Question

A tire of an automobile has a gauge pressure of 30 psi at the top of a mountain, and the air temperature inside the tire is 300 K. The atmospheric pressure at the top of the mountain is 0.7 atm. The car is driven down to sea level, where the atmospheric pressure is 1.0 atm, and the air inside the tire is heated to 350 K. at sea level, assuming no leaks, the gauge pressure of the tire will be?A.)higherB.)the sameC.)lower

Answer 1

Answer: Option (A) is the correct answer.

Explanation:

Relation between gauge pressure ($P_{g}$), absolute pressure ($P_{a}$) and atmospheric pressure ($P_{atm}$ is as follows.

                   $P_{g} = P_{a} - P_{atm}$

In atm units, the gauge pressure at mountain top is calculated as follows.

            $P_{g} = (30 psi) \times (\frac{0.068 atm}{1 psi})$

                       = 2.04 atm

Now, we will calculate the value of absolute pressure $P_{1}$ of the tire at the top as follows.

       $P_{g} = P_{1} - P_{atm}$    

or,      $P_{1} = P_{g} + P_{atm}$

                    = 0.7 atm + 2.04 atm

                    = 2.74 atm

Now, we assume that absolute pressure of the tire at sea level is $P_{2}$. According to ideal gas equation,

             $\frac{P_{1}V}{T_{1}} = \frac{P_{2}V}{T_{2}}$

Hence,   $P_{2} = \frac{2.74 atm \times 350 K}{300 K}$

                         = 3.19 atm

Gauge pressure at the sea level is as follows.

           $(P_{g})_{s} = P_{2} - P_{atm}$

                      = 3.19 atm - 1 atm

                     = 2.19 atm

This means that gauge pressure at the sea level increases.

Thus, we can conclude that at sea level, assuming no leaks, the gauge pressure of the tire will be higher.