Boyle's Law: Formulation

Below is a table of data presented by Robert Boyle in 1662. Boyle and an assistant carried out an experiment involving a large U-shaped tube closed at the top of one side. Enough mercury was poured into the open end to trap some gas in the closed end. Additional mercury was poured into the open end, which had the effect of compressing the air in the closed end. Column A is the height of the column of air in the closed end, in arbitrary units; one can also think of it as a measure of the volume of the air. Column B is the excess height of the mercury column, the difference in height of the mercury in the open and closed tubes; it represents the pressure beyond atmospheric pressure that is applied to the gas. Column C, a constant, is the height of a column of mercury (in the same apparatus) that would balance atmospheric pressure; thus it represents atmospheric pressure. Column D is the sum of columns B and C; thus it represents the total pressure on the gas.
ABCDE
height of airexcess height of mercuryheight of mercury balancing atmospheretotal height of mercury
(volume)("excess" pressure)(atmospheric pressure)(total pressure)
48029.12529.125
461.437529.12530.5625
442.812529.12531.9375
424.37529.12533.5
406.187529.12535.3125
387.87529.12537
3610.12529.12539.3125
3412.529.12541.625
3215.062529.12544.1875
3017.937529.12547.0625
2821.187529.12550.3125
2625.187529.12554.3125
2429.687529.12558.8125
2332.187529.12561.3125
2234.937529.12564.0625
2137.937529.12567.0625
2041.562529.12570.6875
194529.12574.125
1848.7529.12577.875
1753.687529.12582.75
1658.12529.12587.875
1563.937529.12593.0625
1471.312529.125100.4375
1378.687529.125107.8125
1288.437529.125117.5625
(Download a copy of the Boyle data in a spreadsheet file by clicking here.)
Pretend you do not know Boyle's Law, and that you are looking for a relationship between the pressure and the volume (either the excess pressure or the total pressure).
a) Plot the pressure and the excess pressure vs. the volume (on the same plot, preferably). Does there appear to be a smooth relationship or a random one? If there is a smooth relationship, is it positive or negative or neither?
b) Take the natural logs of the volume and the pressure (total and excess). Plot the logs of the pressures vs. the log of the volume. Is the result of either plot a straight line? If so, note the slope m and y-intercept b, and write the resulting equation y = mx + b in terms of the pressure p and the volume V.
c) Boyle's table contained a fifth column, E, a calculation of the total pressure based on the law he hypothesized from the data. Make a fifth column and calculate the pressure based on the pressure-volume relationship you derive in part b.

Reference

Robert Boyle, A Defence of the Doctrine Touching the Spring And Weight of the Air (1662)
Copyright 2003 by Carmen Giunta. Permission is granted to reproduce for non-commercial educational purposes.

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