Q. 2.93 - An Introduction to General, Organic, and Biological Chemistry

Question

The gray cube has a density of $4.5 g/c{m}^3$ . Is the density of the green cube the same, lower than, or higher than that of the gray cube? (2.7)

Step-by-Step Solution

Answer

The new level of both the cylinders are determined as $48.0 mL$

Step 1 : Introduction

The given is two cubes, white and green

The objective is to find if the density is same, lower or higher of the green cube with the density of the grey cube

Step 2 : Explanation

Density:

The ratio of a substance's mass and volume determines its density. A substance's density can be defined as its mass per unit volume.

$\text{ Density }\left(\mathrm{g}/{\mathrm{cm}}^3\right)=\frac{\text{ Mass }\left(\mathrm{g}\right)}{\text{ Volume }\left({\mathrm{cm}}^3\right)}\text{ or }\frac{Mass\left(\mathrm{g}\right)}{\text{ Volume }\left(\mathrm{mL}\right)}$

To begin, determine the volume of a cube of lead and aluminium as follows:

Lead:

$2.0 \mathrm{cm}\times 2.0 \mathrm{cm}\times 2.0 \mathrm{cm}=8.0{ \mathrm{cm}}^3$

Aluminum:

$2.0 \mathrm{cm}\times 2.0 \mathrm{cm}\times 2.0 \mathrm{cm}=8.0{ \mathrm{cm}}^3$

Step 3 : Change volume unit

Change the volume unit from gallons to liters ${\mathrm{cm}}^3$ to $\mathrm{mL}$ as follows:

$$\begin{gathered} 1.0 c{m}^3=1.00 mL \\ 8.0 c{m}^3 \times \frac{1.00 mL}{1.0 c{m}^3} \\ =8.0 mL \end{gathered}$$

Because lead and aluminum have a higher density than water, they will sink in it, and the volume of water will increase by the same amount as the volume of each block.

As a result, calculate the new cylinder volume as follows:

Step 4

$New volume=water volume +volume of lead$

$$\begin{gathered} =40.0 mL+8.0 mL \\ =48.0 mL \end{gathered}$$

With aluminum cube:

$New volume = water volume + volume of aluminum$

$$\begin{gathered} 40.0 mL+8.0 m \\ =48.0 mL \end{gathered}$$

Hence the new level in both cylinders is $48.0 \mathrm{mL}$