Question

A solid with an irregular shape and a mass of \(11.33 \mathrm{~g}\) is added to a graduated cylinder filled with water \((d=\) \(1.00 \mathrm{~g} / \mathrm{mL}\) ) to the 35.0 -mL mark. After the solid sinks to the bottom, the water level is read to be at the 42.3 -mL mark. What is the density of the solid?

Short answer

Answer: The density of the irregularly shaped solid is 1.55 g/mL.

Step-by-step solution

Identify the given information

In this problem, we are given: - mass of the solid (m) = 11.33 g - initial water level in the cylinder (V1) = 35.0 mL - final water level in the cylinder (V2) = 42.3 mL - water's density (d_water) = 1.00 g/mL

Calculate the volume of the solid

The volume of the solid (V_solid) can be found by calculating the difference in water levels in the cylinder before and after adding the solid: V_solid = V2 - V1 V_solid = 42.3 mL - 35.0 mL V_solid = 7.3 mL

Calculate the density of the solid

Now that we have the mass and volume of the solid, we can calculate its density (d_solid) using the formula: d_solid = m / V_solid d_solid = 11.33 g / 7.3 mL d_solid = 1.55 g/mL The density of the irregularly shaped solid is 1.55 g/mL.

Key concepts

Mass and Volume

Understanding the concepts of mass and volume is crucial when studying the properties of matter. Mass represents the amount of matter in an object and is measured in grams (g), kilograms (kg), or other units of weight. Volume, on the other hand, is a measure of the space that an object occupies and is commonly expressed in milliliters (mL), liters (L), or cubic centimeters (cm³).

In the context of this exercise, the mass of the irregular solid is given as 11.33g, which is a direct and clear quantitative measure. The volume, however, cannot be measured as straightforwardly due to the irregular shape of the object. The displacement method, using a graduated cylinder, is a practical way to determine the volume of such an object, as it displaces a volume of water equal to its own.

Graduated Cylinder

A graduated cylinder is a common laboratory equipment used to measure the volume of liquids accurately. It is marked with a series of lines to indicate volume levels. Using a graduated cylinder, one can conduct the displacement method to find the volume of an irregularly shaped solid, as described in our exercise problem. You fill the cylinder with a known amount of water, note the volume, then add the object into it. The rise in water level corresponds to the volume of the object.

For the highest accuracy when reading a graduated cylinder, you should ensure that the measurement is taken at eye level and read from the bottom of the meniscus, which is the curve seen at the top of a liquid column.

Density of Solids

Density is a fundamental property that defines the compactness of a material. For solids, it is the ratio of mass to volume and indicates how tightly matter is packed together. The higher the density, the less space there is between molecules. Knowing the density of a solid can help us predict its buoyancy, stability, and how it interacts with other substances.

To measure the density of a solid with an irregular shape, as in our exercise, we must first determine its mass and volume. Since we cannot measure the volume directly due to the shape, we use water displacement in a graduated cylinder to find it indirectly.

Density Formula

The density formula is a simple but powerful tool in science. It is given by the equation \( \text{density (d)} = \frac{\text{mass (m)}}{\text{volume (V)}} \). The units most commonly used are grams per milliliter (g/mL) or grams per cubic centimeter (g/cm³), as 1 mL is equivalent to 1 cm³.

In applying this formula to our exercise, we divided the mass of the solid (11.33 g) by the volume of water displaced, which is the change in water level measured with the graduated cylinder (7.3 mL). The result is the density of the solid, which is 1.55 g/mL, indicating how dense the material is relative to water.