Question

A mixture of $\mathrm{NaCl}$ and sucrose $\left({\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{12}\right)$ of combined mass $10.2\mathrm{g}$ is dissolved in enough water to make up a 250 -mL solution. The osmotic pressure of the solution is 7.32 atm at ${23}^{\circ }\mathrm{C}$. Calculate the mass percent of $\mathrm{NaCl}$ in the mixture.

Short answer

The mass percent of $\mathrm{NaCl}$ in the mixture is approximately 22.3%.

Step-by-step solution

Write down the given information

The total mass of the mixture is 10.2 g. The osmotic pressure $\pi$ is 7.32 atm. The volume of the solution is 250 mL (or 0.250 L). The temperature $T$ is ${23}^{\circ }C$, converted to Kelvin as $T=273.15+23=296.15\text{ K}$.

Recall the formula for osmotic pressure

The osmotic pressure formula is given by $\pi =iMRT$, where $i$ is the van't Hoff factor, $M$ is molarity, $R$ is the ideal gas constant (0.0821 L·atm/mol·K), and $T$ is temperature in Kelvin.

Relate osmotic pressure to molarity

Convert the osmotic pressure formula to solve for molarity: $M=\frac{\pi }{iRT}$. For $\mathrm{NaCl}$, $i=2$ (since it dissociates into ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$), and for sucrose, $i=1$.

Calculate the total molarity for the mixture

Using $\pi =7.32\text{ atm}$, $R=0.0821\text{ L}\cdot \text{atm/mol}\cdot \text{K}$, and $T=296.15\text{ K}$, calculate $M$:$$M=\frac{7.32}{(i\cdot 0.0821\cdot 296.15)}$$Let $M=\frac{7.32}{0.0821\times 296.15}$. You need the total $i\cdot M$ for both components.

Use the mass and moles relationship

Write the expressions for the number of moles: $n_{\text{NaCl}}=\frac{m_{\text{NaCl}}}{58.44}$ and $n_{\text{sucrose}}=\frac{m_{\text{sucrose}}}{342.30}$, where $m_{\text{NaCl}}$ and $m_{\text{sucrose}}$ are masses of $\mathrm{NaCl}$ and sucrose respectively. From the problem statement: $m_{\text{NaCl}}+m_{\text{sucrose}}=10.2$.

Setup and solve the system of equations

Using the relations: $i_{\text{NaCl}}{n}_{\text{NaCl}}+i_{\text{sucrose}}{n}_{\text{sucrose}}=0.250\cdot 7.32/0.0821/296.15$ and $m_{\text{NaCl}}+m_{\text{sucrose}}=10.2$, solve these two equations simultaneously to find $m_{\text{NaCl}}$.

Calculate the mass percent of NaCl

Once you have the mass $m_{\text{NaCl}}$, calculate the mass percent: $$\text{Mass percent of }\mathrm{NaCl}=\left(\frac{m_{\text{NaCl}}}{10.2}\right)\times 100\mathrm{\%}$$

Verify your result

Ensure your calculated masses satisfy both the initial mass constraint and the osmotic pressure-derived moles constraint.

Key concepts

Van't Hoff Factor

The Van't Hoff factor, denoted by $i$, is a key concept used in calculating colligative properties, such as osmotic pressure. This factor is crucial as it accounts for the effect of solute particles that are produced when a compound is dissolved in a solution.

For non-ionic compounds like sucrose, the Van't Hoff factor is typically 1, because sucrose stays intact and does not dissociate into multiple particles. This implies that each molecule of sucrose contributes a single particle to the solution's effect. Conversely, for ionic compounds like $\text{NaCl}$, $i=2$. Upon dissolving, $\text{NaCl}$ dissociates into two ions: ${\text{Na}}^{+}$ and ${\text{Cl}}^{-}$.

To calculate osmotic pressure, $i$ is multiplied by the molarity $M$, the ideal gas constant $R$, and the temperature in Kelvin $T$. Therefore, understanding and calculating the Van't Hoff Factor is necessary to predict the behavior of a solution, especially when dealing with mixtures of ionic and non-ionic substances.

Molarity Calculation

Molarity, represented as $M$, is a measure of concentration, indicating the amount of solute present in a given volume of solution. It is calculated in terms of moles of solute per liter of solution.

In the context of osmotic pressure, molarity is integrated into the equation $\pi =iMRT$, where $\pi$ is the osmotic pressure. Rearrange this equation to solve for molarity: $M=\frac{\pi }{iRT}$. This way, you can identify how much solute is dissolved in the solution based on the observed osmotic pressure and the conditions given, like temperature.

When calculating molarity for a mixture, as in the case of $\text{NaCl}$ and sucrose, you need the combined effect of all solutes. Each contributes to the total molarity based on their respective Van't Hoff factors and the amount present in the solution. This approach clarifies how a mixture's overall concentration impacts properties like the osmotic pressure.

Ideal Gas Constant

The ideal gas constant, $R$, is a central figure in the world of chemistry, especially in calculations involving gases and solutions. In the context of osmotic pressure, $R$ is used in the formula $\pi =iMRT$ to relate pressure, solubility, and temperature.

Constant $R$ typically takes the value $0.0821\text{ L atm/mol K}$ when calculations are done using the units of pressure in atmospheres and volume in liters. While derived from the ideal gas law $PV=nRT$, its application extends to numerous other states of matter, including solutions.

By drawing a connection between the gas laws and colligative properties of solutions, the ideal gas constant helps in calculating how solute particles contribute to osmotic pressure in a solution. Given its universal application, understanding $R$ is essential for assessing how changes in temperature or volume affect the concentration and pressure in any system.