Question

The density of a \(56.0 \%\) by weight aqueous solution of 1 -propanol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{OH}\right)\) is \(0.8975\) \(\mathrm{g} / \mathrm{cm}^{3}\). What is the mole fraction of the compound ? (a) \(0.292\) (b) \(0.227\) (c) \(0.241\) (d) \(0.276\)

Short answer

The mole fraction of 1-propanol in the solution is approximately 0.292.

Step-by-step solution

Calculate the mass of 1-propanol and water

Given a 56.0% by weight solution, this means that every 100 grams of solution contain 56.0 grams of 1-propanol and 44.0 grams of water.

Calculate the moles of 1-propanol

Calculate the moles of 1-propanol using its molar mass. The molar mass of 1-propanol is approximately 60.1 grams/mol. The moles of 1-propanol are calculated by dividing the mass by its molar mass: moles of 1-propanol = 56.0 grams / 60.1 grams/mol.

Calculate the moles of water

Calculate the moles of water using its molar mass. The molar mass of water (H2O) is approximately 18.015 grams/mol. The moles of water are calculated by dividing the mass of water by its molar mass: moles of water = 44.0 grams / 18.015 grams/mol.

Calculate the mole fraction of 1-propanol

The mole fraction of 1-propanol is the moles of 1-propanol divided by the total moles of the solution. Mole fraction of 1-propanol = moles of 1-propanol / (moles of 1-propanol + moles of water).

Plug in values and calculate

Plug in the calculated moles of 1-propanol and water from Step 2 and Step 3 into the equation from Step 4 and solve for the mole fraction of 1-propanol. Compare this value to the given options to find the correct one.

Key concepts

Molar Mass

Understanding the concept of molar mass is fundamental when it comes to mole fraction calculation. Molar mass, often referred to in units of grams per mole (g/mol), is a property that tells us the mass of one mole of a substance. A mole is a unit that represents Avogadro's number, which is approximately 6.022 \( \times \) 10^23 particles, be they atoms, molecules, or ions. This allows chemists to weigh out amounts of a substance in grams and convert this into an amount of moles, which is more useful for chemical reactions and calculations.

To calculate molar mass, one must sum the atomic masses of every atom in the molec8ular formula. The atomic masses are usually found on the periodic table. For example, for the organic compound 1-propanol ((CH_{3}CH_{2}CH_{2}OH)), the molar mass is the combined atomic masses of three carbon atoms, eight hydrogen atoms, and one oxygen atom. Having molar mass allows for the conversion between the mass of a compound and the number of moles, which is crucial for calculating solution concentrations and mole fractions.

Moles Calculation

When it comes to working with solutions, the calculation of moles provides a bridge between a substance's mass and the number of its molecules. To determine the number of moles, we utilize the formula:\[ \text{number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]

This formula is pivotal because it lets us quantify substances in terms of particles, rather than mass. In practice, say we have a certain mass of 1-propanol, and we want to find out how many moles we have. We would divide the given mass by the molar mass, calculated as provided in the 'Molar Mass' section previously. Similarly, for water with a known molar mass, the same formula applies to find the number of moles of water. These calculations are indispensable in determining the proportions of components in chemical reactions, as well as in calculating concentrations and mole fractions in solutions.

Solution Concentration

Solution concentration quantifies how much solute is present in a given amount of solvent. It can be expressed in various ways, such as percent by weight, molarity, molality, and mole fraction. Each of these measures has specific applications and is useful for various types of chemical calculations.

For instance, a percentage by weight, as used in the given exercise, refers to grams of solute per 100 grams of solution. Molarity and molality are concerned with the relationships between moles of solute and volume or mass of solvent, respectively. The mole fraction, however, expresses the proportion of the moles of one component relative to the total moles in the mixture, thus providing a dimensionless number that's valuable in calculations involving vapor pressure, osmotic pressure, and boiling point elevation. Calculating solution concentration appropriately is indispensable for precise chemical analysis and formulation of mixtures.

Density and Concentration Relationship

Density and concentration are closely related when we consider solutions. Density is defined as mass per unit volume and is often expressed in grams per cubic centimeter (g/cm^3). In the context of solutions, understanding how density relates to concentration can be essential for various calculations.

For example, if the density of a solution is known, you can figure out the total mass of the solution for a given volume, and then using the percentage by weight (concentration), deduce the mass of the solute and solvent present. In laboratory practices, this relationship is often used for preparing solutions where specific concentrations are required. It is also utilized in industrial applications to determine the formulation of chemical products. By mastering the concept of density and its relation to concentration, students can tackle a wide array of problems in chemistry with confidence.