Question

A \(32.65-\mathrm{g}\) sample of a solid is placed in a flask. Toluene, in which the solid is insoluble, is added to the flask so that the total volume of solid and liquid together is \(50.00 \mathrm{~mL}\). The solid and toluene together weigh \(58.58 \mathrm{~g} .\) The density of toluene at the temperature of the experiment is \(0.864 \mathrm{~g} / \mathrm{mL}\). What is the density of the solid?

Short answer

The density of the solid is 1.635 g/mL.

Step-by-step solution

Determine the mass and volume of toluene

The total mass of the solid and toluene together is given as 58.58 g. The mass of the solid is given as 32.65 g. By subtracting the mass of the solid from the total mass, we can obtain the mass of toluene. Mass of toluene = Total mass - Mass of the solid Mass of toluene = 58.58 g - 32.65 g Mass of toluene = 25.93 g We are given the total volume of the mixture as 50.00 mL and the density of toluene as 0.864 g/mL. We can determine the volume of toluene using its mass and density: Volume of toluene = Mass of toluene / Density of toluene Volume of toluene = 25.93 g / 0.864 g/mL Volume of toluene = 30.03 mL

Determine the volume of the solid

Now that we know the volume of toluene, we can subtract it from the total volume of the mixture to find the volume of the solid: Volume of the solid = Total volume - Volume of toluene Volume of the solid = 50.00 mL - 30.03 mL Volume of the solid = 19.97 mL

Calculate the density of the solid

With the mass and volume of the solid determined, we can now find its density using the density formula: Density of the solid = Mass of the solid / Volume of the solid Density of the solid = 32.65 g / 19.97 mL Density of the solid = 1.635 g/mL The density of the solid is 1.635 g/mL.

Key concepts

Mass and Volume Relationship

Understanding the relationship between mass and volume is crucial when working with physical quantities in science. Mass is a measure of the amount of matter in an object and is usually expressed in grams (g) or kilograms (kg). Volume, on the other hand, measures the space that an object occupies and is often expressed in milliliters (mL) or liters (L). These two properties are intimately linked when dealing with the concept of density.

To elucidate this relationship, imagine filling a container with a known volume with a substance. The amount of this substance, measured by its mass, will determine how dense the substance is within that volume. A greater mass in the same volume indicates a higher density, while a lesser mass suggests a lower density. This concept is vital in a wide range of applications, from engineering to medicine, as it affects material properties such as buoyancy and stability.

In the context of the problem at hand, a solid and a liquid (toluene), which are insoluble, coexist in the same flask. To find the respective volumes and calculate the solid's density, the solid's mass is given, and the total volume is known, allowing the calculation of the solid's volume by subtracting the volume occupied by toluene.

Density of Toluene

Toluene, a colorless liquid with a distinctive smell, is frequently used as a solvent in chemical processes and can serve as a reference substance in density determination experiments. The density of a substance is defined as its mass per unit volume, and specifically for toluene, it is given as 0.864 grams per milliliter (g/mL) in the exercise.

Understanding the density of toluene is vital in this context because it allows us to calculate the volume of toluene that takes up a certain mass within the mixture. By using the density formula, we can derive that the volume occupied by a known mass of toluene can be determined by dividing the mass by the liquid's density. This step is essential because it provides us with the remaining volume that can be attributed to the solid, enabling us to calculate the solid's density.

Remember, the density value of toluene could vary with temperature and pressure; however, for a controlled laboratory setting as indicated in the problem, the density can be considered constant, which simplifies the calculations.

Density Formula

The density formula is a fundamental expression that relates mass, volume, and density. It is written as:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
In the given exercise, this formula is the tool that allows us to calculate the density of the solid after determining its mass and volume. Once we deduce the volume of toluene in the flask, we subtract this from the total volume to find the volume of the solid. With this volume and the known mass of the solid, we apply the density formula to determine its density.

It's important to ensure that the mass and volume units are compatible, such as grams (g) for mass and milliliters (mL) for volume, which simplifies the calculation and avoids the need for unit conversion, thus enabling a straightforward application of the density formula. In the educational sense, grasping this relationship develops a deeper understanding of material properties and how they are quantified, which is an invaluable skill in scientific problem-solving.