Problem 33 - General Chemistry: Principles and Modern Applications

Question

Calculate the mole fraction of the solute in the following aqueous solutions:(a) \(0.112 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) \((d=1.006 \mathrm{g} / \mathrm{mL})\) (b) \(3.20 \%\) ethanol,by volume \((d=0.993 \mathrm{g} / \mathrm{mL}\) pure \(\left.\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}, d=0.789 \mathrm{g} / \mathrm{mL}\right)\).

Step-by-Step Solution

Answer

The mole fraction of the solute is \(0.0020\) in (a) and \(0.010\) in (b)

Step 1: Determine the number of moles of solute in (a)

e first part of the problem provides a solution in which the concentration of the solute, glucose, is given as \(0.112 M\). The molarity (M) is defined as the number of moles of the solute divided by the liters of the solution. Thus, one liter solution contains \(0.112 moles\) of glucose.

Step 2: Calculate the number of moles of solvent in (a)

To find the number of moles of the solvent, we should know the mass and the molar mass of the solvent. The mass of the solvent can be found by subtracting the mass of solute from the total mass of the solution. Since this is an aqueous solution, the solvent is water, with a density of \(1.00 g/mL\) and a molar mass of \(18.015 g/mol\). Thus, in one liter of solution (\(1000 mL\)), the mass of water is \(1000 g\). Hence, the number of moles of water is \(1000 g / 18.015 g/mol = 55.5 moles\).

Step 3: Calculate the mole fraction in (a)

The mole fraction is the ratio of the number of moles of the solute to the total number of moles in the solution. Hence, the mole fraction of glucose in the solution is \(0.112 moles / (0.112 moles + 55.5 moles) = 0.0020\). So the mole fraction of the solute in solution (a) is \(0.0020\)

Step 4: Determine the number of moles of solute in (b)

In the second part of the problem, the concentration of the solute, ethanol, is given as 3.20% \(v/v\). This means that 3.20 mL of ethanol is present in 100 mL of solution. The molar mass of ethanol is \(46.07 g/mol\). The density of pure ethanol is \(0.789 g/mL\). Hence, the mass of ethanol in 100 mL of solution is \(3.20 mL * 0.789 g/mL = 2.52 g\). So, the number of moles of ethanol in 100 mL solution is \(2.52 g / 46.07 g/mol = 0.055 moles\).

Step 5: Calculate the number of moles of solvent in (b)

To find the number of moles of the solvent, we should know the mass and the molar mass of the solvent. Since this is also an aqueous solution, the solvent is water, with a density of \(1 g/mL\) and a molar mass of \(18.015 g/mol\). The total volume of solution is 100 mL and we have 3.20 mL of ethanol in it. Hence, the volume of water in solution is \(100 mL - 3.20 mL = 96.8 mL\). So, the mass of water is \(96.8 g\) and the number of moles of water is \(96.8 g / 18.015 g/mol = 5.37 moles\).

Step 6: Calculate the mole fraction in (b)

The mole fraction is the ratio of the number of moles of the solute to the total number of moles in the solution. Hence, the mole fraction of ethanol in the solution is \(0.055 moles / (0.055 moles + 5.37 moles) = 0.010\). So the mole fraction of the solute in solution (b) is \(0.010\).

Key Concepts

Molarity

Molarity (\(M\)) is a measurement that refers to the concentration of a solution. It is defined as the number of moles of solute per liter of solution. For example, if you have a solution with a molarity of \(0.112\ M,\) it means there are \(0.112\) moles of the solute in every liter of the solution. This concept is particularly helpful for chemists because it allows them to understand how concentrated a substance is within the solution.

• To calculate molarity, use the formula: \(M = \frac{n}{V}\), where \(n\) is the number of moles of solute and \(V\) is the volume of solution in liters.
• Molarity is crucial for reactions occurring in solutions, as it helps determine the amount of reactants and products.

Understanding molarity is fundamental when working with aqueous solutions as it directly influences the reactions undertaken. It's an essential concept for accurate scientific experimentation and formulation.

Aqueous Solutions

An aqueous solution is a solution in which water acts as the solvent. Water is a universal solvent and forms the base of many solutions used in chemistry.

• A solution is considered aqueous if water is the major component.
• In aqueous solutions, solutes like salts, acids, or alcohols dissolve well and disperse throughout the water.
• The physical properties of the solvent, such as density and viscosity, are crucial for calculations involving aqueous solutions.

One common property of aqueous solutions is their transparency; they allow light to pass through, making them appear clear. This feature is beneficial for various scientific and industrial applications. Aqueous solutions play critical roles in biological systems and chemical processes due to water's high polarity and ability to stabilize ions and molecules.

Ethanol Solution

Ethanol solutions involve ethanol dissolved in a solvent, often water, to create a mixture that utilizes ethanol's properties. Ethanol, a type of alcohol, is both a solvent and a solute depending on the context.

• Ethanol behaves both as a solvent and a solute, due to its ability to mix with water while altering some of water's properties such as density.
• Understanding its properties like density (0.789 g/mL) is important for calculations.
• Ethanol solutions are used in various fields, including pharmaceuticals, cosmetics, and household products.

When preparing an ethanol solution, knowing the percentage by volume is key to determine concentration; this helps in assessing how much ethanol is present in a given volume of solution. The calculations typically involve understanding the proportion of ethanol to the total solution volume utilizing its volumetric data.

Glucose Solution

Glucose solutions are a mixture of glucose in water, commonly used in both scientific and medical contexts for their energy-supplying capabilities.

• Glucose has the chemical formula \(C_{6}H_{12}O_{6}\), commonly used as a solute in aqueous solutions.
• It's used to provide a source of energy in hospital settings, often administered intravenously.
• In scientific experiments, glucose solutions are used to study biochemical and physiological processes.

To calculate properties like the mole fraction in glucose solutions, knowing the molarity (like \(0.112 M\)) and the density of the solution is helpful. By knowing the number of moles of glucose and comparing it to the entire solution, one can determine the relative concentration of the solute, allowing control over the experiment or treatment being administered.