Question

A 25.00-g sample of a solid is placed in a graduated cylinder, and then the cylinder is filled to the 50.0-mL mark with benzene. The mass of benzene and solid together is 58.80 g. Assuming that the solid is insoluble in benzene and that the density of benzene is \(0.880 \mathrm{g} / \mathrm{cm}^{3},\) calculate the density of the solid.

Short answer

The density of the solid is \(2.16 \mathrm{g/cm^3}\).

Step-by-step solution

Find the mass of benzene in the mixture

We are given the mass of the mixture, which includes both the solid and the benzene. We are also given the mass of the solid. To find the mass of benzene, we need to subtract the mass of the solid from the total mass of the mixture: mass_benzene = mass_mixture - mass_solid Given: mass_mixture = 58.80 g mass_solid = 25.00 g mass_benzene = 58.80 g - 25.00 g = 33.80 g So, the mass of the benzene in the mixture is 33.80 g.

Find the volume of the benzene in the mixture

We are given the density of benzene and we have calculated the mass of benzene in the mixture. We can now use the density formula to find the volume of benzene: density = mass / volume We will rearrange the formula to solve for volume: volume = mass / density Given: density_benzene = 0.880 g/cm³ mass_benzene = 33.80 g volume_benzene = 33.80 g / 0.880 g/cm³ = 38.41 cm³ So, the volume of the benzene in the mixture is 38.41 cm³.

Find the volume of the solid

We know the total volume of the mixture and the volume of the benzene. We can now find the volume of the solid by subtracting the volume of benzene from the total volume: volume_solid = volume_mixture - volume_benzene Given: volume_mixture = 50.0 mL (1 mL = 1 cm³) volume_benzene = 38.41 cm³ volume_solid = 50.0 cm³ - 38.41 cm³ = 11.59 cm³ So, the volume of the solid is 11.59 cm³.

Calculate the density of the solid

Now that we have the mass and volume of the solid, we can calculate its density using the density formula: density_solid = mass_solid / volume_solid Given: mass_solid = 25.00 g volume_solid = 11.59 cm³ density_solid = 25.00 g / 11.59 cm³ = 2.16 g/cm³ So, the density of the solid is 2.16 g/cm³.

Key concepts

Mass and Volume Relationship

Mass and volume are two of the most fundamental properties defining any substance. The mass refers to the amount of matter in an object, while volume measures the space that the object occupies.
In our exercise, we are given a solid with a mass of 25.00 grams. It is mixed with benzene, filling the graduated cylinder to a total volume of 50.0 mL (equivalent to 50.0 cm³).
By understanding the mass and volume relationship, we can dive deeper into concepts like density. The relationship allows us to explore how space (volume) and the amount of substance (mass) connect with one another. Hence, mass and volume are critical for calculating density using the density formula.

• Mass is measured in grams (g).
• Volume can be measured in milliliters (mL) or cubic centimeters (cm³).
• One mL is equivalent to one cm³, which simplifies our calculations.

In every measurement and calculation, precision is key because it directly affects the results, especially when calculating properties like density.

Density Formula

The density formula is a fundamental equation that relates mass and volume. It provides valuable information about the compactness of a substance. The formula is written as:\[ \text{density} = \frac{\text{mass}}{\text{volume}} \]Using this formula, you can determine the density of virtually any material, given that you know the mass and volume.
For this specific exercise, we calculated the density of the unknown solid. With a known mass of 25.00 grams and an identified volume of 11.59 cm³ (found by subtracting the volume of benzene, as detailed later), we used this formula to arrive at the solid's density.
Knowing how to rearrange and apply this formula is crucial. By dividing mass by volume, we get the density. It's a simple yet powerful equation utilized in various fields to understand and compare material properties. When working with any density problems, always keep the units consistent to ensure accurate results.

Subtraction of Volumes

Subtraction of volumes is a technique used to find the actual volume of one component within a mixture. This comes in handy when dealing with a system containing multiple substances.
In the exercise, we determined the total volume of the solid by subtracting the volume of the benzene from the total volume in the graduated cylinder:\[ \text{volume}_{\text{solid}} = \text{volume}_{\text{mixture}} - \text{volume}_{\text{benzene}} \]The total mixture was 50.0 cm³, and after calculating the volume of the benzene as 38.41 cm³, we easily got the volume of the solid alone by this subtraction. With these calculations, we found the volume of the solid to be 11.59 cm³.
This process exemplifies how crucial accurate subtraction is when determining the characteristics of individual substances from a mixture. It's a straightforward step that unveils essential information about composition and density, allowing for targeted analysis of each element within a compound.