Problem 27 - General Chemistry: Principles and Modern Applications

Question

How many grams of iodine, \(I_{2}\), must be dissolved in \(725 \mathrm{mL}\) of carbon disulfide, \(\mathrm{CS}_{2}(d=1.261 \mathrm{g} / \mathrm{mL}),\) to produce a \(0.236 \mathrm{m}\) solution?

Step-by-Step Solution

Answer

Approximately 54.763 grams of iodine must be dissolved in the carbon disulfide to produce a 0.236 m solution.

Step 1: Calculate the mass of the solvent

First, find the mass of the carbon disulfide (\(CS_2\)). The mass (\(m\)) can be calculated from the density (\(d\)) and volume (\(V\)) using the following equation: \(m = d \cdot V\). Substitute \(d = 1.261 g/mL\) and \(V = 725 mL\): \(m = (1.261 g/mL) * (725 mL) = 914.225 g\). Convert the mass from grams to kilograms by dividing by 1000: \(914.225 g = 0.914225 kg\).

Step 2: Use the molality equation

The equation for molality (\(m\)) is defined as the number of moles of solute (\(n_s\)) divided by the mass of solvent (\(m_s\)) in kilograms: \(m = n_s/m_s\). Rearrange it to solve for \(n_s\): \(n_s = m \cdot m_s\). The given molality is \(0.236 m\), and from Step 1, we know that \(m_s = 0.914225 kg\). So, \(n_s = (0.236 m) * (0.914225 kg) = 0.2157771 moles\).

Step 3: Convert moles to grams

Finally, convert the moles of iodine to grams using its molar mass. The molar mass of iodine (\(I_2\)) is \(253.80894 g/mole\). Thus, the mass of iodine can be calculated by multiplying the number of moles by the molar mass: \(Mass = n_s \cdot Molar\,Mass = (0.2157771 moles) * (253.80894 g/mole) = 54.763 g\). This means that one must dissolve approximately 54.763 grams of iodine to produce the desired solution.

Key Concepts

Molarity

Molarity is a way of expressing the concentration of a solution. It measures the number of moles of a solute per liter of solution. This is different from molality, which uses the mass of the solvent instead of the volume of the whole solution. Molarity is useful when working with reactions that occur in liquid solutions because it directly relates the volume of the solution to the amount of substance in it.

To calculate molarity (\( M \)), use the formula:

• \( M = \frac{n}{V} \)

where \( n \) is the number of moles and \( V \) is the volume of the solution in liters.

Molarity can vary with temperature as the volume of the solution expands or contracts, hence it is temperature-dependent.

Density

Density is a fundamental property of matter that relates mass to volume. It tells us how much mass is contained in a specific volume of a substance.This is particularly important in solution preparation when we need to know the mass of a solvent that occupies a given volume, as seen in the problem where the density of carbon disulfide was used.

The formula for density (\( d \)) is:

• \( d = \frac{m}{V} \)

where \( m \) is mass and \( V \) is volume.

In the problem, the density of carbon disulfide helps us find out exactly how much it weighs for a given volume, which is crucial to determining the solution's properties.

Molar Mass

Molar mass is the mass of one mole of a given substance. It is measured in grams per mole (\( g/mol \)) and is numerically equal to the atomic mass of the substance in atomic mass units.The molar mass allows us to convert between grams and moles, which is crucial in chemical calculations.

For iodine (\( I_{2} \)), the molar mass is approximately 253.81 g/mol. This information is used in the problem to convert the number of moles of iodine, which is calculated from molality,into grams to find out the amount needed to be added to the solvent.

Understanding molar mass allows us to relate the number of particles in a substance to its measurable quantity, a key concept in chemistry calculations.

Solution Preparation

Solution preparation involves dissolving a solute in a solvent to make a desired concentration. In our problem, iodine is the solute and carbon disulfide is the solvent. The goal is to achieve a specific molality, a measure of concentration in moles of solute per kilogram of solvent.

Here are simple steps involved in solution preparation based on the problem:

• Calculate the required mass of the solvent using volume and density, like we did with carbon disulfide.
• Determine the amount of solute needed using desired molality and calculated mass of the solvent.
• Convert moles of solute to grams using its molar mass, ensuring accurate measurements for your solution.

Being precise in these calculations ensures the solution has the correct concentrations for experimental work or application. Understanding these concepts is necessary for making solutions that are accurate for research and industrial needs.