Problem 8 - Chemical Principles: The Quest for Insight

Question

When a piece of metal of mass \(5.21 \mathrm{~g}\) is dropped into a graduated cylinder containing, \(16.7 \mathrm{ml}\). of watcr, the water level rises to \(18.2 \mathrm{mt}\). What is the density of the metal (in grams per cubic centinucter)?

Step-by-Step Solution

Answer

The density of the metal is \frac{5.21 g}{18.2 mL - 16.7 mL} = \frac{5.21 g}{1.5 mL} = 3.47 g/mL or 3.47 g/cm³.

Step 1: Calculate the Volume of the Metal

To find the volume of the metal, subtract the initial water level from the final water level in the graduated cylinder. Volume of metal = Final water level - Initial water level = 18.2 mL - 16.7 mL.

Step 2: Calculate the Density of the Metal

Density is mass divided by volume. Use the mass of the metal and the volume you calculated in the previous step to calculate the metal's density. Density = Mass / Volume.

Key Concepts

Mass-to-Volume Ratio

Understanding the mass-to-volume ratio is essential in determining the density of an object. This ratio simply tells us how much mass is contained in a unit volume of a material. In the context of a metal or any other solid substance, by knowing its mass and the volume it occupies, we can find out how dense it is.

For a piece of metal, the mass is typically given in grams while volume - the space the metal occupies - is measured in cubic centimeters or milliliters. To calculate the mass-to-volume ratio, which is the density, you divide the mass of the metal by its volume. This fundamental concept is used in various fields including material science, chemistry, and engineering.

Density of Metal

The density of a metal is a reflection of its compactness; it’s crucial for identifying materials and predicting their behavior in different applications. Density is often measured in grams per cubic centimeter (g/cm³) for solids. Metals, being usually dense substances, have high mass-to-volume ratios compared to materials like wood or plastic.

In the exercise, the density of the metal is calculated by dividing its mass by its displaced volume of water in the graduated cylinder. A metal with a higher density will displace less water for the same mass, resulting in a smaller change of the water level.

Chemical Principles

Chemical principles, like the conservation of mass and the behavior of substances in different conditions, underpin the calculations involving density. The buoyant force exerted by fluids, like water, is applied equally across all materials - which allows for the method of displacement used in the given exercise to determine volume.

Furthermore, understanding the intermolecular forces and atomic structure of metals helps explain why they have higher densities compared to substances with less tightly packed atoms. These chemical properties and behaviors provide a rich context for calculations involving density and guide practical applications and predictions in real-world scenarios.

Graduated Cylinder Measurement

A graduated cylinder is a common laboratory equipment used to measure the volume of liquids with good accuracy. When a solid object is submerged in a liquid within a graduated cylinder, we can measure the displacement of the liquid to find the volume of the solid.

This method is based on the Archimedes' principle which states that the volume of the fluid displaced is equal to the volume of the solid that caused the displacement. It is critical the graduated cylinder is read at eye level and the bottom of the meniscus, which is the curve seen at the top of the liquid layer, is aligned with the measurement marking for accurate reading.