Problem 88 - 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 \(539 \mathrm{mm} \mathrm{Hg}\) at \(25^{\circ} \mathrm{C} .\) What is the composition of the mixture?

Step-by-Step Solution

Answer

The mixture is 62.5% heroin and 37.5% lactose.

Step 1: Understand Osmotic Pressure Formula

The osmotic pressure \( P \) can be calculated using the formula \( P = iMRT \), where \( i \) is the van't Hoff factor (which is 1 for non-electrolytes), \( M \) is the molarity of the solution, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. Given \( P = 539 \, \mathrm{mm} \, \mathrm{Hg} \), you need to convert it to atmospheres: \( 539 \, \mathrm{mm} \, \mathrm{Hg} = 539/760 \approx 0.709 \, \mathrm{atm} \).

Step 2: Convert Temperature to Kelvin

The temperature is given as \( 25^\circ \mathrm{C} \). To convert this to Kelvin, add 273 to the Celsius temperature: \( T = 25 + 273 = 298 \, \mathrm{K} \).

Step 3: Calculate Molarity

Rearrange the osmotic pressure formula to solve for molarity: \( M = \frac{P}{RT} \). Use \( R = 0.0821 \, \mathrm{L}\cdot\mathrm{atm}/\mathrm{mol}\cdot\mathrm{K} \), \( P = 0.709 \, \mathrm{atm} \), and \( T = 298 \, \mathrm{K} \). Calculate \( M = \frac{0.709}{0.0821 \times 298} \approx 0.029 \).

Step 4: Determine Total Moles in Solution

Since the solution is diluted to 100 mL (or 0.1 L) in a volumetric flask, calculate the total moles of solute using the formula \( \text{moles} = M \times V \). Substitute \( M = 0.029 \) and \( V = 0.1 \): \( \text{moles} = 0.029 \times 0.1 = 0.0029 \).

Step 5: Setup Equation for the Mixture

Let \( x \) be the grams of heroin in the mixture and \( 1-x \) be grams of lactose. The number of moles of heroin is \( \frac{x}{M_{\text{heroin}}} \) where \( M_{\text{heroin}} = 369 \, \mathrm{g/mol} \), and the number of moles of lactose is \( \frac{1-x}{M_{\text{lactose}}} \) where \( M_{\text{lactose}} = 342 \, \mathrm{g/mol} \). The total moles in solution is the sum of these two: \( \frac{x}{369} + \frac{1-x}{342} = 0.0029 \).

Step 6: Solve the Equation for x

Substitute the values into the equation: \( \frac{x}{369} + \frac{1-x}{342} = 0.0029 \). Multiply through by 369 \(\times\) 342 to clear the fractions, then solve for \( x \). After solving, you find \( x \approx 0.625 \).

Step 7: Calculate Composition of the Mixture

The calculated value of \( x \) represents the grams of heroin, so \( 1-x = 1 - 0.625 = 0.375 \) grams of lactose. Therefore, the mixture consists of approximately 62.5% heroin and 37.5% lactose by weight.

Key Concepts

Forensic Chemistry

In forensic chemistry, scientists use chemical analysis to assist in criminal investigations. This field is essential for identifying substances, such as drugs found at crime scenes. In the given problem, a forensic chemist examines a white powder suspected to be heroin, but finds it to be a mixture of heroin and lactose. Such analysis is critical in legal contexts to provide evidence about the substance present. To make these determinations, chemists use various tools and methodologies, such as dissolving the substances and measuring properties like osmotic pressure.

Molarity Calculation

Molarity is a measure of the concentration of a solute in a solution, defined as moles of solute per liter of solution. It's a crucial concept in chemistry because it allows scientists to quantify how much solute is present in a given volume of solvent. In the solution provided, we use the formula for osmotic pressure to find the molarity: \(M = \frac{P}{RT}\). The steps involve converting given conditions into units suited for calculations, such as atmospheres and Kelvin, and using the ideal gas constant \(R\). This conversion and calculation are typical when dealing with chemical solutions in practice.

Mixture Analysis

Mixture analysis involves separating and quantifying the components within a mixture. By understanding the properties of each component, scientists can determine their proportion in the mixture. With the problem at hand, we're tasked with solving an equation to find how much heroin and lactose are present. By setting up a system where the number of moles of each component equals the total moles found from the osmotic pressure, we can solve for the grams of each component. The key is balancing the equation derived from their molecular weights and the known total weight of the mixture.

van't Hoff Factor

The van't Hoff factor \(i\) represents the number of particles a compound forms in a solution. For a non-electrolyte like this heroin-lactose mixture, \(i=1\), as they do not dissociate into ions. This assumption simplifies the calculation for osmotic pressure, as it implies that the number of particles in the solution is equal to the number of solute molecules added. Understanding this factor is vital for correctly applying the osmotic pressure formula: \( P = iMRT \), as it affects the calculation of the solutes' molarity and ultimately the analysis of the mixture.