Question

An automobile tire at \(22^{\circ} \mathrm{C}\) with an internal volume of \(20.0 \mathrm{~L}\) is filled with air to a total pressure of 30 psi (pounds per square inch). \(\left[1 \mathrm{~atm}=14.696 \mathrm{lb} / \mathrm{in} .^{2}\right]\) (a) What is the amount in moles of air in the tire? (b) If the air were entirely nitrogen \(\left(\mathrm{N}_{2}\right)\), how many grams of it would be in the tire? How many pounds of it would be in the tire? \([453.6 \mathrm{~g}=1 \mathrm{lb}]\)

Short answer

(a) The amount of air in the tire is \(1.68\,\mathrm{moles}\). (b) If the air was entirely nitrogen, there would be \(47.1\,\mathrm{grams}\) (or \(0.1038\,\mathrm{pounds}\)) of nitrogen in the tire.

Step-by-step solution

Convert temperature from Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, simply add 273.15: \(T_{Celsius} = 22^{\circ}\mathrm{C}\) \(T_{Kelvin} = 22 + 273.15 = 295.15\mathrm{K}\)

Convert pressure from psi to atm

To convert the pressure from psi to atm, use the conversion factor given: \(P_{psi} = 30\,\mathrm{psi}\) \(P_{atm} = P_{psi} \times \frac{1\,\mathrm{atm}}{14.696\,\mathrm{lb}/\mathrm{in}^{2}} = 30\,\mathrm{psi} \times \frac{1\,\mathrm{atm}}{14.696\,\mathrm{lb}/\mathrm{in}^{2}} = 2.04\,\mathrm{atm}\)

Determine the moles of air in the tire

Using the ideal gas equation and the values we have found, we can now solve for the number of moles (n): \(PV = nRT\) We can rearrange the equation to solve for \(n\): \(n = \frac{PV}{RT}\) Plug in the values: \(n = \frac{(2.04\,\mathrm{atm})(20.0\,\mathrm{L})}{(0.0821\,\mathrm{L\cdot atm}\, \mathrm{mol}^{-1}\mathrm{K}^{-1})(295.15\,\mathrm{K})} = 1.68\,\mathrm{mol}\)

Determine the mass of nitrogen inside the tire

We will first calculate the mass in grams by using the molar mass of nitrogen gas (\(\mathrm{N_2}\)), which is 28.02 g/mol: mass_N2 = n * molar mass_N2 mass_N2 = 1.68 mol * 28.02 g/mol = 47.1 g

Convert grams of nitrogen to pounds

To convert the mass of nitrogen from grams to pounds, use the conversion factor given: \(47.1\,\mathrm{g} \times \frac{1\,\mathrm{lb}}{453.6\,\mathrm{g}} = 0.1038\,\mathrm{lb}\) Answer: (a) The amount of air in the tire is 1.68 moles. (b) If the air was entirely nitrogen, there would be 47.1 grams (or 0.1038 pounds) of nitrogen in the tire.

Key concepts

Moles of gas

The concept of moles of gas is crucial in understanding various gas-related calculations. A mole is a unit in chemistry that represents a specific number of particles, usually atoms or molecules. In the context of gases, a mole corresponds to Avogadro's number, which is approximately 6.022 × 10²³ particles. This number is significant because it allows chemists to count entities on the atomic scale and equate them to a macroscopic amount that can be measured practically.

When dealing with the Ideal Gas Law, the number of moles ( ) is directly proportional to the volume ( V ) and pressure ( P ) of the gas, at a given temperature and constant. If we keep the temperature and pressure constant, increasing the number of moles will increase the volume. This relation is essential when determining the amount of substance within a container, like the air in a tire in our exercise.

Gas pressure conversion

Understanding gas pressure conversion is an integral part of working with gases in various units of pressure measurement. Pressure can be expressed in atmospheres (atm), pounds per square inch (psi), pascals (Pa), and more. It is frequently necessary to convert between these units to use the Ideal Gas Law, where the standard unit is atmospheres.

Typically, conversion factors are used to convert between units. For example, as with the exercise, 1 atm is equivalent to 14.696 psi. This means that by multiplying or dividing by this value, we can switch between atmospheric pressure and the psi measurement. This is crucial for many real-world applications, such as inflating car tires, where pressure is often given in psi.

Molar mass of nitrogen

The molar mass of a substance is defined as the mass of one mole of its particles and is expressed in grams per mole (g/mol). For nitrogen gas ( N_2 ), which is diatomic, meaning each molecule consists of two nitrogen atoms, the molar mass is calculated by adding the mass of the two nitrogen atoms.

The atomic mass of nitrogen is approximately 14.01 g/mol; since there are two atoms in a nitrogen molecule, the molar mass of nitrogen gas is 28.02 g/mol. Knowledge of the molar mass is important when converting between the mass of a substance and the number of moles, as we've applied in the exercise to find the mass of nitrogen in the tire.

Avogadro's law

Avogadro's Law is a principle stating that equal volumes of all gases at the same temperature and pressure contain the same number of molecules. It is a direct consequence of this law that the volume of a gas is proportional to the number of moles of the gas when the temperature and pressure are held constant. Therefore, by knowing the volume, we can determine the number of moles.

In practice, Avogadro's Law allows us to understand the underlying molecular nature of gas volumes and to perform calculations that relate volume to the number of moles. It supports the Ideal Gas Law that binds together pressure, volume, temperature, and moles into a single, unifying equation. Avogadro's Law is fundamental in the exercise for determining the number of moles of air in a tire under given conditions of temperature and pressure.