Problem 90 - Chemistry and Chemical Reactivity

Question

In a police forensics lab, you examine a package that may contain heroin. However, you find the white powder is not pure heroin but a mixture of heroin \(\left(\mathrm{C}_{21} \mathrm{H}_{23} \mathrm{O}_{5} \mathrm{N}\right)\) and lactose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right) .\) To determine the amount of heroin in the mixture, you dissolve \(1.00 \mathrm{g}\) of the white powdery mixture in water in a 100.0 -mL volumetric flask. You find that the solution has an osmotic pressure of \(539 \mathrm{mm}\) Hg at \(25^{\circ} \mathrm{C}\). What is the composition of the mixture?

Step-by-Step Solution

Answer

The mixture contains 21.39% heroin and 78.61% lactose.

Step 1: Calculate Osmotic Pressure in atm

We convert the given osmotic pressure from millimeters of mercury to atmospheres. The conversion factor is 1 atm = 760 mmHg.\[P = \frac{539 \, \text{mmHg}}{760 \, \text{mmHg/atm}} = 0.7092 \, \text{atm}\]

Step 2: Apply the Osmotic Pressure Formula

Osmotic pressure (\( \Pi \)) is given by the formula \( \Pi = iMRT \). Here, \(i\) is the van 't Hoff factor (assumed to be 1 for non-electrolytes), \(M\) is molarity, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.First, convert 25°C to Kelvin:\[ T = 25 + 273.15 = 298.15 \, \text{K} \]Rearrange the formula to find molarity (\( M \)):\[M = \frac{\Pi}{RT} = \frac{0.7092}{0.0821 \times 298.15} = 0.02877 \, \text{mol/L}\]

Step 3: Determine Total Moles of Solute in Solution

The solution is in a 0.100 L volumetric flask, so calculate the total moles in the solution:\[\text{Total moles of solute} = 0.02877 \, \text{mol/L} \times 0.100 \, \text{L} = 0.002877 \, \text{mol}\]

Step 4: Let Variables Represent Moles of Components

Let \( x \) be the moles of heroin \( \text{C}_{21}\text{H}_{23}\text{O}_{5}\text{N} \) and \( y \) be the moles of lactose \( \text{C}_{12}\text{H}_{22}\text{O}_{11} \).Using the expression for total moles of solute, we have:\[ x + y = 0.002877 \]

Step 5: Molar Mass Calculation

Find the molar mass of each substance:- Heroin: \( 21(12.01) + 23(1.01) + 5(16.00) + 14.01 = 369.4 \, \text{g/mol} \)- Lactose: \( 12(12.01) + 22(1.01) + 11(16.00) = 342.3 \, \text{g/mol} \)

Step 6: Set Up Mass Equation

The total mass of 1 gram is given by:\[ 369.4x + 342.3y = 1.00 \]

Step 7: Solve the System of Equations

We have the following system of equations:1. \( x + y = 0.002877 \)2. \( 369.4x + 342.3y = 1.00 \)Solve equation 1 for \( y \):\[ y = 0.002877 - x \]Substitute into equation 2:\[369.4x + 342.3(0.002877 - x) = 1.00\]Simplify and solve for \( x \):\[369.4x + 0.9843 - 342.3x = 1.00\]\[27.1x = 0.0157\]\[x = 0.000579336 \, \text{mol}\]Substituting \( x \) in \( y = 0.002877 - x \):\[ y = 0.002877 - 0.000579336 = 0.002297664 \, \text{mol}\]

Step 8: Calculate Mass of Each Component

Calculate the mass of each component:- Mass of heroin: \( 369.4 \, \text{g/mol} \times 0.000579336 \, \text{mol} = 0.2139 \, \text{g} \)- Mass of lactose: \( 342.3 \, \text{g/mol} \times 0.002297664 \, \text{mol} = 0.7861 \, \text{g} \)

Step 9: Find Composition of the Mixture

The composition of the mixture is:- Heroin: \( \frac{0.2139}{1.00} \times 100\% = 21.39\% \)- Lactose: \( \frac{0.7861}{1.00} \times 100\% = 78.61\% \)

Key Concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution. This unit allows chemists to describe how much of a substance is present in a given volume of liquid. When dealing with osmotic pressure, knowing the molarity of a solution is essential.

• Definition: Molarity \( M \ = \ \frac{\text{moles of solute}}{\text{liters of solution}} \)
• Importance: Molarity helps determine how concentrated a chemical solution needs to be for different reactions or processes.

In the exercise example, molarity is used to find out the total number of moles of solute in the solution based on the osmotic pressure given. The calculation starts by rearranging the osmotic pressure formula, which involves converting the given temperature from degrees Celsius to Kelvin and thereby solving for molarity.

Volumetric Flask

A volumetric flask is a type of laboratory glassware that is used to prepare solutions with a precise volume. This flask is characterized by a narrow neck with an etched line around it, signifying the exact level to which it should be filled to achieve the desired volume. Volumetric flasks are essential in experiments requiring accuracy in solution concentration.

• Purpose: To contain a precise volume of liquid when making up a chemical solution.
• Material: Usually made from glass or plastic for stability and resistance to chemical reactions.

In the context of the exercise, a 100.0 mL volumetric flask is used to dissolve the white powdery mixture into water, ensuring that the resulting solution is precisely 100.0 mL. This precision is significant for calculating the molarity of the solution accurately, impacting the determination of osmotic pressure and, eventually, the composition of the mixture in question.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms present in a chemical formula. Molar mass acts as a bridge between the mass of a substance and the amount in moles.

• Application: Used to convert between grams and moles, enabling calculations involving the mass and amount of substances.
• Example: For heroin, the molar mass is calculated as \( 369.4 \, \text{g/mol} \), and for lactose, it is \( 342.3 \, \text{g/mol} \).

In forensic chemistry, determining the molar mass of substances like heroin and lactose helps in setting up equations for calculating the mass of each component based on their molar concentrations. This is crucial when analyzing mixtures where precise composition needs to be known, like the example involving the powdery mixture in the exercise.

Forensics Chemistry

Forensics chemistry involves the application of chemical techniques and analyses to solve crimes. It plays a critical role in the investigation and prosecution of criminal activities by examining substances and biological materials. Forensic chemists must be adept at various chemical methods to profile and identify substances found at crime scenes, assisting in legal investigations.

• Role: Helps law enforcement understand and form links to potential illegal activities through scientific evidence.
• Methods: Includes techniques such as chromatography, spectroscopy, and chemical analysis for drug identification, among others.

In the exercise scenario, forensics chemistry is applied to determine the composition of a suspected drug mixture through quantitative analysis. By dissolving the mixture and measuring the osmotic pressure, forensic chemists can unravel the proportions of components, such as heroin and lactose, confirming the nature of the substance in question.