Problem 30 - Chemistry: An Atoms-Focused Approach

Question

Alpetin is a compound found in Alpinia speciosa, a tropical evergreen used historically for treating cold, flu, fever, flatulence, and indigestion. A \(54.1 \mathrm{mg}\) sample of alpetin is dissolved in \(75.0 \mathrm{mL}\) of water. The osmotic pressure of the solution is 0.0657 atm at \(27^{\circ} \mathrm{C}\) Assuming that alpetin is a nonelectrolyte, calculate the molar mass of alpetin.

Step-by-Step Solution

Answer

Answer: The molar mass of alpetin is approximately \(270 \, \mathrm{g/mol}\).

Step 1: 1. Convert temperature to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15: \(T = 27^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{K} \)

Step 2: 2. Rearrange the osmotic pressure formula for molarity

We have the osmotic pressure and temperature, so we can rearrange the formula to calculate molarity: \(c = \dfrac{\Pi}{RT}\)

Step 3: 3. Calculate molarity

Now, plug in the osmotic pressure, temperature, and gas constant into the formula: \(c = \dfrac{0.0657 \mathrm{atm}}{(0.0821 \frac{\text{atm} \cdot \text{L}}{\text{mol} \cdot \text{K}})(300.15 \mathrm{K})} = 0.00267 \, \mathrm{mol/L}\)

Step 4: 4. Calculate moles of alpetin

Now that we have the molarity, we can calculate the moles of alpetin in the solution using the volume of the water: \(n = c \cdot V = (0.00267 \, \mathrm{mol/L})(0.0750 \mathrm{L}) = 2.0025 \times 10^{-4} \, \mathrm{mol}\)

Step 5: 5. Calculate the molar mass of alpetin

Finally, we can calculate the molar mass using the given mass of alpetin and the moles calculated in the previous step: \(M = \dfrac{54.1 \, \mathrm{mg}}{2.0025 \times 10^{-4} \, \mathrm{mol}}\) First, convert the mass from milligrams to grams: \(54.1 \, \mathrm{mg} = 0.0541 \, \mathrm{g}\) Now, calculate the molar mass: \(M = \dfrac{0.0541 \, \mathrm{g}}{2.0025 \times 10^{-4} \, \mathrm{mol}} = 270 \, \mathrm{g/mol}\) So, the molar mass of alpetin is approximately \(270 \, \mathrm{g/mol}\).

Key Concepts

Osmotic Pressure

Osmotic pressure is a fundamental concept in chemistry and biology, especially when dealing with solutions. It refers to the pressure required to prevent the flow of a solvent across a semipermeable membrane due to osmosis. Osmosis itself is the movement of a solvent (like water) from a region of lower solute concentration to a higher one, to balance the solute concentrations on both sides of the membrane.

In the context of the exercise, we are given the osmotic pressure to help us find the molar mass of alpetin. The osmotic pressure is linked to several factors, including the molarity of the solution and its temperature. The formula used is:\[ \Pi = cRT \]where:

• \( \Pi \) is the osmotic pressure (in atm)
• \( c \) is the molarity of the solution (in mol/L)
• \( R \) is the ideal gas constant (0.0821 atm·L/mol·K)
• \( T \) is the temperature in Kelvin

Rearranging this formula can determine the molarity of a solution when osmotic pressure and temperature are known.

Nonelectrolytes

Nonelectrolytes are substances that, when dissolved in water, do not dissociate into ions. This means they do not conduct electricity in solution. Typical examples of nonelectrolytes include sugars and alcohols. Alpetin, as stated in the exercise, is a nonelectrolyte.

When dealing with nonelectrolytes in molar mass calculations through osmotic pressure, we assume that each molecule stays intact in solution. This contrasts with electrolytes, which dissociate and create more particles in the water, affecting colligative properties like osmotic pressure.

Understanding that alpetin is a nonelectrolyte simplifies the calculation because the van 't Hoff factor (which is usually considered for ionizing solutes) is equal to 1. Thus, the osmotic pressure directly relates to the molarity of the nonelectrolyte without any adjustment for ionization.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions or, in this context, the quantitative relationship between substances in a solution. It is crucial when determining the molar mass of a compound based on experimental data like mass, volume, and osmotic pressure.

The exercise used stoichiometry to calculate the molar mass of alpetin. Here's a quick breakdown:

• Converting the mass of alpetin from milligrams to grams ensures consistency with the molarity, which measures moles per liter.
• Finding the molarity involved using the osmotic pressure formula, already adjusted for nonelectrolytes.
• By using the molarity and the known volume of water, we calculated the number of moles of alpetin dissolved.
• Finally, using the ratio of mass to moles, we could find the molar mass of alpetin.

Stoichiometry shapes how we approach these calculations, ensuring accuracy in the conversion and manipulation of units throughout the process.