Question

The Problem:

The air inside an inflatable stand-up paddle board is 68.0°F and a gauge measures the pressure of this air to be 10.0 psi. If the volume of the air inside the board is reduced by 12.0% and the temperature increases to 102.2°F, what is the final gauge pressure of this air, in psi? (1.00 psi = 0.0680 atm)

Intermediate steps:

a.) This problem requires use of the ideal gas law, which can be expressed separately for the two relevant snapshots in time. What is the ideal gas law? [Consider: Which variables will not change between the two snapshots in time, that is between the initial and final states?]

b.) Only absolute values are used in the ideal gas law: absolute temperature, absolute volume, and absolute pressure. (“Absolute” means that a zero value is a true zero temp/pressure/volume.) Convert the initial and final temperature values (68.0°F and 102.2°F, respectively) along with the initial gauge pressure (10.0 psi) into absolute temperature and absolute pressure (in atm) values. Assume the air pressure outside of the paddleboard is 1.00 atm, or “standard pressure”. [Think: What is the reading on the pressure gauge when it is not being used to make a pressure measurement?]

c.) The volume of the board is not provided, only a relationship between the initial and final volumes. State this relationship in an equation (e.g. V_final = ___ V_initial). [Check: Which volume, final or initial, is greater? Does your equation/relationship correspond?]

d.) Without entering numerical values, write an expression for the final absolute pressure. [Recommended steps: (i) Using the ideal gas law, put all values that will change (the variables) on one side of the equal sign, and all values that are constant on the other. (ii) Equate the expressions for the variables from the initial and final states, as they equal the same constant expression. (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 inside the paddle board, in psi? [Please enter this answer as your final value. How many significant figures should be submitted?]

Answer 1

DATA:

SOLUTION.

Convert initial temperature to Kelvin, as follows:

Replacing data given value into equation (1) we have:

Solving we obtain:

Similarly, convert the final temperature to Kelvin:

Solving we obtain:

Now, convert the initial pressure to   as follows:

Then, convert this value of gauge pressure to absolute pressure using:

Replacing (4) and 1 atm for atmospheric pressure into equation (5) we have:

Solving we obtain:

We need to find now, the final volume, which is reduced by 12.0% of the initial volume:

or

The gas ideal law is defined as:

For, the initial state we have:

Isolating the number of moles we have:

Similarly, for the final state we have:

isoalting the number of moles from equation (11) we have:

Equaling equations (10) and (12) we have:

or

Isolating from equation (13) we have:

Replacing (7) into equation (14) we obtain:

Inserting (2) , (3) and (6) into equation (15) we have:

Solving we obtain:

Now, the final gauge pressure is:

Solving we obtain:

Converting to psi we obtain: