At 25°C the density of ether is 72.7 kg/m3 and the density of iodine is 4930 kg/m3. A cylinder is filled with iodine to a depth of 1.6 m. 1) How tall would a cylinder filled with ether need to be so that the pressure at the bottom is the same as the pressure at the bottom of the cylinder filled with iodine? (Express your answer to two significant figures.)
The correct answer and explanation is :
To solve this problem, we use the hydrostatic pressure equation:
[
P = \rho g h
]
where:
- ( P ) is the pressure at the bottom,
- ( \rho ) is the density of the fluid,
- ( g ) is the acceleration due to gravity ((9.81 \, \text{m/s}^2)),
- ( h ) is the height of the fluid column.
Step 1: Calculate the Pressure at the Bottom of the Iodine Cylinder
The pressure due to the iodine column is:
[
P_{\text{iodine}} = \rho_{\text{iodine}} g h_{\text{iodine}}
]
Substituting values:
[
P_{\text{iodine}} = (4930 \, \text{kg/m}^3) (9.81 \, \text{m/s}^2) (1.6 \, \text{m})
]
[
P_{\text{iodine}} = 77214.72 \, \text{Pa} \approx 7.7 \times 10^4 \, \text{Pa}
]
Step 2: Determine the Required Height of Ether
For the ether column to exert the same pressure:
[
P_{\text{ether}} = \rho_{\text{ether}} g h_{\text{ether}}
]
Setting ( P_{\text{ether}} = P_{\text{iodine}} ):
[
(72.7 \, \text{kg/m}^3) (9.81 \, \text{m/s}^2) h_{\text{ether}} = 77214.72 \, \text{Pa}
]
Solving for ( h_{\text{ether}} ):
[
h_{\text{ether}} = \frac{77214.72}{(72.7 \times 9.81)}
]
[
h_{\text{ether}} = \frac{77214.72}{713.187}
]
[
h_{\text{ether}} \approx 108.3 \, \text{m}
]
Step 3: Round to Two Significant Figures
[
h_{\text{ether}} \approx 110 \, \text{m}
]
Explanation:
The key concept here is that the pressure at the bottom of a liquid column depends on the density and height of the liquid. Since iodine is much denser than ether (4930 kg/m³ vs. 72.7 kg/m³), ether needs to be filled to a much greater height to produce the same pressure. This height is approximately 110 meters, significantly larger than the 1.6-meter iodine column. The reason is that for a less dense fluid, more depth is required to achieve the same pressure exerted by a denser fluid at a smaller height.