Question

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.)

Answer 1

1) 1.296 mol

2)
`2.69\cdot 10^5 Pa`

3) Yes

4) 234 kPa

Step-by-step explanation:

1)

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

`pV=nRT`

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:

`p=240 kPa = 2.40\cdot 10^5 Pa`

`V=13.2 L = 0.0132 m^2`

`R=8.314 J/mol K`

`T=21^(\circ)C+273=294 K`

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

`n=(pV)/(RT)=((2.40\cdot 10^5)(0.0132))/((8.314)(294))=1.296mol`

2)

The equation of state can be rewritten as

`(p)/(T)=(nR)/(V)`

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

`(p_1)/(T_1)=(p_2)/(T_2)`

Where in this problem:

`p_1 = 2.40\cdot 10^5 Pa` is the initial pressure

`T_1=294 K` is the initial temperature

`p_2` is the final pressure in Death Valley

`T_2=56^(\circ)C+273=329 K` is the temperature in Death Valley

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

`p_2 = (p_1 T_2)/(T_1)=((2.40\cdot 10^5)(329))/(294)=2.69\cdot 10^5 Pa`

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

`p_2 = 2.69\cdot 10^5 Pa`

Which can be rewritten as

`p_2 = 269 kPa`

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

`p_b=262 kPa`

We observe that

`p_2>p_b`

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

`p_2 = p_b = 262 kPa`

We can use again the equation used in part 2:

`(p_1)/(T_1)=(p_2)/(T_2)`

In order to find
`p_1`, 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

`T_1=294 K` is the initial temperature

`T_2=56^(\circ)C+273=329 K` is the temperature in Death Valley

So the initial pressure must be

`p_1 = (p_2 T_1)/(T_2)=((262)(294))/(329)=234 kPa`