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

Cylinders A and B have equal heights.

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

Cylinder B is filled with glycerin.

The diameter of cylinder B = 1/2 A,

Explanation

Density is calculated as

$Density==\frac{\text{Mass}}{\text{Volume}}\Rightarrow \rho =\frac{M}{V}....................\left(1\right)$

Find the volume to be used in equation (1)

Volume is calculated as $V=\pi r^{2}hwherehisheight,andrisradius$

From equation (1) we can find mass

$$\begin{gathered} M_A={\rho }_A\times V_A={\rho }_A\times \pi r_A^{2}h.......................\left(2\right) \\ {M}_B={\rho }_B\times V_B={\rho }_B\times \pi r_B^{2}h.........................\left(3\right) \end{gathered}$$

Get the density

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

Substitute the value and r_B = r_A

To find the ratio divide equation(1) by equation (2), we get

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

So the ratio is 1760:1.