The Problem:
The air inside an inflatable stand-up paddle board is 68.0°F anda gauge measures the pressure of this air to be 10.0 psi. If thevolume of the air inside the board is reduced by 12.0% and thetemperature increases to 102.2°F, what is the final gauge pressureof this air, in psi? (1.00 psi = 0.0680 atm)
Intermediate steps:
a.) This problem requires use of the ideal gas law, which can beexpressed separately for the two relevant snapshots in time. Whatis the ideal gas law? [Consider: Which variables will notchange between the two snapshots in time, that is between theinitial and final states?]
b.) Only absolute values are used in the ideal gas law: absolutetemperature, absolute volume, and absolute pressure. (“Absoluteâ€means that a zero value is a true zerotemp/pressure/volume.) Convert the initial and final temperaturevalues (68.0°F and 102.2°F, respectively) along with the initialgauge pressure (10.0 psi) into absolute temperature and absolutepressure (in atm) values. Assume the air pressure outside of thepaddleboard is 1.00 atm, or “standard pressureâ€. [Think:What is the reading on the pressure gauge when it is not being usedto make a pressure measurement?]
c.) The volume of the board is not provided, only a relationshipbetween the initial and final volumes. State this relationship inan equation (e.g. Vfinal = ___Vinitial). [Check: Which volume, final orinitial, is greater? Does your equation/relationshipcorrespond?]
d.) Without entering numerical values, write an expression forthe final absolute pressure. [Recommended steps: (i) Usingthe ideal gas law, put all values that will change (the variables)on one side of the equal sign, and all values that are constant onthe other. (ii) Equate the expressions for the variables from theinitial and final states, as they equal the same constantexpression. (iii) Solve for the final absolute pressure.] Now,numerically solve for final absolute pressure.
The solution:
e.) What is the final gauge pressure of the air insidethe paddle board, in psi? [Please enter this answer asyour final value. How many significant figures should besubmitted?]
