Problem 174 - General Chemistry

Question

An experimenter places a piece of a solid metal weighing \(255 \mathrm{~g}\) into a graduated cylinder, which she then fills with liquid mercury. After weighing the cylinder and its contents, she removes the solid metal and fills the cylinder with mercury. She now finds that the cylinder and its contents weigh \(101 \mathrm{~g}\) less than before. The density of mercury is \(13.6 \mathrm{~g} / \mathrm{cm}^{3}\). What is the density of the solid metal?

Step-by-Step Solution

Answer

The density of the solid metal is approximately 34.31 g/cm³.

Step 1: Understand the problem

The experimenter conducts an experiment where a piece of solid metal is submerged in mercury and then weighed. We later re-weigh the cylinder with only mercury. We need to find the density of the metal using the weight difference.

Step 2: Calculate the submerged mercury volume

The weight difference between the two measurements is given as 101 g. This difference represents the weight of the volume of mercury that the metal displaced when submerged.

Step 3: Use mercury's density to find displaced volume

Using the weight difference and mercury's density, find the displacement volume: \ \( \text{Volume of metal} = \frac{101}{13.6} \; \text{cm}^3 \approx 7.43 \; \text{cm}^3\).

Step 4: Calculate the density of the solid metal

Density is calculated as mass divided by volume. The mass of the metal is 255 g and the volume displaced is approximately 7.43 cm³. \ \( \text{Density of metal} = \frac{255}{7.43} \; \text{g/cm}^3 \approx 34.31 \; \text{g/cm}^3\).

Key Concepts

Density of Solids

Density is a fundamental concept in physics that tells us how much mass is packed into a given volume of space. For solids, this can be visualized as the amount of matter contained within a certain size object. In the experiment, the density of the solid metal was determined through a series of weighing processes. Knowing the metal's mass, which was given as 255 g, is just part of the process.

To find the density, we need to determine its volume in addition to its mass. Density is mathematically defined as:

• Density = Mass / Volume

For the solid metal, its density was calculated by first determining the volume of mercury it displaced when submerged—all of this combined allows calculations to be made without directly measuring the metal's volume. Remember, typically denser objects will sink in a liquid, reinforcing why finding the right material density is crucial in many fields like material science and engineering.

Archimedes' Principle

Archimedes' principle is a scientific law that explains why objects float or sink when placed in a fluid. It states that an object submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object. In simpler terms, it helps us understand how volume displacement and buoyancy work.

In the given experiment, when the solid metal is submerged into the mercury, it displaces a certain volume of mercury. The weight difference (101 g in this case) between when the cylinder is filled with only mercury and when it contains the metal accounts for the displacement volume. This phenomenon is due to Archimedes’ principle and is crucial in determining the volume of the solid indirectly.

By understanding this principle, one can find the volume of irregularly shaped objects through displacement, allowing for the calculation of its density, without needing to physically measure its dimensions. This makes Archimedes' principle especially useful in experiments involving irregular shapes and incomprehensively sized samples.

Mass and Volume Relationship

The relationship between mass and volume is pivotal when determining the density of an object. Mass is a measure of how much matter is in an object, whereas volume indicates how much space that matter occupies. Together, they define an object's density.

In the context of the experiment, the metal's mass was straightforwardly given as 255 g. However, its volume was not directly measured. Instead, it was calculated using the volume of mercury displaced, thanks to Archimedes' principle. This indirect measurement confirms the inherently linked nature of mass and volume when solving for density.

The equation to calculate density looks like this:

• Density = Mass / Volume

Understanding how these quantities interrelate allows for solving complex problems in physics and materials science. They help us comprehend why materials behave differently when interacting with various fluids, which has practical implications across numerous scientific and engineering disciplines.