Density: Question 9

Worked solution

Part (a): Gradient of the graph

Using the two most widely separated points, $(20.0, 17.0)$ and $(80.0, 68.0)$:

$\text{gradient} = \frac{\Delta m}{\Delta V} = \frac{68.0 - 17.0}{80.0 - 20.0} = \frac{51.0}{60.0}$

$\text{gradient} = \boxed{0.85\text{ g/cm}^3}$

Since $\rho = \dfrac{m}{V}$, the gradient of a mass–volume graph is exactly the density of the liquid.

Part (b): Why the line is straight and passes through the origin

Every row of the table gives the same ratio, $\dfrac{m}{V} = 0.85$, so mass is directly proportional to volume, doubling the volume doubles the mass, and so on. A directly proportional relationship always plots as a straight line through the origin, which also makes physical sense here: a volume of $0\text{ cm}^3$ of liquid (i.e. none at all) must have a mass of $0\text{ g}$.

Part (c): Predicting the mass of $150\text{ cm}^3$

Use the gradient (density) found in part (a):

$m = \rho V = 0.85\text{ g/cm}^3 \times 150\text{ cm}^3$

$m = \boxed{127.5\text{ g}}$

Final answers

• (a) Gradient $=$ $0.85\text{ g/cm}^3$, the density of the liquid.
• (b) Mass is directly proportional to volume, so $m/V$ is constant and the line passes through $(0, 0)$.
• (c) Predicted mass $=$ $127.5\text{ g}$