Question

A piece of silver (Ag) metal weighing \(194.3 \mathrm{~g}\) is placed in a graduated cylinder containing \(242.0 \mathrm{~mL}\) of water. The volume of water now reads \(260.5 \mathrm{~mL}\). From these data calculate the density of silver.

Short answer

The density of silver is 10.505 g/mL.

Step-by-step solution

Identify Volume Displacement

The volume of the graduated cylinder initially is 242.0 mL. After placing the silver metal, the volume reads 260.5 mL. The volume displacement is the difference between the final and initial volumes, and it represents the volume of the silver metal. \[\text{Volume of silver} = 260.5 \ \text{mL} - 242.0 \ \text{mL} = 18.5 \ \text{mL}\]

Recall Density Formula

Density is mass per unit volume. The formula for density \( \rho \) is:\[\rho = \frac{\text{mass}}{\text{volume}}\]In this exercise, the mass of silver is given as 194.3 g, and the volume we calculated in Step 1 is 18.5 mL.

Calculate Density of Silver

Using the density formula, substitute the mass of silver (194.3 g) and the volume of silver (18.5 mL):\[\rho = \frac{194.3 \ \text{g}}{18.5 \ \text{mL}} = 10.505 \ \text{g/mL}\]Therefore, the density of silver is 10.505 g/mL.

Key concepts

Volume Displacement

Volume displacement occurs when an object is submerged in a fluid, causing the fluid to rise. It's the difference between the final and initial fluid levels. In this exercise, a piece of silver is placed in water, raising the water level. To find the volume of the silver, you subtract the initial volume of the water from the final volume. If the initial water level is 242.0 mL and the final reading is 260.5 mL, the silver displaces 18.5 mL of water.

• Initial Volume of Water: 242.0 mL
• Final Volume of Water: 260.5 mL
• Volume of Silver: Final Volume - Initial Volume = 18.5 mL

This measurement gives us a direct way to identify the volume of the irregularly shaped silver metal.

Mass of Silver

The mass of an object is a measure of how much matter it contains. Here, the mass of the silver is provided directly, which simplifies the calculation. The silver in this instance weighs 194.3 grams. Knowing this mass is crucial as it is a key component in calculating its density. Whenever you engage in density calculations, make sure to use consistent units. In this case, grams (g) for mass, as it pairs seamlessly with milliliters (mL) for volume in density equations. Always ensure that your measuring tools are calibrated and provide accurate measures to avoid errors in your final results.

Density Formula

Density is a fundamental property that relates the mass of an object to its volume. It is expressed as mass per unit volume. The formula for calculating density is: \[\rho = \frac{\text{mass}}{\text{volume}}\]To find the density of silver, substitute the known values: the mass of the silver (194.3 grams) and the volume of the silver (18.5 mL). Performing the calculation: \[\rho = \frac{194.3 \text{ g}}{18.5 \text{ mL}} = 10.505 \text{ g/mL}\]Density not only identifies the compactness of a substance but also assists in identifying materials, comparing them, and conducting quality control processes.

Graduated Cylinder Measurement

A graduated cylinder is a common laboratory tool used to measure the volume of liquids. It offers a clear view of how much fluid is present and how much has been displaced when an object is submerged. When using a graduated cylinder, it's important to place it on a flat surface and view the liquid's meniscus at eye level to ensure an accurate measurement. The meniscus is the curve seen at the top of a liquid in response to its container; always take the measurement from the lowest point of the curve. By accurately reading the initial and final volumes, we can effectively calculate the volume of an immersed object, such as the piece of silver in this example.