Question

Cylinders A and B have equal heights. Cylinder A is filled with helium gas at 1.0 atm pressure and 0^oC. The diameter of cylinder B is half that of cylinder A, and cylinder B is filled with glycerin. What is the ratio of the fluid mass in cylinder B to that in cylinder A?

Short answer

The ratio is 1760: 1

Step-by-step solution

Given information

Cylinder A is filled with helium gas at 1.0 atm pressure and 0^oC.

The diameter of cylinder B is half that of cylinder A,

Cylinder B is filled with glycerin.

Explanation

The density is defined as Mass/Volume

ρ =M/V

Volume of cylinder is

$V=\pi r^{2}h$

$$\begin{gathered} M_A={\rho }_A\times V_A={\rho }_A\times \pi r_A^{2}h \\ {M}_B={\rho }_B\times V_B={\rho }_B\times \pi r_B^{2}h \end{gathered}$$

As given

$r_B=\frac{r_A}{2}$

Density of helium gas, ρ_A=0.179 kg/m³
Density of glycerine, ρ_B=1260 kg/m³

Now substitute values

$$\begin{gathered} \frac{M_B}{M_A}=\frac{{\rho }_B\times \pi r_B^{2}h}{{\rho }_A\times \pi r_A^{2}h}=\frac{{\rho }_B\times r_B^2}{{\rho }_A\times r_A^2} \\ \frac{M_B}{M_A}=\frac{1260\times {\left(\frac{r_A}{2}\right)}^2}{0.179\times r_A^2}=1760:1 \end{gathered}$$