On a summer day, you take a road trip through Death Valley, California, in an antique car. You start out at a temperature of 21°C, but the temperature in Death Valley will reach a peak of 56°C. The tires on your car hold 13.2 L of nitrogen gas at a starting pressure of 240 kPa. The tires will burst when the internal pressure (Pb) reaches 262 kPa. Answer the following questions and show your work.
• How many moles of nitrogen gas are in each tire?
• What will the tire pressure be at peak temperature in Death Valley?
• Will the tires burst in Death Valley? Explain.
• If you must let nitrogen gas out of the tire before you go, to what pressure must you reduce the tires before you start your trip? (Assume no significant change in tire volume.)

1) 1.296 mol

2)

3) Yes

4) 234 kPa

Explanation:

1)

In order to find the number of moles inside each tire, we can use the equation of state for an ideal gas:

where:

p is the pressure of the gas

V its volume

n the number of moles

R the gas constant

T the absolute temperature

At the beginning, we have:

Therefore, the number of moles of nitrogen in each tire is:

2)

The equation of state can be rewritten as

For the nitrogen gas inside the tires, the quantity nR/V remains constant, so we can write:

Where in this problem:

is the initial pressure

is the initial temperature

is the final pressure in Death Valley

is the temperature in Death Valley

Solving for p2, we find the final pressure of the tires:

3)

As we have calculated in part 2, the pressure of the gas inside the tires when the car reaches the Death Valley will be

Which can be rewritten as

The text of the problem states that the tires will burst if the internal pressure reaches a value of

We observe that

Which means that the internal pressure is larger than the breaking pressure: so, the tires will burst.

4)

Here we want the final pressure in the tires to be at most equal to the breaking pressure: so it must be

We can use again the equation used in part 2:

In order to find , the initial pressure at which the tires should be in order not to burst when the car arrives in the Death Valley.

The temperatures are

is the initial temperature

is the temperature in Death Valley

So the initial pressure must be