Problem 97 - Chemistry

Question

Lignum vitae is a hard, durable, and extremely dense wood used to make ship bearings. A sphere of this wood with a diameter of \(7.60 \mathrm{~cm}\) has a mass of 313 g. (a) What is the density of the lignum vitae sphere? (b) Will the sphere float or sink in water? (c) Will the sphere float or sink in chloroform? (The density of chloroform is \(1.48 \mathrm{~g} / \mathrm{mL} .\) )

Step-by-Step Solution

Answer

(a) Density = 1.36 g/cm³ (b) Sink in water (c) Float in chloroform

Step 1: Calculate the Volume of the Sphere

To find the density of the sphere, we first need to calculate its volume. The formula for the volume of a sphere is \( V = \frac{4}{3}\pi r^3 \). The diameter of the sphere is given as \( 7.60 \text{ cm} \), so the radius \( r \) is \( \frac{7.60}{2} \text{ cm} = 3.80 \text{ cm} \). Therefore, the volume is:\[V = \frac{4}{3} \pi (3.80)^3 \approx 230.95 \text{ cm}^3.\]

Step 2: Determine the Density of the Sphere

Density \( \rho \) is calculated with the formula \( \rho = \frac{m}{V} \). We know the mass \( m \) is \( 313 \text{ g} \) and we calculated the volume \( V \) as \( 230.95 \text{ cm}^3 \). Substituting these values in, we have: \[\rho = \frac{313 \text{ g}}{230.95 \text{ cm}^3} \approx 1.36 \text{ g/cm}^3.\]

Step 3: Determine if the Sphere Will Float or Sink in Water

The density of water is \( 1.00 \text{ g/cm}^3 \). A sphere will float in water if its density is less than that of water. Since the density of the lignum vitae sphere is \( 1.36 \text{ g/cm}^3 \), which is greater than \( 1.00 \text{ g/cm}^3 \), the sphere will sink in water.

Step 4: Determine if the Sphere Will Float or Sink in Chloroform

The density of chloroform is given as \( 1.48 \text{ g/cm}^3 \). A sphere will float in a liquid if its density is less than that of the liquid. Since the density of the lignum vitae sphere is \( 1.36 \text{ g/cm}^3 \), which is less than \( 1.48 \text{ g/cm}^3 \), the sphere will float in chloroform.

Key Concepts

Sphere Volume

To understand how to calculate the density of a spherical object, we first need to calculate its volume. A sphere's volume is determined by the formula \( V = \frac{4}{3}\pi r^3 \). Here, \( r \) is the radius of the sphere. Calculating the radius is easy—it's simply half the diameter of the sphere. For our lignum vitae wood sphere with a diameter of 7.60 cm, the radius will be:\[ r = \frac{7.60}{2} = 3.80 \text{ cm} \]Substituting this radius into the sphere volume formula gives us the volume as:\[ V = \frac{4}{3} \pi (3.80)^3 \approx 230.95 \text{ cm}^3 \]This volume will then be used along with the sphere's mass to determine its density. Understanding these volume calculations allows us to further explore the relationship between mass and density.

Mass and Density Relationship

Density is a property that relates the mass of an object to the volume it occupies. It's represented by \( \rho \) and calculated using the formula \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume. The mass tells us how much matter is in the object, while the volume informs us about the space it occupies.

• If an object has a large mass for a small volume, it is more dense.
• If it has a small mass for a large volume, it is less dense.

In the exercise, with a lignum vitae sphere mass of 313 grams and a volume of approximately 230.95 cm³, the density calculation turns out to be:\[ \rho = \frac{313 \text{ g}}{230.95 \text{ cm}^3} \approx 1.36 \text{ g/cm}^3 \]Knowing the density is essential for understanding whether the object will float or sink in a particular liquid, which we'll explore next.

Floating and Sinking in Liquids

Whether an object floats or sinks in a liquid depends on its density compared to the liquid's density. An object will float if its density is less than the liquid it is placed in. Conversely, it will sink if its density is greater.

• Water has a standard density of 1.00 g/cm³. Objects with density less than this will float. Those with greater density will sink.
• Chloroform has a density of 1.48 g/cm³, a bit higher than that of water. Similarly, compare the object’s density to decide if it will float or sink.

In this case, the lignum vitae sphere has a density of 1.36 g/cm³. Since this is greater than the density of water, it will sink. However, because its density is less than that of chloroform, the sphere will float. These observations are crucial in real-world applications, such as making ships or designing materials that need specific buoyancy characteristics.