Question

An osmotic pressure of 5.15 atm is developed by a solution containing \(4.80 \mathrm{~g}\) dioxane (a nonelectrolyte) dissolved in \(250 . \mathrm{mL}\) water at \(15.0^{\circ} \mathrm{C}\). The empirical formula of dioxane is \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O} .\) Use the osmotic pressure data to show that the empirical formula and the molecular formula of dioxane are not the same.

Short answer

The empirical formula mass (44 g/mol) differs from molar mass (88 g/mol), indicating different molecular formulas.

Step-by-step solution

Understanding Osmotic Pressure

Osmotic pressure for a solution can be calculated using the formula\[ \pi = iMRT \]where \(\pi\) is the osmotic pressure, \(i\) is the van 't Hoff factor (which is 1 for nonelectrolytes), \(M\) is the molarity of the solution, \(R\) is the ideal gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in Kelvin. For this problem, \(\pi = 5.15 \) atm, \(i = 1\), and \(T\) is converted to Kelvin: \(15.0 + 273.15 = 288.15\) K.

Calculate Molarity

The molarity \(M\) is calculated from the osmotic pressure formula \(\pi = MRT\) rearranging to \(M = \frac{\pi}{RT}\), substituting in the values:\[ M = \frac{5.15\, \text{atm}}{0.0821\, \text{L atm/mol K} \times 288.15\, \text{K}} \approx 0.219\, \text{mol/L} \]

Calculate Moles of Dioxane

Calculate the moles of dioxane using molarity:\[ \text{moles of dioxane} = M \times V \]where \(V\) is the volume in liters (0.250 L in this case). Therefore:\[ \text{moles of dioxane} = 0.219\, \text{mol/L} \times 0.250\, \text{L} = 0.05475\, \text{mol} \]

Calculate Molar Mass of Dioxane

The molar mass is found by dividing the mass of dioxane by the calculated number of moles:\[ \text{molar mass of dioxane} = \frac{4.80\, \text{g}}{0.05475\, \text{mol}} \approx 87.7\, \text{g/mol} \]

Compare with Empirical Formula Mass

The empirical formula of dioxane \(\text{C}_2\text{H}_4\text{O}\) has an empirical formula mass of \(2 \times 12.01 + 4 \times 1.01 + 16.00 = 44.08\, \text{g/mol}\). This empirical mass differs from the molar mass of 87.7 g/mol found previously, indicating that the molecular formula is not the same as the empirical formula.

Key concepts

Nonelectrolyte

A nonelectrolyte is a substance that, when dissolved in water, does not ionize or produce charged particles. This means it does not conduct electricity in solution. Dioxane, the substance in our problem, is a nonelectrolyte. Unlike electrolytes, which dissociate into ions, nonelectrolytes dissolve as whole molecules.
This is why the van 't Hoff factor (\(i\)) is 1 for nonelectrolytes—it represents the number of particles the compound produces in solution. Since nonelectrolytes do not break apart, this factor remains 1. This property is crucial when calculating osmotic pressure, a key part of understanding the behavior of solutions.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of elements in a compound. For dioxane, the empirical formula is \(\text{C}_2\text{H}_4\text{O}\). This formula tells us that for every 2 carbon atoms, there are 4 hydrogen and 1 oxygen atom in the simplest form of dioxane.

• It does not necessarily reveal how many of these groups are in a molecule.
• An empirical formula can be derived from experimental data like combustion analysis or, in this problem, differences in calculated molar mass.

Understanding how empirical formulas differ from molecular formulas is key to comprehending compound structures.

Molarity

Molarity is the concentration of a solution expressed as the number of moles of solute per liter of solution. It is denoted by \(M\) and provides a way to relate the amount of solute to the volume of the solution.
In our example, we calculated the molarity from the osmotic pressure using the rearranged formula:

• The formula \(M = \frac{\pi}{RT}\) was applied.
• Values were substituted to get the molarity of dioxane in the solution.

This information helps understand how concentrated the solution is, which is vital for further calculations such as determining the number of moles.

Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It is crucial for converting between the amount of substance in moles and its mass in grams.
To find the molar mass of dioxane, we used the formula:

• Molar mass = \(\frac{\text{mass of dioxane}}{\text{moles of dioxane}}\)
• This calculation helps us understand the substance's properties and compare them to potential empirical formulas.

Dioxane's calculated molar mass was approximately 87.7 g/mol, indicating more complexity than the empirical formula suggests.

Molecular Formula

The molecular formula provides the actual number of each type of atom present in a molecule. It can be different from the empirical formula if the molecule contains more units of the empirical formula than one.
For dioxane, the calculated molar mass indicates a molecular formula differing from the empirical one.

• The empirical mass was 44.08 g/mol compared to the calculated 87.7 g/mol.
• The fact that this is roughly double suggests the molecular formula might be \(\text{C}_4\text{H}_8\text{O}_2\)

This means the molecule has twice the number of atoms indicated by the empirical formula, demonstrating a more complex molecular structure.