Question

A mountain biker goes for a ride in the desert. The air temperature is 21°C at the start of the ride, but the temperature in the desert will reach a peak of 51°C. The tires on the bike hold 15.6 L of nitrogen gas at a starting pressure of 249 kPa. The tires will burst when the internal pressure (Pb) reaches 269 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 the peak temperature in the desert?

Will the tires burst at the peak temperature? Explain.

To what pressure should the tire pressure be reduced before starting the ride to avoid bursting of the tires in the desert heat? (Assume no significant change in tire volume.)

Answer:

Answer 1

Under standard conditions, 1 mol of nitrogen occupies 22.39 Liters of space
https://www.aqua-calc.com/calculate/mole-to-volume-and-weight/substance/nitrogen-blank-gas
Standard conditions are defined as temperature of 273.15 Kelvin and pressure of 101.3 kPa. Volume changes inversely in proportion to pressure at constant temperature and expands in proportion to temperature at constant pressure

Answer 2

First, to find out how many moles of Nitrogen we're working with, let's convert the conditions in the problem to standard conditions and see how the volume changes:
Holding temperature constant and reducing pressure from 269 to 101.3 kPa is a factor of about 2.69, so we can multiply the volume by 2.69. Then, holding pressure constant and reducing temperature from 21 C (294.15 K) to 273.15 K is dividing by a factor of about 1.07.
From this, we can figure that the volume of Nitrogen at standard conditions would be 15.6 Liters * 2.69 / 1.07, which is about 40.2 Liters. Divide that by the 22.39 Liters of Nitrogen per mole to get about 1.8 moles of Nitrogen gas

Answer 3

Temperature and pressure are also proportionally related, assuming that volume stays the same. In this example, the tires will not expand much so that is a safe assumption.
As the temperature increases from 21C (294.15K) to 51C(324.15K), which is an increase by a factor of (324.15/294.15 = 1.1)
The pressure will also increase by the same proportion, from 249 to 249*1.1 = 274.4 kPa

Answer 4

Of course, this exceeds the pressure limit of the tires, so the answer to the third part of the question should be clear

Answer 5

I trust you can apply these concepts to puzzle out that last part? Let me know if you need more help, and I'll hop back in!

Answer 6

How many moles of nitrogen gas are in each tire?

PV = nRT

for initial conditions: P = 249 kPa, V = 15.6 L, n = moles nitrogen, R = gas constant, 8.314 L kPa mol−1 K−1 since we're in kPa, T = 294.15

solve for n

Answer 7

What will the tire pressure be at the peak temperature in the desert?

PV = nRT

for peak temperature conditions: V = 15.6 L, n = moles nitrogen from the previous step, R = gas constant, 8.314 L kPa mol−1 K−1 since we're in kPa, T = 324.15K

solve for P

Answer 8

Will the tires burst at the peak temperature? Explain.

from the previous step, determine whether your pressure is greater than, or less than, the bursting pressure 269 kPa

Answer 9

We know from our previous calculations that the temperature increases by a ratio of about 1.1
So, to avoid the pressure exceeding the burst pressure, we need the starting pressure to be less than (burst pressure)/1.1, assuming Volume stays the same (as the question states)

Answer 10

From this, we know that the starting pressure must be equal to or less than 269/1.1 = 244.5

Answer 11

I love you guys.

Answer 12

We love you too <3