Question

a cylindrical glass tube 15.0 cm in length is filled with ethanol. the mass of the ethanol needed to fill the tube is found to be 9.64 g. calculate the inner diameter of the tube in cm, if the density of ethanol is 0.789 g/ml.

Explanation:

Step1: Calculate the volume of ethanol

We know that density $
ho=\frac{m}{V}$, where $
ho$ is density, $m$ is mass and $V$ is volume. Rearranging for $V$, we get $V = \frac{m}{
ho}$. Given $m = 9.64\ g$ and $
ho=0.789\ g/mL$, so $V=\frac{9.64\ g}{0.789\ g/mL}=12.22\ mL$. Since $1\ mL = 1\ cm^{3}$, the volume $V = 12.22\ cm^{3}$.

Step2: Use the volume formula for a cylinder

The volume formula for a cylinder is $V=\pi r^{2}h$, where $h$ is the height (length) of the cylinder and $r$ is the radius. We know $V = 12.22\ cm^{3}$ and $h=15.0\ cm$. Rearranging the formula for $r^{2}$, we have $r^{2}=\frac{V}{\pi h}$. Substituting the values, $r^{2}=\frac{12.22\ cm^{3}}{\pi\times15.0\ cm}\approx0.26\ cm^{2}$. Then $r=\sqrt{0.26\ cm^{2}}\approx0.51\ cm$.

Step3: Calculate the diameter

The diameter $d = 2r$. So $d=2\times0.51\ cm = 1.02\ cm$.

Answer:

$1.02\ cm$