Problem 110 - General Chemistry: Principles and Modern Applications

Question

A solution contains \(750 \mathrm{g}\) of ethanol and \(85.0 \mathrm{g}\) of sucrose \(\left(180 \mathrm{g} \mathrm{mol}^{-1}\right) .\) The volume of the solution is \(810.0 \mathrm{mL} .\) Determine (a) the density of the solution (b) the percent of sucrose in the solution (c) the mole fraction of sucrose (d) the molality of the solution (e) the molarity of the solution

Step-by-Step Solution

Answer

The detailed calculations are needed to give precise results. However, the density would be roughly \(1.03 g/mL\), the percent of sucrose around \(10.18%\), and the molarity around \(0.583 M\). The mole fraction and molality cannot be calculated without additional information about the moles or mass of ethanol.

Step 1: Calculation of density

The density of a solution is defined as its mass divided by its volume. The total mass of the solution is the sum of the masses of ethanol and sucrose, which is \(750 g + 85 g = 835 g = 0.835 kg\). The volume of the solution is given as \(810.0 mL = 0.810 L\). Thus, the density of the solution can be calculated using the formula: \[ Density = \frac{mass}{volume} \]

Step 2: Calculation of percent of sucrose

Percentage of sucrose in the solution is calculated as the ratio of the mass of sucrose to the total mass of the solution, multiplied by 100. So, \[ Percentage = \frac{mass_{sucrose}}{mass_{total}} \times 100\%\]

Step 3: Calculation of mole fraction of sucrose

The mole fraction of sucrose is calculated as the ratio of the number of moles of sucrose to the total number of moles (sucrose + ethanol). The number of moles of a compound is given by the ratio of its mass to its molar mass. For sucrose, the molar mass is given as \(180 g/mol\). Thus, the number of moles of sucrose is \(85 g / 180 g/mol = 0.4722 mol\). Unfortunately, the number of moles of ethanol is not given, so it is not possible to calculate the mole fraction of sucrose.

Step 4: Calculation of molality of the solution

Molality is defined as the number of moles of the solute per kilogram of the solvent. In this case, the solute is sucrose and the solvent is ethanol. It can be calculated using the formula: \[ Molality = \frac{moles_{sucrose}}{mass_{ethanol}} \]

Step 5: Calculation of molarity of the solution

Molarity is defined as the number of moles of the solute per liter of the solution. It is calculated using the formula: \[ Molarity = \frac{moles_{sucrose}}{volume_{solution}} \]

Key Concepts

Density of a Solution

The density of a solution is a fundamental concept in chemistry that helps determine how concentrated a solution is. It is defined as the mass of the solution divided by its volume. In practical terms, think about density as the "heaviness" or "lightness" of a solution in a given space. To calculate it:

• Add up the total mass of all components in the solution. In our example, we added the mass of ethanol (750 g) and sucrose (85 g) to get 835 g.
• Convert this mass to kilograms if necessary (as in this example) and divide by the volume in liters. Here, we used 0.810 L to find our solution's density.

The result gives us the density which can help in further analysis, but remember, a low-density means a lighter solution, whereas a high density indicates a heavier one.

Percent Composition

Percent composition is a way to express the concentration of a solute in a solution in terms of mass percentage. It's pretty straightforward:

• Divide the mass of the component (in this case sucrose) by the total mass of the solution.
• Then, multiply by 100 to convert that ratio to a percent.

This percentage answers the question: "What ending part of this solution's mass is due to this particular solute?" For our sucrose example, you would calculate the mass of sucrose (85 g) divided by the total mass of the solution (835 g), and multiply by 100. It's a handy tool for quickly evaluating how much of a given substance is present.

Mole Fraction

The mole fraction lets you compare how many moles of a solute are present compared to the total moles in a solution, without worrying about volume or mass units. Here's how you get there:

• First, convert the mass of your solute and solvent into moles by dividing each by its respective molar mass. For sucrose with a molar mass of 180 g/mol, it turned into 0.4722 moles.
• Sum up all the moles present in the solution.
• The mole fraction of sucrose is then the moles of sucrose divided by the total moles in the solution.

This calculation helps in understanding the proportion of sucrose relative to the whole mix, providing a "mole-centered" perspective on composition.

Molality

Molality provides insight into concentration based on the mass of solvent rather than the total volume of the solution, which makes it convenient under varying temperature conditions. To find molality:

• Calculate the moles of solute present. For sucrose, as before, that’s 0.4722 moles.
• Find the mass of the solvent in kilograms. In this case, that's the mass of ethanol, 0.750 kg.
• Divide the moles of solute by the mass of the solvent in kg.

Molality thus gives you a concentration value that remains constant regardless of temperature changes, making it very useful when working with solutions that might get heated or cooled.

Molarity

Molarity tells us how many moles of solute are contained in one liter of the solution. It's one of the most common ways to describe solution concentration because it's straightforward and directly relates to volume, which is easy to measure. Steps to find molarity include:

• Calculate the moles of sucrose, which we have as 0.4722 moles.
• Use the volume of the entire solution in liters, 0.810 L in this case.
• Divide the moles of solute by the volume in liters.

The resulting number gives a clear picture of the concentration, allowing easy comparison between different solutions based on their potential reactive effects.