Question

The density of lead is \(11.3 \mathrm{~g} / \mathrm{mL}\). The water level in a graduated cylinder initially at \(215 \mathrm{~mL}\) rises to \(285 \mathrm{~mL}\) after a piece of lead is submerged. What is the mass, in grams, of the lead?

Short answer

791 g

Step-by-step solution

Determine the Volume of the Lead

Find the difference between the final water level and the initial water level in the graduated cylinder. This will give the volume of lead submerged in the water.Initial water level: 215 mLFinal water level: 285 mLVolume of lead: 285 mL - 215 mL = 70 mL

Use Density Formula

Use the formula for density, which is \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\). Rearrange the formula to solve for mass: \(\text{Mass} = \text{Density} \times \text{Volume}\).

Calculate the Mass

Substitute the known values into the formula: \(\text{Density} = 11.3 \text{ g/mL}\) and \(\text{Volume} = 70 \text{ mL}\).\(\text{Mass} = 11.3 \text{ g/mL} \times 70 \text{ mL} = 791 \text{ g}\)

Key concepts

density formula

Density is a measure of how much mass is contained in a given volume. The density formula is: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]. Density is usually expressed in units like grams per milliliter (g/mL) for liquids and solids. It helps us understand how tightly matter is packed together. If you know two of the three variables (density, mass, and volume), you can always compute the third using this formula.
In our example, we used the given density of lead, which is 11.3 g/mL, to compute the mass once we determined the volume.

volume displacement

Volume displacement is a method used to measure the volume of an object by observing the amount of fluid it displaces when submerged. This technique is especially helpful for irregularly shaped objects.
Here's how it works in our exercise:

• Measure the initial volume of water in a graduated cylinder (215 mL).
• Submerge the object (a piece of lead, in this case).
• Measure the new water level (285 mL).
• Calculate the volume of the object by finding the difference between the initial and final water levels (285 mL - 215 mL = 70 mL).

This provided us with the submerged volume of the lead, which is crucial for further calculations.

mass calculation

To calculate mass when you know the density and volume, you can use the rearranged density formula: \[\text{Mass} = \text{Density} \times \text{Volume}\]. Substituting the known values from our problem:

• Density of lead = 11.3 g/mL
• Volume of lead = 70 mL

We multiply these to get the mass:
\[\text{Mass} = 11.3 \text{ g/mL} \times 70 \text{ mL} = 791 \text{ g} \]. Thus, the lead has a mass of 791 grams. Understanding how to manipulate these basic formulas allows you to solve a wide range of science problems!