Problem 38 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) After the label fell off a bottle containing a clear liquid believed to be benzene, a chemist measured the density of the liquid to verify its identity. A \(25.0-\mathrm{mL}\) portion of the liquid had a mass of 21.95 g. A chemistry handbook lists the density of benzene at \(15^{\circ} \mathrm{C}\) as \(0.8787 \mathrm{~g} / \mathrm{mL}\). Is the calculated density in agreement with the tabulated value? (b) An experiment requires \(15.0 \mathrm{~g}\) of cyclohexane, whose density at \(25^{\circ} \mathrm{C}\) is \(0.7781 \mathrm{~g} / \mathrm{mL}\). What volume of cyclohexane should be used? (c) A spherical ball of lead has a diameter of \(5.0 \mathrm{~cm}\). What is the mass of the sphere if lead has a density of \(11.34 \mathrm{~g} / \mathrm{cm}^{3} ?\) (The volume of a sphere is \((4 / 3) \pi r^{3},\) where \(r\) is the radius.)

Step-by-Step Solution

Answer

(a) The calculated density of the liquid is \(0.878 \mathrm{~g/mL}\), and the tabulated value is \(0.8787 \mathrm{~g/mL}\). Since they are very close, the liquid is likely to be benzene. (b) The volume of cyclohexane required is \(19.3\mathrm{~mL}\). (c) The mass of the spherical lead ball is approximately \(741.48 \mathrm{~g}\).

Step 1: Calculate the Density of the Liquid

To calculate the density, use the formula: Density = \(\frac{Mass}{Volume}\) Given: \(Mass = 21.95 g\) and \(Volume = 25.0 mL\) \(Density = \frac{21.95}{25.0}\) = \(0.878 \mathrm{\ g/mL}\)

Step 2: Comparing Calculated and Tabulated Density Values

The calculated density is \(0.878 \mathrm{\ g/mL}\), and the tabulated value is \(0.8787 \mathrm{\ g/mL}\). The calculated density is very close to the tabulated value, indicating that the liquid is likely to be benzene. #(b) Calculate the Volume of Cyclohexane#

Step 3: Using the Density Formula to Find Volume

Using the density formula: \(Density = \frac{Mass}{Volume}\) We can solve for volume: \(Volume = \frac{Mass}{Density}\) Given: \(Mass = 15.0 g\) and \(Density = 0.7781 \mathrm{\ g/mL}\) \(Volume = \frac{15.0}{0.7781}\) = \(19.3\mathrm{\ mL}\) The volume of cyclohexane required is \(19.3\mathrm{\ mL}\). #(c) Calculate the Mass of the Sphere#

Step 4: Calculate the Volume of the Sphere

To find the volume of a sphere, we use the formula: \(Volume =\frac{4}{3} πr^{3}\) Given diameter = \(5.0 cm\) Radius (r) = \(\frac{Diameter}{2}\) = \(\frac{5.0}{2}\) = \(2.5 cm\) \(Volume = \frac{4}{3} π(2.5)^{3}\) ≈ \(65.45 \mathrm{\ cm^3}\)

Step 5: Calculate the Mass using Density

Now, let's use the density formula to find the mass: \(Density = \frac{Mass}{Volume}\) Given: \(Volume = 65.45 \mathrm{\ cm^3}\) and \(Density = 11.34 \mathrm{\ g/cm\xa03}\) Mass = \(Density × Volume\) Mass = \(11.34 × 65.45\) ≈ \(741.48 \mathrm{\ g}\) The mass of the spherical lead ball is approximately \(741.48 \mathrm{\ g}\).

Key Concepts

Mass and Volume Relationships

Understanding the relationship between mass and volume is crucial when it comes to determining the density of a substance. Density is defined as mass per unit volume, and it helps us identify substances by comparing their measured density to known values.

• Mass is the amount of matter in an object, measured in grams (g) or kilograms (kg).
• Volume is the space that a substance or object occupies, often measured in milliliters (mL) for liquids or cubic centimeters (cm³) for solids.
• Density, symbolized by the Greek letter rho (\(\rho\)), is calculated using the formula: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\).

In the exercise, a liquid's mass and volume were measured to calculate its density. By dividing mass by volume, we could compare it to known density values to verify the liquid's identity.

Density of Liquids

The density of liquids can vary with temperature and is often used to identify or verify samples. Knowing the density can also assist in finding out how much of a liquid is needed for a reaction, as explored in the exercise with cyclohexane.

• Density of liquids is typically measured in grams per milliliter (g/mL).
• To find a liquid's volume when only the mass and density are known, rearrange the density formula: \(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\).
• An example from the exercise showed how to calculate the volume of cyclohexane required, based on a specified mass and its density.

Such calculations are critical in industries where precise amounts of chemicals are necessary for reactions or formulations.

Spherical Volume Calculations

Calculating the volume of a sphere is a common task in geometry and can also apply in practical scenarios involving spherical objects. The formula utilized is \[\text{Volume of Sphere} = \frac{4}{3} \pi r^3\], where \(r\) is the radius.

• The radius is half of the diameter, making it essential to convert the diameter to radius when given.
• In the exercise, the volume of a lead sphere was calculated this way, using its diameter to find the radius and then applying the sphere volume formula.

Once the volume is determined, it can be multiplied by the density (\(\text{Mass} = \text{Density} \times \text{Volume}\)) to find the mass, highlighting the integration of geometry with mass-density relationships for practical problem-solving.