Question

At \(25^{\circ} \mathrm{C}\), the vapor pressure of pure water is \(23.76 \mathrm{mmHg}\) and that of seawater is \(22.98 \mathrm{mmHg}\). Assuming that seawater contains only \(\mathrm{NaCl}\), estimate its molal concentration.

Short answer

The molal concentration of NaCl in seawater is approximately 0.133 mol/kg.

Step-by-step solution

Understand Raoult's Law

Raoult's Law states that the vapor pressure of a solvent above a solution \(P_{\text{solution}}\) is equal to the mole fraction of the solvent \(X_{\text{solvent}}\) times the vapor pressure of the pure solvent \(P_{\text{pure solvent}}\). The formula is: \[P_{\text{solution}} = X_{\text{solvent}} \times P_{\text{pure solvent}}\]

Calculate the Change in Vapor Pressure

The change in vapor pressure (\(\Delta P\)) is the difference between the vapor pressure of pure water and seawater. \[\Delta P = 23.76 - 22.98 = 0.78\, \text{mmHg}\]

Express Mole Fraction in Terms of Molality

The mole fraction of the solvent \(X_{\text{solvent}}\) is related to molality \(m\). For a dilute solution:\[X_{\text{solvent}} \approx \frac{1}{1 + m \cdot i \cdot M_{\text{solute}}}\]where \(i\) is the van't Hoff factor for \(\text{NaCl}\) and \(M_{\text{solute}}\) is the molar mass of the solute.

Set Up Raoult's Law Equation for Seawater

Rearrange Raoult's law equation with known values:\[22.98 = \left(\frac{1}{1 + m \cdot 2 \cdot \text{M}_{\text{NaCl}}} \right) \cdot 23.76\]Since \(i = 2\) for \(\text{NaCl}\), we approximate \(M_{\text{NaCl}}\) as 58.44 g/mol.

Solve for Molality (m)

Rearrange and solve the equation for \(m\):\[0.78 = \left(1 - \frac{22.98}{23.76}\right) = \frac{m \cdot 2 \cdot 58.44}{1000}\]Solve for \(m\): \[m \approx \frac{0.78}{2 \cdot 58.44} \approx 0.133\]

Interpret the Result

The molality of \(\text{NaCl}\) in the seawater is approximately \(0.133\, \text{mol/kg}\). This indicates that there are about 0.133 moles of \(\text{NaCl}\) per kilogram of solvent in seawater.

Key concepts

Vapor Pressure

The concept of vapor pressure plays a key role in understanding various physical phenomena, including the behavior of solutions. Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form in a closed system. At a given temperature, it is a measure of the tendency of molecules to escape from the liquid or solid to the gas phase. For pure substances like water, vapor pressure is an intrinsic property dependent solely on temperature.

Raoult's Law helps us connect vapor pressure with solutions. It states that the vapor pressure of a solvent in a solution is reduced compared to that of the pure solvent. This reduction is due to the presence of solute particles that hinder or interact with solvent molecules, thus lowering their evaporation rate.

• For example, when salt (NaCl) is added to water, the vapor pressure of the resultant seawater is lower than that of pure water at the same temperature.
• This reduction in vapor pressure illustrates the colligative property of solutions where it depends on solute concentration rather than solute type.

Understanding vapor pressure helps in various applications like determining boiling points and predicting the behavior of different solutions in varying conditions.

Molality

Molality is a measure of solute concentration in a solution, defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which is volume-based, molality is mass-based, making it useful when examining properties that involve temperature changes, as mass doesn't change with temperature.

For instance, when calculating the molal concentration of seawater with NaCl, it provides insight into how solute presence affects solution properties like vapor pressure. By establishing a relationship between molality and mole fraction through Raoult's Law, the equation can help estimate the concentration of solutes based on observed changes in physical properties.

• In our specific scenario with seawater having a lower vapor pressure than pure water, molality helps quantify the effect of salt in water.
• The calculation ultimately showed a molality of approximately 0.133, reflecting how much solute—here NaCl—is present per kilogram of water in the solution.

Molality is an essential concept when exploring solutions' thermodynamic properties and their reactions to changes in temperature or pressure.

van't Hoff Factor

The van't Hoff factor, symbolized as "i," is a dimensionless value that indicates the number of particles into which a compound dissociates in solution. This factor is crucial when examining colligative properties, such as boiling point elevation, freezing point depression, and vapor pressure lowering.

For salts like sodium chloride ( NaCl), the van't Hoff factor is 2 because NaCl dissociates into two ions: Na^+ and Cl^-. This dissociation impacts how solute concentration is calculated, which in turn affects physical properties like vapor pressure.

• In the exercise, the van't Hoff factor of 2 is used when applying Raoult's Law to derive the seawater solution's molal concentration.
• This factor is particularly significant for electrolytes that dissociate in solution, differing from non-electrolytes, which typically have a factor of 1.

Learning about the van't Hoff factor is vital for fully understanding how real solutions behave, especially those containing ionic compounds.

Mole Fraction

The mole fraction is another important concentration measure used to describe the proportions of components in a solution. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the solution.

This concept comes into play in Raoult's Law, connecting it to vapor pressure. Specifically, the law expresses the relationship between the solvent's vapor pressure and its mole fraction in the solution.

• For a solution like seawater, the mole fraction of water (solvent) is directly related to its vapor pressure compared to the pure solvent.
• The interaction of solute molality with the mole fraction reveals the changes in vapor pressure, emphasizing the subtle shifts in solvent behavior due to solute presence.

A solid understanding of mole fractions not only supports comprehension of fundamental chemistry but also enhances the ability to manipulate and predict solution behaviors in practical scenarios.