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 the blimp?

Short answer

The mass of helium in the blimp is approximately 809,211.9 grams, given the volume, temperature, and pressure conditions provided. This is calculated using the Ideal Gas Law equation, conversions of units for volume and temperature, and the molar mass of helium.

Step-by-step solution

Write down the given information

We have the following information about the helium in the blimp: Volume (V) = 175,000 ft³ Temperature (T) = 23°C Pressure (P) = 1.0 atm First, we need to convert the given information into the required units: Volume: Convert ft³ to L (liters) using the conversion factor: 1 ft³ = 28.3168 L. Temperature: Convert Celsius to Kelvin, adding 273.15 to the Celsius temperature.

Convert volume to liters

Convert the volume from ft³ to L. \(V_{L} = 175{,}000 \mathrm{ft}^{3} \times \frac{28.3168 \mathrm{L}}{1 \mathrm{ft}^3}\) \(V_{L} = 4{,}955{,}440 \mathrm{L}\)

Convert temperature to Kelvin

Convert the temperature from Celsius to Kelvin. \(T_{K} = 23^{\circ} \mathrm{C} + 273.15\) \(T_{K} = 296.15 \mathrm{K}\)

Apply the Ideal Gas Law

The Ideal Gas Law equation is: \(PV = nRT\) Where \(P\) is the pressure (atm), \(V\) is the volume (L), \(n\) is the number of moles of gas, \(R\) is the ideal gas constant (0.0821 L×atm/mol×K), and \(T\) is the temperature in Kelvin. Solve for \(n\). \(n = \frac{PV}{RT}\) Plug in values: \(n = \frac{(1.0 \mathrm{atm})(4{,}955{,}440 \mathrm{L})}{(0.0821 \mathrm{L \cdot atm/mol \cdot K})(296.15 \mathrm{K})}\)

Calculate the number of moles

Calculate the number of moles of helium by performing the division. \(n = 202{,}128.7 \mathrm {mol}\)

Calculate the mass of helium

Now, we calculate the mass of helium by multiplying the number of moles by the molar mass of helium (4.0026 g/mol). Mass (m) = Number of moles (n) × Molar mass of helium \(m = 202{,}128.7 \mathrm{mol} \times 4.0026 \mathrm{g/mol}\)

Find the mass

Calculate the mass of helium in the blimp. \(m = 809{,}211.9 \mathrm{g}\) So, the mass of helium in the blimp is approximately 809,211.9 grams.

Key concepts

Helium

Helium is one of the noble gases, known for its lack of reactivity due to its full electron shell. It is second on the periodic table, right after hydrogen. When it comes to its unique properties:

• It's the lightest noble gas.
• Has an atomic number of 2.
• Is less dense than air, which is why it is used in blimps and balloons for lift.

Helium is not only lighter than air, but also non-flammable, making it a safer choice for lifting compared to other gases like hydrogen. As it doesn't react easily with other elements, helium is very stable. This stability makes it a perfect option for applications needing non-reactive environments. Additionally, being a noble gas means helium is used in applications like cryogenics and in scientific equipment like MRI machines. Helium's discovery heralded the dawn of astronomical chemical analysis, emphasizing how multifaceted this inert gas truly is.

Molar Mass

Molar mass is a crucial concept in chemistry as it relates the mass of a substance to the amount of substance present in moles. For helium, this value is precisely 4.0026 g/mol. This means:

• 1 mole of helium atoms weighs exactly 4.0026 grams.
• This allows for easy conversion between mass and moles, especially when calculating the amount of helium needed for applications like filling a blimp.

Understanding molar mass is critical for using the ideal gas law effectively. It allows you to determine the mass of a gas when the number of moles is known or vice versa. This concept is widely employed in stoichiometry, where balanced chemical equations are used to calculate the masses of reactants or products in a reaction. In our problem, using the molar mass of helium, we straightforwardly calculated the mass of helium in the blimp, guiding us to an efficient and precise solution.

Volume Conversion

Volume conversion is essential in seamlessly transitioning between different units of measure. In many applications, you may find yourself needing to convert units to solve problems accurately. For instance, the volume of helium in the blimp was initially given in cubic feet. Here’s how the conversion works:

• 1 cubic foot equals 28.3168 liters.
• By multiplying the volume in cubic feet by 28.3168, you get the volume in liters.

In the original exercise, the blimp’s volume of 175,000 ft³ was converted to 4,955,440 liters in order to simplify the use of the ideal gas law. Working in consistent units, particularly in the metric system, generally lends itself to more straightforward calculations as it harmonizes with most science-based equations and constants.

Temperature Conversion

Converting temperature ensures calculations involving temperature-dependent variables, like those in gas laws, are accurate. In the process of solving our exercise, the temperature initially provided in Celsius had to be converted to Kelvin, which is the standard unit for temperature in scientific calculations due to its direct relation to absolute temperature.

• To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
• This conversion is crucial because the Kelvin scale starts at absolute zero, the point where theoretically, atoms stop moving.

Thus, the original temperature of 23°C became 296.15 K, enabling accurate use of the ideal gas law. Converting to Kelvin helps ensure that the proportional relationships defined by gas laws hold true, avoiding errors in calculations that result from using an incompatible temperature scale.