Question

Lauryl alcohol is obtained from coconut oil and is used to make detergents. A solution of \(5.00 \mathrm{~g}\) of lauryl alcohol in \(0.100 \mathrm{~kg}\) of benzene freezes at \(4.1^{\circ} \mathrm{C}\). What is the approximate molar mass of lauryl alcohol?

Short answer

The approximate molar mass of lauryl alcohol is \(625\mathrm{~g/mol}\).

Step-by-step solution

Recall the freezing point depression formula and its constants

The freezing point depression formula is given by: \(ΔT_f = K_f × molality\) where \(ΔT_f\) is the freezing point depression, given by \(4.1^{\circ} \mathrm{C}\) in our case. \(K_f\) is the cryoscopic constant for benzene (Freezing Point Depression Constant), \(5.12 \mathrm{~K \cdot kg/mol}\). \(molality\) is the molality of the solution, which we will calculate later.

Calculate the molality of the solution

The molality of the solution is the number of moles of solute (lauryl alcohol) per kilogram of solvent (benzene). Since we need to find the molar mass of lauryl alcohol, let's call the molar mass \(M_L\). First, we will calculate the number of moles and then the molality of the solution. Moles of lauryl alcohol = \(\frac{mass}{molar\_mass} = \frac{5.00\mathrm{~g}}{M_L\mathrm{~g/mol}}\) Substitute the mass of benzene (in kg) in the molality formula: \(molality = \frac{moles\_of\_solute}{mass\_of\_solvent}= \frac{\frac{5.00\mathrm{~g}}{M_L\mathrm{~g/mol}}}{0.100\mathrm{~kg}}\)

Rewrite the freezing point depression formula

Plug the molality expression into the freezing point depression formula: \(4.1^{\circ} \mathrm{C}= 5.12 \mathrm{~K \cdot kg/mol} × \frac{\frac{5.00\mathrm{~g}}{M_L\mathrm{~g/mol}}}{0.100\mathrm{~kg}}\) Now our goal is to solve this equation for \(M_L\).

Solve for the molar mass of lauryl alcohol

We can now isolate the term with \(M_L\) and solve for it: \(M_L = \frac{5.00\mathrm{~g} × 5.12\mathrm{~K \cdot kg/mol} × 0.100\mathrm{~kg}}{4.1^{\circ} \mathrm{C}}\) Calculate the value of \(M_L\): \(M_L \approx \frac{2.56\mathrm{~g \cdot kg}}{4.1^{\circ} \mathrm{C}} \approx 0.625\mathrm{~kg/mol}\) Convert the molar mass back to grams per mole: \(0.625\mathrm{~kg/mol} × \frac{1000\mathrm{~g}}{1\mathrm{~kg}} \approx 625\mathrm{~g/mol}\) Thus, the approximate molar mass of lauryl alcohol is \(625\mathrm{~g/mol}\).

Key concepts

Molar Mass Calculation

Calculating the molar mass of a substance is an essential task in chemistry that helps us understand the mass of one mole of a given compound. The molar mass allows us to convert between the amount of substance in moles and its mass in grams.
In our problem, we need to find the molar mass of lauryl alcohol using the freezing point depression method. Here are the steps we follow:

• Identify the Mass: We have a mass of lauryl alcohol given as 5.00 g.
• Moles of Solute: The moles of lauryl alcohol can be expressed as \( \frac{5.00 \mathrm{~g}}{M_L \mathrm{~g/mol}} \), where \( M_L \) is the molar mass of lauryl alcohol.
• Use the Freezing Point Depression: Using the equation \( \Delta T_f = K_f \cdot \text{molality} \), we equate it with the molality formulation to solve for \( M_L \).
• Calculate: Once \( M_L \) is isolated, plug in the values to compute its approximate numerical result, converting the final outcome to grams per mole for ease of understanding.

Remember, accurate molar mass calculation assists in comprehending how much of a compound participates in chemical reactions.

Cryoscopic Constant

The cryoscopic constant, often denoted as \( K_f \), is a specific characteristic of each solvent used in calculations of freezing point depression. It indicates how much the freezing point temperature of a solution is lowered per molal concentration of a solute.

• Definition: It is expressed in units of \( \text{K} \cdot \text{kg/mol} \) and quantifies the degree of freezing point depression due to the solute.
• Specific Values: Each solvent has a unique cryoscopic constant. For benzene, used in our exercise, \( K_f = 5.12 \text{ K} \cdot \text{kg/mol} \).
• Usage: \( K_f \) is a critical factor to calculate the molality and thus the molar mass when solving for a solute's resulting impact on freezing point.

Understanding the cryoscopic constant allows you to predict how a solvent's freezing point is affected by different solutes, leveraging it in various practical chemistry situations to determine properties of unknown substances.

Molality

Molality is a measure of solute concentration that plays a key role in colligative properties like freezing point depression. It is defined as the number of moles of solute per kilogram of solvent.

• Formula: Molality (m) is given by \( \text{molality} = \frac{\text{moles of solute}}{\text{kg of solvent}} \).
• Unit: It is expressed in units of moles per kilogram (mol/kg).
• Application in Exercise: For the task at hand, using the mass of benzene as 0.100 kg, we find the molality from the moles of lauryl alcohol formula provided, and it contributes to calculating the freezing point depression.

Why choose molality? Unlike molarity, molality doesn’t change with temperature as it depends solely on the mass, making it more reliable in scenarios with temperature fluctuations. It is crucial in determining colligative properties reliably.