### Density of soap

Laboratory 01 in SC 130 Physical Science focused on the linear relationship between volume and density for soap. The class began with a return look at Monday's demonstration.

Measuring the mass of the vials did clearly demonstrate that the sinking vial was slightly more massive than the floating vials. Density calculations were complicated by the cap being slightly larger than the glass vial. Volume calculations attempted at 8:00 were plagued by measurement and mathematical errors. Misreading of the calipers and forgetting to halve the diameter both contributed to a confusing presentation. At 11:00 an attempt was made to determine the volume by displacement using the graduated cylinder.

The complication is that the masses do not yield densities that predict floating and sinking.

The floating vials massed at 13.2 and 13.4 grams. The sinking vial massed at 14.0g. By displacement, the volume of a vial is 14 cm3.

Differences in the 1.575 cm diameter of the vials was less than 0.025 centimeters between vials, thus volumetric differences in the vials were on the order of, at most, 2%. The mass difference was the larger effect at 6%.

I also massed an empty 100 milliliter beaker, filled the beaker with 80 ml of water, and remassed the beaker to work out the density of the tap water in the laboratory.

A 43.7  gram beaker massed 126.4 g with 80 ml of water for a calculated density of 1.03 g/cm3 for the tap water. The complication with this approach is that few, if any, of my students have worked with milliliters, thus this calculation is little better than magic.

The above implies a water mass displacement of 14 times 1.03 or 14.42 grams of water, suggesting that all three vials should float. As they do not all float, errors in measurement must remain. The differences are small, only a 6% spread in mass, a difference of less than one gram. This is a rather sensitive system.

Cutting the soap into a rectangular slab

Harmony beauty soap and Safeguard anti-bacterial soap both with a density greater than one gram per cubic centimeter was used by six lab pairs in each section. Two lab pairs in each section had Ivory soap with a density of less than one gram per cubic centimeter. The soap was carved into square chunks so the volume could be calculated from length × width × height.

The class began with the Freeman Dyson quote:
For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created... ...equations are quite miraculous in a certain way. ... the fact that nature talks mathematics, I find it miraculous. ... I spent my early days calculating very, very precisely how electrons ought to behave. Well, then somebody went into the laboratory and the electron knew the answer. The electron somehow knew it had to resonate at that frequency which I calculated. So that, to me, is something at the basic level we don't understand. Why is nature mathematical? But there's no doubt it's true. And, of course, that was the basis of Einstein's faith. I mean, Einstein talked that mathematical language and found out that nature obeyed his equations, too.
At the start of the class I noted that if the density of the soap was less than one gram per cubic centimeter, the soap would float. If the density of the soap was more than one gram per cubic centimeter, then the soap would sink. The density would be determined from the slope of the graph later in the class period.

Connie masses Harmony soap

I also included a quote from William Gilbert on the need to perform experiments and when experimenting to "handle the bodies carefully, skilfully, and deftly, not heedlessly and bunglingly." The above difficulties with the vials suggests that I too am not heedful enough of Gilbert's cautionary words. Somewhere in the vial measurements I bungled.

This class also provides an opportunity to introduce the mass balance and measuring in centimeters. After gathering the data on the length, width, height, and mass of the soap, the class moved upstairs to the computer laboratory to plot the data and determine the slope of the best fit line.

The class does not presume prior knowledge of linear regressions, on the contrary the class teaches the concepts of slope and intercept through the science encountered in the class.

Only after the students work out their slope data do I ask them to predict what their soap chunks will do when dropped into a beaker of water. Each makes a prediction. Then I drop the Harmony soap chunks into the water, followed by the Ivory soap chunks. As Dyson noted, the students made a mathematically based prediction and the soap knew what to do.

The chart is based on student data for the Ivory, Harmony, and Safeguard soaps. The blue line is a density of one gram per cubic centimeter, the theoretic density of water. Obviously this experiment could be mathematically simplified by having the students directly calculate the density per chunk. The densities could then be averaged. The point, however, is to move the students towards mathematical models. Laboratory three and section 041 will feature non-linear models. Slope is mathematically a farther reach for the students, the gain is worth the mind stretching.