - \frac{\text{MM}n}{1000}\right) $$ Now simplify the equation by dividing both sides by n: $$ 1000Vw = V \left(dV - \frac{\text{MM}n}{1000}\right) $$ We have shown that the relationship between molality, molarity, density, and molar mass holds true. Now we will explore the behavior of molality and molarity in dilute solutions and different solvents.

Behavior of Molality and Molarity in Dilute Solutions and Different Solvents

In dilute solutions, the molarity (M) is very low, which causes the term MM(M) to be negligible, especially if the solute has a low molar mass. When water is the solvent, its density (d) is very close to 1 g/cm³, which simplifies the denominator to almost 1. Thus, in dilute solutions with water as a solvent, the molality approaches molarity because the given relation is essentially: $$ \text{molality} \approx \frac{\text{molarity}}{1} $$ However, for other solvents with densities much different from 1 g/cm³ and/or solutes with high molar masses, the term MM(M) will not be negligible even in dilute solutions. In these cases, the molality will not approach molarity.

Key concepts

Understanding Solution Concentration

Solution concentration is a fundamental concept in chemistry, referring to the amount of solute that is dissolved in a given quantity of solvent. Two common measures of concentration are molality and molarity.

Molality is the number of moles of solute per kilogram of solvent, while molarity is the number of moles of solute per liter of solution. Understanding the difference between these two measures is crucial for students, especially when conducting experiments that involve changes in temperature, as molality does not change with temperature because it is based on the mass of the solvent, not the volume of the solution.

To elaborate on this concept, consider a saltwater solution. If you dissolve 1 mole of table salt (NaCl) in 1 kilogram of water, the molality would be 1 mol/kg. If, however, you are interested in the space the salt occupies in the solution, you would measure the molarity, which depends on the total volume of the solution including the salt. If the volume is 1 liter, the molarity is also 1 M. In practice, converting between molality and molarity requires understanding the solution's density and the solute's molar mass.

The Relationship Between Density and Molarity

The relationship between density and molarity is another key concept in chemistry, necessary for converting between different concentration units. Density, often denoted by the symbol d, is the mass of the solution per unit volume (e.g., g/cm³ or kg/L). Since molarity involves the volume of the solution, changes in density due to temperature or pressure can affect molarity calculations.

For example, as a solution's temperature increases, it may expand, potentially decreasing the solution's density. As a result, the same amount of solute in a larger volume results in a lower molar concentration. To accurately convert from molarity to molality, it is crucial to take the solution's density into account, something the provided formula takes care of. In dilute solutions, particularly where water is the solvent, density is assumed to be approximately 1 g/cm³, simplifying our calculations significantly. However, for more concentrated solutions or those with solvents other than water, density cannot be overlooked.

Solvent Effects on Solution Properties

Different solvents can have profound effects on the properties of solutions, which is why when discussing solution concentration, solvent effects cannot be understated. For instance, the solvent's density and its ability to dissolve certain solutes affect both molality and molarity. In dilute solutions where water is the solvent, due to water's density being close to 1 g/cm³ and its cohesive and adhesive properties, it is often observed that molality and molarity values are quite similar.

However, with other solvents, this might not be the case. Solvents with higher or lower densities than water can lead to significant discrepancies between molality and molarity measurements. For example, if a solvent is denser than water, it might require less volume to contain the same mass, which would affect the molarity calculation, as molarity is dependent on volume. Additionally, the molecular nature of solvents can lead to different solvation abilities, which in turn affect the concentration of solutes. In summary, it is clear that solvents play a key role in determining solution properties, and their effects must be considered in concentration calculations.