Problem 82 - Chemistry The Central Science In Si Units

Question

A dilute aqueous solution of fructose in water is formed by dissolving \(1.25 \mathrm{~g}\) of the compound in water to form \(0.150 \mathrm{~L}\) of solution. The resulting solution has an osmotic pressure of \(112.8 \mathrm{kPa}\) at \(20^{\circ} \mathrm{C}\). Assuming that the organic compound is a nonelectrolyte, what is its molar mass?

Step-by-Step Solution

Answer

The molar mass of fructose is approximately 180 g/mol.

Step 1: Understand the Formula for Osmotic Pressure

The osmotic pressure  \( \Pi \)  of a solution is given by the formula \( \Pi = iMRT \), where \( M \) is the molarity of the solution, \( R \) is the ideal gas constant \( 0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \), \( T \) is the temperature in Kelvin, and \( i \) is the van't Hoff factor which is 1 for nonelectrolytes. First, convert the osmotic pressure from kPa to atm.

Step 2: Convert Units

To convert the osmotic pressure from kPa to atm, use the conversion factor \( 1 \, \text{atm} = 101.3 \, \text{kPa} \). Thus, \( 112.8 \, \text{kPa} = \frac{112.8}{101.3} \, \text{atm} \approx 1.113 \, \text{atm} \). Also, convert the temperature from Celsius to Kelvin: \( T = 20 + 273.15 = 293.15 \, \text{K} \).

Step 3: Calculate Molarity of the Solution

Using the converted osmotic pressure in the equation \( \Pi = MRT \), rearrange to solve for molarity \( M \): \( M = \frac{\Pi}{RT} \). Substitute \( \Pi = 1.113 \, \text{atm} \), \( R = 0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \), and \( T = 293.15 \, \text{K} \) into the equation to compute the molarity.

Step 4: Compute the Molar Mass

The molarity tells us moles per liter, and thus the moles of solute present. Since \( M = \frac{n}{V} \) and \( V = 0.150 \, \text{L} \), calculate \( n \) (moles of fructose): \( n = M \times V \). Then use \( \text{molar mass} = \frac{\text{mass}}{\text{moles}} \) to find molar mass, with mass given as \( 1.25 \, \text{g} \).

Key Concepts

Molarity

Molarity is an essential concept in chemistry which measures the concentration of a solute in a solution. Specifically, it is defined as the number of moles of solute per liter of solution. This measurement helps us understand how much of a substance is dissolved in a given volume of liquid.
When calculating molarity, the unit is expressed in moles per liter (mol/L), often denoted as "M." The formula to calculate molarity is:

• \[ M = \frac{n}{V} \]

where:

• \( n \) is the number of moles of the solute.
• \( V \) is the volume of the solution in liters.

To find the molarity in the context of osmotic pressure, the formula is rearranged from the osmotic pressure equation \( \Pi = iMRT \). By substituting the values such as the osmotic pressure from experiments and converting it using appropriate constant values, the molarity can be established.

Van't Hoff Factor

The Van't Hoff factor, symbolized as \( i \), is pivotal when dealing with solutions, especially those involving osmotic pressure. It describes the number of particles a single unit of solute separates into when dissolved in a solution. For nonelectrolytes, substances that do not dissociate into ions in solution, the Van’t Hoff factor is typically 1.
This factor is crucial in calculations involving colligative properties, including osmotic pressure, boiling point elevation, and freezing point depression, since these depend on the number of particles dissolved and not on their identity.
For the calculation of osmotic pressure, the Van’t Hoff factor modifies the ideal conditions: osmotic pressure \( \Pi \) shows how much a given solute alters a property of the solvent.

• For instance, in the formula \( \Pi = iMRT \), substituting the Van't Hoff factor for \( i \) provides accurate results even for substances that can ionize or dissociate.

In most educational exercises concerning nonelectrolytes, \( i \) is equal to 1, simplifying calculations and focusing attention on the primary concentration-related aspects.

Ideal Gas Constant

The Ideal Gas Constant, often denoted as \( R \), bridges various concepts in thermodynamics and solution chemistry. It is a critical part of the ideal gas law equation, which serves as a good approximation for the behavior of gases under many conditions. In the context of solutions and osmotic pressure, \( R \) helps quantify how solute molecules exert pressure when dissolved in a solvent.
The ideal gas constant is expressed as:

• \( R = 0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \)

Using \( R \) in the osmotic pressure equation \( \Pi = iMRT \) is crucial for converting theoretical mole-based expressions into measurable amounts such as atmospheric pressure.
To apply this in practice, the temperature must be converted to Kelvin, a standard international unit for thermodynamic temperature measurements. This ensures consistent results in the calculations because Kelvin accounts for absolute zero - the point where all molecular motion ceases.
In summary, \( R \) helps link the theoretical calculations to real-world measurements, supporting a broader understanding of gaseous behavior and its principles in chemical solutions.