Density, Volume, and Mass What is the volume of 100.0 g of copper? The density of copper is 8.9 g/cm^3.

Question

Density, Volume, and Mass

What is the volume of 100.0 g of copper? The density of copper is 8.9 g/cm³.

Accepted Answer

We need to carry out the following conversion:

[latex]\fbox{Mass in grams}\overset{?}{\longrightarrow } \fbox{Volume in milliliters}[/latex]

The relationship between mass and volume is given by density:

[latex]\fbox{Mass in grams}\overset{Density \frac{mass}{volume} }{\longrightarrow } \fbox{Volume in milliliters}[/latex]

First, we rearrange the density equation to get volume on one side by itself. This manipulation involves cross multiplication, which is described in Math Toolbox 1.3 (Ratio Approach). In the expression for density there is an implied 1:

[latex]Density = \frac{mass}{volume} [/latex]

[latex]\frac{Density}{1} = \frac{mass}{volume} [/latex]

Cross multiplying this density expression, we get:

Density × volume = mass × 1

Since we are trying to find the volume, we want to isolate it on one side of the equation. We can do this by dividing both sides by the density. (We’ll also drop the “× 1” because any quantity times 1 is that quantity.)

[latex]\frac{\cancel{Density} imes volume}{\cancel{Density}} = \frac{mass}{Density} [/latex]

Now we have an expression that solves for the volume:

[latex]Volume= \frac{mass}{density} [/latex]

Then, we substitute the known values of mass and density into the equation and solve for the value of volume:

[latex]Volume= \frac{100.0 g}{8.9 g/ cm^{3}}= 11 cm^{3}[/latex]

In a second approach to this problem, consider that since the density of copper is always the same, the ratio of mass to volume is the same for both what we know and what we don’t:

[latex]\frac{8.9 g}{1 cm^{3}}=\frac{100.0 g}{x cm^{3}}[/latex]

Cross multiply to solve for x:

[latex]x cm^{3}= \frac{(1 cm^{3}) imes (100.0 \cancel{g})}{8.9 \cancel{g}}= 11 cm^{3} [/latex]

In both approaches, the gram units cancel to give the expected volume unit of cm³.
There is yet another approach to solving this problem that involves using density as a conversion factor:

[latex]Volume = 100.0 \cancel{g} imes \frac{1 cm^{3}}{8.9 \cancel{g}}= 11 cm^{3}[/latex]

Does the answer make sense? Yes. The density tells us that 8.9 g of copper occupy a volume of 1 cm³. The mass given, 100.0 g, is over 10 times greater than 8.9 so we would expect it to occupy a volume that is over 10 times greater than 1 cm³.

Question 1.7: Density, Volume, and Mass What is the volume of 100.0 g of c...

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