Water and Mercury in U-Tube (2B40.53)

Description:

Measure heights of liquid boundaries and use the data to calculate relative densities.

Procedure or Operation Notes:

Instead of using the heights of the Mercury and water columns to calculate their relative densities, most instructors have been using this demo to calculate the height of the water column given the density and height of the Mercury column. Both approaches are valid but they have very different potential errors.

Method 1: Calculate the relative densities of water and Mercury.

From the above pictures:
Δhw = 77.3cm - 7.3cm = 70cm.
ΔhM = 12.2cm - 7.3cm = 4.9cm.

Setting ρgΔh for water equal to Mercury, we get ρM = (70/4.9) * ρw = 14.29 * ρw. This is within 5.5% of the actual relative density of water at 13.546 * ρw.

Method 2: Calculate the height of the water column given the density and height of the Mercury column.

Again, from the above pictures:
ΔhM = 12.2cm - 7.3cm = 4.9cm, and given ρM = 13.546 * ρw.

The height of the water colum is 13.546 * 4.9cm = 66.4cm. The bottom of the water column is at 7.3cm so the final height should be 73.7cm.

Even though this is off by several centimeters, it's still within 5.2% of the actual water column height. This is because an error in reading the Mercury height is multiplied by ~13.6 when applied to the water column, e.g., every 1mm error for Mercury gives an extra 1.3 cm of error for water.

Considering the convex meniscus of Mercury, the concave meniscus of water, and trying to do this quickly during a lecture; getting errors within 10% is still pretty good for this demonstration.