Problem 47 - Chemistry: Principles and Reactions

Question

A metal slug weighing \(25.17 \mathrm{~g}\) is added to a flask with a volume of \(59.7 \mathrm{~mL}\). It is found that \(43.7 \mathrm{~g}\) of methanol \((d=0.791 \mathrm{~g} / \mathrm{mL})\) must be added to the metal to fill the flask. What is the density of the metal?

Step-by-Step Solution

Answer

Answer: The density of the metal slug is 5.65 g/mL.

Step 1: Determine the volume of methanol

Using the mass and density of methanol, we can find its volume by dividing the mass by the density: Volume of methanol = \(\frac{mass}{density}\) Plug in the given values: \(43.7 \mathrm{~g}\) (mass of methanol) \(0.791 \mathrm{~g/mL}\) (density of methanol) Volume of methanol = \(\frac{43.7 \mathrm{~g}}{0.791 \mathrm{~g/mL}} = 55.25 \mathrm{~mL}\)

Step 2: Find the volume of the metal slug

To find the volume of the metal slug, subtract the volume of methanol from the total volume of the flask: Volume of metal slug = Total volume of flask - Volume of methanol Volume of metal slug = \(59.7 \mathrm{~mL} - 55.25 \mathrm{~mL} = 4.45 \mathrm{~mL}\)

Step 3: Calculate the density of the metal slug

Now that we have the mass and volume of the metal slug, we can find its density by using the formula: Density of metal slug = \(\frac{mass}{volume}\) Plug in the given values: \(25.17 \mathrm{~g}\) (mass of metal slug) \(4.45 \mathrm{~mL}\) (volume of metal slug) Density of metal slug = \(\frac{25.17 \mathrm{~g}}{4.45 \mathrm{~mL}} = 5.65 \mathrm{~g/mL}\) The density of the metal slug is \(5.65 \mathrm{~g/mL}\).

Key Concepts

Density of Materials

The density of a material is a measure of how much mass it contains per unit of volume. It's a fundamental property that can tell us quite a lot about the substance, such as its composition and whether it will float or sink in a fluid.

Density is calculated using the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \].

For instance, materials like lead or gold have high densities which means for a given volume, they have a higher mass compared to materials like aluminium or plastic. This concept becomes crucial when identifying materials, contemplating buoyancy, or designing objects for specific applications.

Volume Calculation

Volume refers to the amount of space occupied by an object or fluid. It's crucial in many fields, including chemistry, where reactions occur in three-dimensional spaces. To calculate the volume of a regular-shaped object, one might use mathematical formulas involving the object's dimensions. For irregular objects, displacement methods — like the one used in the provided exercise to find the volume of the metal slug — are practical.

In practical chemistry, the ability to precisely measure volume is essential for creating solutions, administering reagents, and analyzing the outcomes of experiments. Knowing the volume helps in determining the right concentrations for chemical reactions, which is crucial for both the effectiveness and safety of these processes.

Mass-Volume Relationship

The relationship between mass and volume is directly captured by the concept of density. Given a constant mass, an object’s volume can determine whether it will be dense or sparse. Conversely, for a set volume, the mass can define whether the material inside is heavy or light.

In chemistry and other sciences, this mass-volume relationship explains a variety of phenomena, from the buoyancy of objects in fluids to the design of materials with desired properties. It is also essential in stoichiometry calculations (the calculation of reactants and products in chemical reactions) and in industries that need to package or transport materials efficiently.