Question

A forensic chemist is given a white solid that is suspected of being pure cocaine \(\left(\mathrm{C}_{17} \mathrm{H}_{21} \mathrm{NO}_{4}, \text { molar mass }=303.35 \mathrm{g} / \mathrm{mol}\right)\) She dissolves \(1.22 \pm 0.01 \mathrm{g}\) of the solid in \(15.60 \pm 0.01 \mathrm{g}\) benzene. The freezing point is lowered by \(1.32 \pm 0.04^{\circ} \mathrm{C}\) a. What is the molar mass of the substance? Assuming that the percent uncertainty in the calculated molar mass is the same as the percent uncertainty in the temperature change, calculate the uncertainty in the molar mass. b. Could the chemist unequivocally state that the substance is cocaine? For example, is the uncertainty small enough to distinguish cocaine from codeine \(\left(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{NO}_{3}, \text { molar }\right.\) mass \(=299.36 \mathrm{g} / \mathrm{mol}\) )? c. Assuming that the absolute uncertainties in the measurements of temperature and mass remain unchanged, how could the chemist improve the precision of her results?

Short answer

The calculated molar mass of the substance is approximately 305.43 ± 9.25 g/mol. Considering the uncertainty range, it is not possible to unequivocally state that the substance is cocaine because the molar mass of codeine (299.36 g/mol) also falls within the uncertainty range. To improve the precision of the results, the chemist could use more accurate measuring devices, perform multiple trials, or use a solution with a lower freezing point depression constant.

Step-by-step solution

Write down the freezing point depression equation

The freezing point depression equation is given by: ∆T = Kf * m where ∆T is the change in freezing point, Kf is the cryoscopic constant (5.12 °C·kg/mol for benzene), and m is the molality of the solution. The molality is given by: m = (moles of solute) / (mass of solvent in kg) Since we need to calculate the molar mass of the substance, let's first express the moles of solute in terms of molar mass: moles of solute = (mass of solute) / (molar mass) Now, the molality can be written as: m = (mass of solute) / ((molar mass) * (mass of solvent in kg))

Find the molality of the solution

Solve the freezing point depression equation for molality: m = ∆T / Kf m = (1.32 ± 0.04 °C) / (5.12 °C·kg/mol) = 0.258 ± 0.0078 mol/kg

Determine the molar mass

Using the expression for molality from step 1, we can now find the molar mass (M): M = (mass of solute) / (m * (mass of solvent in kg)) M = (1.22 ± 0.01 g) / (0.258 ± 0.0078 mol/kg * 0.0156 ± 0.0001 kg) M ≈ 305.43 g/mol

Calculate the uncertainty of molar mass

The percent uncertainty of the calculated molar mass can be found using the percent uncertainty of the temperature change: Percent uncertainty of ∆T = (0.04 °C) / (1.32 °C) * 100% ≈ 3.03% Now, apply the same percent uncertainty to the calculated molar mass: Uncertainty in molar mass = (3.03%) * (305.43 g/mol) ≈ 9.25 g/mol The molar mass of the substance is approximately 305.43 ± 9.25 g/mol.

Compare the molar mass and uncertainty with cocaine and codeine

Using the calculated molar mass of the substance (305.43 ± 9.25 g/mol), we cannot say for certain that the substance is cocaine as the molar mass of codeine (299.36 g/mol) also falls within the range of uncertainty.

Improving the precision of the experiment

To improve the precision of the results, the chemist could: 1. Utilize a more accurate measuring device for the mass and temperature measurements to reduce their uncertainties. 2. Perform several trials of the experiment and calculate the average molar mass. 3. Employ a method with less environmental factors influencing the results, such as using a solution with a lower freezing point depression constant.

Key concepts

Freezing Point Depression

The concept of freezing point depression is a fundamental principle in chemistry that helps us understand how adding a solute to a solvent reduces the freezing point of the mixture. This phenomenon is useful in determining properties of unknown substances.

When a solute is dissolved in a solvent, the solution's freezing point becomes lower than that of the pure solvent. This effect occurs because solute molecules disrupt the formation of the solid phase, requiring a lower temperature to achieve the same structure. The extent of the freezing point decrease, known as freezing point depression, is given by the equation:

\[\Delta T = K_f \times m\]

where:

• \(\Delta T\) is the change in freezing point.
• \(K_f\) is the cryoscopic constant, characteristic of the solvent.
• \(m\) is the molality of the solution, defined as moles of solute per kilogram of solvent.

This equation allows chemists to determine unknown properties like molar mass of a solute based on measurable changes in the freezing point.

Cryoscopic Constant

The cryoscopic constant, \(K_f\), pertains to the property of a solvent that quantifies its susceptibility to freezing point depression when a solute is added. This constant is unique for each solvent and is typically reported in units of °C·kg/mol.

In the case of benzene, a common solvent used in various chemical analyses, the cryoscopic constant is 5.12 °C·kg/mol. This means that for every one molal increase in solute concentration, the freezing point of benzene decreases by 5.12 °C.

Understanding the cryoscopic constant is crucial because it allows chemists to calculate the molarity of the solute from the observable change in the freezing point. This principle plays a vital role in forensic chemistry when determining the identity of substances, such as distinguishing between cocaine and other similar compounds.

Uncertainty Analysis

Uncertainty analysis is critical in scientific experiments to estimate the reliability and accuracy of measured data. This involves assessing the possible errors or variations in measurements, quantifying them, and then applying them to the results.

For this exercise, we assessed the uncertainty in the temperature change, which was \(0.04\, ^{\circ}\mathrm{C}\), relative to the overall change of \(1.32\, ^{\circ}\mathrm{C}\). The percent uncertainty was found to be approximately 3.03%. This same percentage was applied to the calculated molar mass, aiding in error propagation to yield \(305.43 \pm 9.25 \) g/mol.

The importance of uncertainty analysis cannot be overlooked as it provides chemists with a range within which they can confidently discuss their findings, offering critical insights into whether the identified substance aligns with its suspected identity.

Forensic Chemistry

Forensic chemistry applies chemical principles and techniques to the law, particularly in the analysis of evidence and detection of illegal substances. Here, the exercise highlights forensic chemistry through the determination of the white solid's identity.

Forensic chemists often use methods like freezing point depression not just to identify molecular characteristics but also to confirm the purity or identity of a substance. Considering the molar masses in this problem, uncertainty plays a key role in determining whether the white solid is cocaine or possibly another substance like codeine.

These calculations and analyses assist in forensic investigations by providing solid scientific backing to claims, making it crucial to ensure accuracy and precision in every step of the analysis.