Problem 30 - Chemistry: The Central Science

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 \(21.95 \mathrm{~g} .\) 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 \(\left(\frac{4}{3}\right) \pi r^{3}\) where \(r\) is the radius.)

Step-by-Step Solution

Answer

(a) The density of the unknown liquid is calculated by dividing its mass by its volume, i.e., \(Density = \frac{21.95 g}{25.0 mL} = 0.878 g/mL\). The calculated density is in agreement with the tabulated value for benzene (\(0.8787 g/mL\)). (b) The volume of cyclohexane needed is calculated using the formula \(Volume = \frac{mass}{density}\), i.e., \(Volume = \frac{15.0 g}{0.7781 g/mL} = 19.3 mL\). (c) The radius of the lead sphere is \(r = \frac{5.0 cm}{2} = 2.5 cm\). The volume of the sphere is \(V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (2.5 cm)^3 = 65.45 cm^3\). The mass of the lead sphere is given by \(Mass = Density \times Volume = 11.34 g/cm^3 \times 65.45 cm^3 = 742.3 g\).

Step 1: Part (a): Calculate the given liquid's density and compare it with benzene's density

To determine if the given density is in accordance with benzene's tabulated value, we'll perform the following steps: 1. Find the density of the given liquid by dividing its mass by its volume: Density = mass / volume 2. Compare the calculated density with the tabulated value of benzene's density to see if they are the same.

Step 2: Part (b): Determine the volume of cyclohexane needed

To find the volume of cyclohexane required, given its mass and density, perform the following step: 1. Use the formula for volume by rearranging the density formula, that is: Volume = mass / density

Step 3: Part (c): Calculate the mass of the lead sphere

To find the mass of the lead sphere, given its diameter and the density of lead, perform the following steps: 1. Find the radius of the sphere by dividing the diameter by 2: radius = diameter / 2 2. Calculate the volume of the sphere using the volume formula, V = (4/3)πr^3 3. Calculate the mass of the sphere using the formula: Mass = Density * Volume

Key Concepts

Density Determination

Understanding how to determine the density of a substance is an essential skill in chemistry, as it allows you to identify substances and predict their behavior. Density is defined as the mass of a substance per unit volume, which can be explored through the formula \( \text{Density} = \frac{\text{mass}}{\text{volume}} \).

To determine whether a liquid is benzene, chemists measure its density and compare it to a known value. In the exercise, the density of a liquid thought to be benzene was calculated by taking the mass of a \( 25.0 \, \text{mL} \) sample (\( 21.95 \, \text{g} \) and dividing it by its volume (\( 25.0 \, \text{mL} \) to find its density. If this calculated density matches the handbook listing for benzene at \(0.8787 \, \text{g/mL}\), we can confirm its identity.

When performing density determination, it's crucial to ensure that the units are correct and that temperature is considered, as density can change with temperature. Precise measurements and calculations will enable a chemist to verify the identity of substances accurately. This process is quintessential in fields such as quality control and materials science.

Volume-to-Mass Relationship

The volume-to-mass relationship is directly derived from the concept of density. This relationship is especially useful when you need to identify how much of a substance is required or produced in a reaction. Given a substance's density and the desired mass, you can calculate the required volume using the rearranged density formula \( \text{Volume} = \frac{\text{mass}}{\text{density}} \).

For instance, if you need \( 15.0 \, \text{g} \) of cyclohexane and know that its density is \(0.7781 \, \text{g/mL}\), you can calculate the necessary volume. By dividing the mass of cyclohexane by its density, the result reveals the amount of cyclohexane in milliliters needed for an experiment. The clear understanding of this relationship allows laboratory professionals and students to measure the correct amounts of chemicals for their work, ensuring successful and safe experiments.

Spherical Volume Calculations

Calculating the volume of a sphere is key in several fields of science and engineering, particularly when it concerns the mass of spherical objects based on their density. The volume \( V \) of a sphere can be calculated using the formula \( V = (\frac{4}{3})\pi r^3 \), where \( r \) is the radius of the sphere.

Let's apply this to a practical scenario: if a lead sphere has a diameter of \(5.0 \, \text{cm}\), we first determine the radius of the sphere by halving the diameter, resulting in a radius of \(2.5 \, \text{cm}\). Using the spherical volume formula, we can now find the volume of the sphere. Once we have the volume and the density of lead (\(11.34 \, \text{g/cm}^3\)), we can calculate the mass by multiplying the sphere's volume by its density (\( \text{Mass} = \text{Density} \times \text{Volume} \)).

This process not only aids in scientific inquiry and material engineering but also has practical applications in everyday life, such as calculating the weight of sports equipment or the amount of material needed to craft spherical objects.