Question

The Goodyear blimps, which frequently fly over sporting events, hold approximately \(175,000 \mathrm{ft}^{3}\) of helium. If the gas is at \(23^{\circ} \mathrm{C}\) and \(1.0 \mathrm{~atm}\), what mass of helium is in a blimp?

Short answer

The mass of helium in a Goodyear blimp is approximately \(816,539 \mathrm{g}\) or \(816.54 \mathrm{kg}\).

Step-by-step solution

Convert given temperature to Kelvin

Given temperature is \(23^{\circ}\mathrm{C}\). To convert it to Kelvin, we add 273.15. \(T = 23 + 273.15 = 296.15 \mathrm{K}\)

Write out the Ideal Gas Law

Now, we can write the Ideal Gas Law equation: \(PV = nRT\) Given values: - \(P = 1.0 \mathrm{atm}\) - \(V = 175,000 \mathrm{ft}^{3}\) - \(R = 0.0821\frac{\mathrm{atm}\cdot\mathrm{L}}{\mathrm{mol}\cdot\mathrm{K}}\) - \(T = 296.15 \mathrm{K}\)

Convert volume from cubic feet to liters

We need to convert the volume from cubic feet to liters for the ideal gas law because we have the value of R in liters. There are approximately \(28.3168 \mathrm{L}\) in 1 cubic foot. So, \(V = 175,000 \mathrm{ft^{3}} \times 28.3168 \frac{\mathrm{L}}{\mathrm{ft}^{3}} = 4,955,440 \mathrm{L}\)

Solve for moles of helium

Now, we can plug the known values into the ideal gas law equation: \(1.0 \mathrm{atm} \times 4,955,440 \mathrm{L} = n \times 0.0821 \frac{\mathrm{atm}\cdot\mathrm{L}}{\mathrm{mol}\cdot\mathrm{K}} \times 296.15 \mathrm{K}\) Rearranging the equation to solve for n: \(n = \frac{1 \mathrm{atm} \times 4,955,440 \mathrm{L}}{0.0821 \frac{\mathrm{atm}\cdot\mathrm{L}}{\mathrm{mol}\cdot\mathrm{K}} \times 296.15 \mathrm{K}}\) Now, calculating the value of n: \(n \approx 203,916 \mathrm{moles}\)

Convert moles of helium to mass

To find the mass of helium, multiply the number of moles we just calculated by the molar mass of helium, which is \(4.0026 \frac{\mathrm{g}}{\mathrm{mol}}\). \(mass = 203,916 \mathrm{moles} \times 4.0026 \frac{\mathrm{g}}{\mathrm{mol}} \approx 816,539 \mathrm{g}\) Thus, the mass of helium in a Goodyear blimp is approximately \(816,539 \mathrm{g}\) or \(816.54 \mathrm{kg}\).

Key concepts

Gas Volume Conversion

Gas volume conversion is a critical step when working with gases in chemistry, as measurements can be required in different units. In the scenario with the Goodyear blimp, the volume of helium is initially given in cubic feet. However, to apply the Ideal Gas Law effectively, the volume must be in liters. This is because the constant (R) in the Ideal Gas Law is defined with volume in liters.

The conversion factor between cubic feet and liters is key: 1 cubic foot equals approximately 28.3168 liters. Therefore, multiplying the given volume in cubic feet by this factor provides us with the volume in liters, necessary for further calculations.

Moles of Gas Calculation

Understanding moles of gas calculation is essential for quantifying the amount of substance in a gas. The mole is a unit that represents a standardized number of particles, allowing chemists to convert between mass, particles, and volume. In the exercise, after converting volume to liters, the next step was to determine the number of moles of helium.

Using the Ideal Gas Law, with the known values for pressure, volume, temperature, and the gas constant, we solve for the variable 'n', which represents the number of moles. The balanced equation gives us a straightforward way to calculate the moles of helium within the blimp.

Gas Laws in Chemistry

The gas laws in chemistry encompass several principles that describe how gases behave under various conditions. The Ideal Gas Law is the most comprehensive of these, combining the relationships of pressure, volume, temperature, and the number of moles in a single equation: \(PV = nRT\). This law assumes gases behave ideally, meaning they have perfectly elastic collisions and no intermolecular forces.

This assumption allows us to make accurate predictions for most gases under normal conditions. In the Ideal Gas Law, 'P' stands for pressure, 'V' for volume, 'n' for the amount in moles, 'T' for temperature in Kelvin, and 'R' is the gas constant. In a practical sense, it provides a mathematical framework to interrelate these properties and calculate one if the others are known.

Molar Mass

The molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a fundamental concept, as it links the mass of a substance to its amount in moles, which in chemistry is the essential bridge between the laboratory scale and the molecular scale.

In our blimp example, once moles of helium were calculated, the molar mass of helium (\(4.0026 \frac{\text{g}}{\text{mol}}\)) was used to convert moles to grams. Every element has a unique molar mass, which can be found on the periodic table. This characteristic allows chemists to quantify the amount of an element or compound in a chemical reaction or, as in this case, contained within a particular volume of gas.