Problem 45 - Chemistry and Chemical Reactivity

Question

You have a 100.0 -mL graduated cylinder containing \(50.0 \mathrm{mL}\) of water. You drop a \(154-\mathrm{g}\) piece of brass \(\left(d=8.56 \mathrm{g} / \mathrm{cm}^{3}\right)\) into the water. How high does the water rise in the graduated cylinder? (a) A graduated cylinder with 50.0 ml of water. (b) A piece of brass is added to the cylinder.

Step-by-Step Solution

Answer

The water rises to 67.991 mL after the brass is added.

Step 1: Calculate the Volume of Brass

To find the volume of the brass piece, we need to use the formula for density, which is given as \( d = \frac{m}{V} \). Here, \( d \) is the density, \( m \) is the mass, and \( V \) is the volume. We can rearrange this to find \( V = \frac{m}{d} \). Given \( m = 154 \text{ g} \) and \( d = 8.56 \text{ g/cm}^3 \), the volume \( V \) can be calculated as follows: \[ V = \frac{154 \text{ g}}{8.56 \text{ g/cm}^3} = 17.991 \text{ cm}^3 \].

Step 2: Convert Brass Volume to Milliliters

Since 1 cm³ is equivalent to 1 mL, the volume of the brass in mL is the same as its volume in cm³. Therefore, the volume of the brass is 17.991 mL.

Step 3: Calculate Total Water Level in the Graduated Cylinder

The initial water level in the graduated cylinder is 50.0 mL. The brass piece displaces an equivalent volume of water equal to its own volume, which is 17.991 mL. Therefore, the new water level in the graduated cylinder will be the initial water level plus the volume of the brass: \[ 50.0 \text{ mL} + 17.991 \text{ mL} = 67.991 \text{ mL} \].

Key Concepts

Volume Displacement

Understanding volume displacement is crucial for solving problems related to changes in liquid levels when objects are immersed. It's based on a simple principle:

• An object submerged in a fluid pushes aside or "displaces" a volume of fluid equal to its own volume.
• This happens regardless of whether the object sinks or floats.

When you drop an object into a liquid, like in our example where a brass piece is added to water in a cylinder, the water level rises. This rise is exactly equivalent to the volume of the displaced water.
In this exercise, after dropping a 154 g brass piece into a graduated cylinder filled with 50.0 mL of water, the volume of water displaced (and thus the rise) is equal to the volume of the brass itself, calculated to be 17.991 mL. Therefore, the water level increase reflects the additional space the object occupies underwater.

Graduated Cylinder

A graduated cylinder is a common laboratory tool used for measuring liquid volumes. But how is it applied practically?

• It is usually made from glass or plastic and features a series of markings along its side to indicate volume, typically in milliliters (mL).
• These markings make it easy to measure volumes with a reasonably high level of accuracy.

In our problem, a graduated cylinder is used to measure the initial volume of water (50.0 mL) and then measure the new volume after the brass piece is added.
Graduated cylinders are preferred in experiments where precise measurement is crucial, as they offer a far more accurate reading than beakers or flasks. This precision is essential for calculating the volume displacement accurately, based on how much the water level increases.
Remember, when reading a graduated cylinder, you should view it at eye level. The correct reading is often taken from the bottom of the meniscus, which is the dip you see at the top of the liquid.

Density Formula

The density formula provides a vital link between mass and volume. This formula is \[ d = \frac{m}{V} \] where:

• \( d \) is density, usually measured in grams per cubic centimeter \((\text{g/cm}^3)\).
• \( m \) is mass, typically measured in grams \((\text{g})\).
• \( V \) is volume, measured in cubic centimeters \((\text{cm}^3)\) or milliliters \((\text{mL})\).

By rearranging this formula to solve for volume, \[ V = \frac{m}{d} \], we can determine how much space an object occupies. This is what we did for the brass piece in the exercise, using its given mass and density.
The ability to manipulate the density formula allows one to compute the volume from a known mass and density, which is instrumental in finding out things like the water displacement caused by an object in our graduated cylinder scenario.
Understanding how density relates mass to how much volume an object takes up is foundational for many science and engineering applications, not just in classroom examples but also in real-world situations such as material selection in manufacturing.