Question

The density of pure silver is 10.5 \(\mathrm{g} / \mathrm{cm}^{3}\) at \(20^{\circ} \mathrm{C} .\) If 5.25 \(\mathrm{g}\) of pure silver pellets is added to a graduated cylinder containing 11.2 \(\mathrm{mL}\) of water, to what volume level will the water in the cylinder rise?

Short answer

The new volume level of the water in the cylinder after adding the pure silver pellets is 11.7 cm³ or 11.7 mL.

Step-by-step solution

Recall the formula for density

Recall that the formula for density is: Density = \(\frac{mass}{volume}\) We rearrange the formula to solve for volume: Volume = \(\frac{mass}{density}\)

Calculate the volume of silver pellets

Now, we will calculate the volume of the silver pellets added to the graduated cylinder. We are given the mass (5.25g) and the density (10.5 g/cm³) of the silver: Volume (silver) = \(\frac{mass}{density} = \frac{5.25\,g}{10.5\,g/cm^3}\) Volume (silver) = \(0.5\,cm^3\) So, the volume of silver pellets is \(0.5\,cm^3\).

Convert the initial volume of water from mL to cm³

We need to convert the initial volume of water in the graduated cylinder from mL to cm³. Note that 1 mL is equal to 1 cm³: Volume (water) = 11.2 mL = 11.2 cm³

Calculate the new volume level

To find the new volume level after adding the silver to the graduated cylinder, we simply add the initial volume of water and the volume of silver pellets: New volume level = Volume (water) + Volume (silver) = 11.2 cm³ + 0.5 cm³ New volume level = 11.7 cm³ The water in the cylinder will rise to a volume level of 11.7 cm³ after adding the pure silver pellets. Please note that we can change it to mL if needed: New volume level = 11.7 mL

Key concepts

Volume Displacement: A Key Concept

Volume displacement is a simple yet vital concept in understanding how objects interact with liquids. Imagine you drop an object into a container of water; the level of water rises. This rise is due to the volume of the object displacing an equivalent volume of liquid. In the exercise, we see silver pellets displacing water in a graduated cylinder. The displaced volume directly corresponds to the volume of the silver pellets. This concept is extremely practical and forms the basis for Archimedes' principle, which states that the volume of fluid displaced by an object is equal to the volume of the object immersed.

**Importance of Volume Displacement**
- Helps determine the volume of irregularly shaped objects by measuring the change in water level.
- Facilitates density calculations, which are crucial in identifying substances.
- Used across various fields from chemistry to engineering for precise measurements.

So, understanding volume displacement not only assists with simple tasks like determining the new water level but is also foundational in scientific inquiries and industrial applications.

Understanding Specific Gravity

Specific gravity is a dimensionless quantity that indicates how dense a substance is relative to water. The specific gravity of a substance is calculated by dividing its density by the density of water (usually at 4°C where water's density is approximately 1 g/cm³). It's represented as:\[ \text{Specific Gravity} = \frac{\text{Density of Substance}}{\text{Density of Water}} \]
In the context of the exercise, the density of silver is 10.5 g/cm³. This means that silver is much denser than water, having a specific gravity of 10.5.

**Why Specific Gravity Matters**
- It helps in comparing the densities of different materials without the need for units, making it more straightforward.
- Used in various industries to determine the purity and quality of a substance, such as in metallurgy and brewing.
- Provides insights on whether a substance will float or sink when placed in water. For example, a specific gravity above 1 indicates it will sink.

Understanding specific gravity enables one to assess material properties feasibly and comprehensively, paving the way for numerous scientific techniques and everyday applications.

Graduated Cylinder Measurement and Its Precision

The graduated cylinder is a standard laboratory instrument used to measure the volume of liquids with high precision. In the given problem, it's crucial for accurately measuring the initial 11.2 mL of water and any subsequent changes due to volume displacement from the silver. These cylinders have graduated markings that allow for precise volume measurements, typically in milliliters or cubic centimeters (cm³).

**Using a Graduated Cylinder Efficiently **
- Ensure the cylinder is on a flat surface, and your eye is level with the meniscus (the curved surface of the liquid) to avoid parallax error.
- Read the bottom of the meniscus for an accurate measurement.
- Take note of the scale; different cylinders have different levels of precision, often indicated by the smallest marking.
In our exercise, understanding how to use the cylinder allows for precise volume measurements, assisting in calculating the final water level. Despite its simplicity, the graduated cylinder is invaluable in both educational settings and professional laboratories for acquiring accurate volume-based data.