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 1.63 g/mL.

Step-by-step solution

Calculate the mass of toluene

First, we'll find the mass of toluene. To do this, we will subtract the mass of the solid from the total mass of the solid and toluene together. Total mass = Mass of solid + Mass of toluene Mass of toluene = Total mass - Mass of solid = 58.58 g - 32.65 g = 25.93 g

Calculate the volume of toluene

Next, we will determine the volume of toluene using the given density of toluene and the mass we calculated in step 1. Density of toluene = Mass of toluene / Volume of toluene Volume of toluene = Mass of toluene / Density of toluene = 25.93 g / 0.864 g/mL = 30.02 mL

Calculate the volume of the solid

Now we can find the volume of the solid. Since the total volume of the solid and liquid is given, we need to subtract the volume of the toluene from the total volume to get the volume of the solid. Total volume = Volume of solid + Volume of toluene Volume of solid = Total volume - Volume of toluene = 50.00 mL - 30.02 mL = 19.98 mL

Calculate the density of the solid

Finally, we will calculate the density of the solid using the formula: Density of solid = Mass of solid / Volume of solid = 32.65 g / 19.98 mL = 1.63 g/mL The density of the solid is 1.63 g/mL.

Key concepts

Mass and Volume Relationship

Understanding the relationship between mass and volume is fundamental to grasping the concept of density. Mass refers to the amount of matter in an object, commonly measured in grams (g) or kilograms (kg). Volume, on the other hand, is the space that an object occupies, generally quantified in milliliters (mL) or cubic centimeters (cm³) for smaller amounts, or liters (L) and cubic meters (m³) for larger quantities. The two properties are directly linked when calculating the density of a substance, which is a measure of how much mass is contained within a specific volume.

Density is expressed by the following formula:
\textbf{Density} = $\frac{\text{Mass}}{\text{Volume}}$
In the context of the exercise, the solid's mass is given as 32.65 g, and after a series of operations that involve toluene, the volume of the solid is deduced to be 19.98 mL. Utilizing these two key pieces of information allows us to calculate the density of the solid.

Density of Toluene

Toluene, a common organic solvent, has specific properties that are essential when working with density calculations. For the exercise in question, the density of toluene is given as 0.864 g/mL. This figure is crucial as it forms the basis for subsequent calculations.

Why Is the Density of Toluene Important?

The density of toluene allows us to determine the volume of toluene used in the experiment. In the real world, this is significant because the density of a liquid can be affected by temperature and pressure. For most classroom exercises, however, these variables are considered constant, and a single density value is used. Computing the volume of the toluene provides insight into the total volume occupied by the toluene in the mixture with the solid, which further leads to finding the volume of the solid itself.
When you encounter a substance with a known density like toluene, you can use that information to find an unknown volume (or mass, if that's what you're looking for) by rearranging the density formula.

Calculating Density of Solids

The procedure for calculating the density of a solid is exemplified in the exercise provided. With solids, the process entails finding both the mass and volume separately, then applying the density formula. It's noteworthy that the solid's volume is often not straightforward to measure, especially if the object has an irregular shape, which is why the method applied in the exercise is useful.

Displacement as a Tool to Find Volume

In our example exercise, we used the displacement of toluene by the solid to find the solid's volume. When the solid is added to a known volume of toluene, it displaces an amount of the liquid equal to its own volume. By subtracting the volume of toluene after displacement from the initial total volume, we can derive the volume of the solid. This method is incredibly helpful when dealing with irregularly shaped objects.
Once the volume is identified, as was done in the exercise, the density formula comes into play:
\textbf{Density of the solid} = $\frac{\text{Mass of the solid}}{\text{Volume of the solid}}$
Using the mass of the solid obtained directly from measurement and the volume derived through displacement, the calculation yields the density of the solid. In practical terms, understanding the density of materials is pivotal in fields like material science, quality control, and various engineering applications.