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3. An automobile tire contains 3730 cu. in. of air at 32 psig and 80 degrees F.

(a) What mass of air is in the tire?

(b) In operation, the air temperature increases to 145 degrees F. If the tire is inflated to the same volume, what is the new pressure?

Answer :

Final answer:

The mass of air in the tire is approximately 0.251 grams.

Explanation:

To calculate the mass of air in the tire, we need to use the ideal gas law equation:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

First, let's convert the given temperature from Fahrenheit to Kelvin:

T(K) = (T(°F) + 459.67) * (5/9)

Given:

  • Pressure (P) = 32 psig
  • Temperature (T) = 80°F
  • Volume (V) = 3730 cm³

Converting the temperature to Kelvin:

T(K) = (80 + 459.67) * (5/9) = 299.82 K

Next, we need to convert the volume from cm³ to m³:

V(m³) = V(cm³) / 1000000

V(m³) = 3730 / 1000000 = 0.00373 m³

Now, we can rearrange the ideal gas law equation to solve for the number of moles (n):

n = PV / RT

Substituting the given values:

n = (32 * 6894.76) / (8.3145 * 299.82)

n ≈ 0.0087 moles

Finally, we can calculate the mass of air using the equation:

m = n * M

Where M is the molar mass of air, which is approximately 28.97 g/mol:

m = 0.0087 * 28.97 ≈ 0.251 g

Therefore, the mass of air in the tire is approximately 0.251 grams.

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Final answer:

To find the mass of air in the tire, we can use the ideal gas law equation PV = nRT. First, we need to convert the pressure and temperature to Kelvin and the volume to cubic meters. Then, we can rearrange the equation to solve for the number of moles of air in the tire. Finally, we can multiply the number of moles by the molar mass of air to find the mass of air in the tire.

Explanation:

To find the mass of air in the tire, we can use the ideal gas law equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. First, we need to convert the pressure and temperature to Kelvin and the volume to cubic meters. Then, we can rearrange the equation to solve for the number of moles of air (n) in the tire. Finally, we can multiply the number of moles by the molar mass of air to find the mass of air in the tire.

Given: Pressure (P) = 32 psig (gauge pressure), Temperature (T) = 80 degrees F (Fahrenheit), Volume (V) = 3730 cubic inches.

Conversion: 1 atm = 14.7 psi, 1 inch = 0.0254 meters, 1 pound = 0.4536 kilograms.

Calculation:

  1. Convert pressure to absolute pressure in atm: (32 psig + 14.7 psi) / 14.7 psi/atm = 3.16 atm.
  2. Convert temperature to Kelvin: (80 degrees F + 459.67) * (5/9) = 299.82 K.
  3. Convert volume to cubic meters: 3730 cubic inches * (0.0254 m/inch)^3 = 0.0611 cubic meters.
  4. Rearrange the ideal gas law equation to solve for n: n = PV / RT.
  5. Plug in the values: n = (3.16 atm)(0.0611 cubic meters) / (0.0821 atm·L/mol·K)(299.82 K) = 0.0733 moles.
  6. Find the molar mass of air: 1 mole of air = 28.97 g.
  7. Calculate the mass of air in the tire: 0.0733 moles * 28.97 g/mole = 2.12 g.

Therefore, the mass of air in the tire is approximately 2.12 grams.

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