Example 1.7: Experimental Determination of Density Using Water Displacement

A piece of rebar is weighed and then submerged in a graduated cylinder partially filled with water, with results as shown.

(a) Use these values to determine the density of this piece of rebar.
(b) Rebar is mostly iron. Does your result in (a) support this statement? How?

Solution

\(m_{\mathrm{rebar}}\) \(= 69.658\ \mathrm{g}\)


The volume of the piece of rebar is equal to the volume of the water displaced:

\(V_{\mathrm{initial}}\) \(= 13.5\ \mathrm{mL}\)


\(V_{\mathrm{final}}\) \(= 22.4\ \mathrm{mL}\)


\(V_{\mathrm{rebar}}\) \(= V_{\mathrm{final}} - V_{\mathrm{initial}}\)

\(\ \ \ =22.4\ \mathrm{mL} - 13.5\ \mathrm{mL}\)

\(\ \ \ =8.9\ \mathrm{mL}\)


(rounded to the nearest 0.1 mL, per the rule for addition and subtraction)
The density is the mass-to-volume ratio:

\(ρ_{\mathrm{rebar}}\) \(= \dfrac{m_{\mathrm{rebar}}}{V_{\mathrm{rebar}}}\)

\(\ \ \ =\dfrac{69.658\ \mathrm{g}}{8.9\ \mathrm{mL}}\)

\(\ \ \ =7.8\ \frac{\mathrm{g}}{\mathrm{mL}}\)


(rounded to two significant figures, per the rule for multiplication and division)
From Table 4, the density of iron is 7.9 g/cm^3, very close to that of rebar, which lends some support to the fact that rebar is mostly iron.