Question

The Goodyear blimps, which frequently fly over sporting events, hold approximately \(4955 \mathrm{~m}^{3}\) of helium. If the gas is at \(23{ }^{\circ} \mathrm{C}\) and \(101.33 \mathrm{kPa},\) what mass of helium is in a blimp?

Short answer

The mass of helium in the blimp can be calculated using the Ideal Gas Law formula, \(PV = nRT\), with the given conditions of volume, temperature, and pressure. After converting the temperature to Kelvin and the pressure to Pascals, we find the number of moles (n) to be approximately \(203183.8\). Using the molar mass of helium, we calculate the mass of helium to be approximately \(812.74 kg\).

Step-by-step solution

Convert temperature to Kelvin

To start, we need to convert the given temperature in Celsius to Kelvin. We can do this using the formula: \(T(K) = T(°C) + 273.15\) Given temperature, \(T = 23 ^{\circ}C\) So, in Kelvin: \(T(K) = 23 + 273.15 = 296.15 K\)

Convert pressure to Pascals

Next, we need to convert the given pressure in kilopascals to Pascals using the formula: \(1 kPa = 1000 Pa\) Given pressure, \(P = 101.33 kPa\) So, in Pascals: \(P = 101.33 * 1000 = 101330 Pa\)

Plug values into Ideal Gas Law

Now that we have all the required values, we can plug them into the Ideal Gas Law formula: \(PV = nRT\) Given volume, \(V = 4955 m^3\) So, to find the number of moles, n: \(\frac{PV}{RT} = n\) Therefore, \(n = \frac{101330 * 4955}{8.314 * 296.15}\)

Calculate the number of moles

To calculate the number of moles, we can plug in the values: \(n = \frac{101330 * 4955}{8.314 * 296.15} = 203183.8\)

Convert number of moles to mass

Finally, we can use the molar mass of helium to convert the number of moles to mass. The molar mass of helium is approximately 4 g/mol. Molar mass of helium: \(M_{He} = 4 g/mol\) So, to find the mass of helium, m: \(m = n * M_{He}\) Therefore, \(m = 203183.8 * 4 = 812735.2 g\)

Convert mass to kilograms (if necessary)

Our answer is in grams, but we can convert it to kilograms if necessary by using the conversion factor: \(1 kg = 1000 g\) So, converting the mass to kilograms: \(m = \frac{812735.2}{1000} = 812.74 kg\) Thus, there are approximately 812.74 kg of helium in the blimp.

Key concepts

Molar Mass

The concept of molar mass is crucial in converting between the number of moles of a substance and its mass. Molar mass is defined as the mass of one mole of a substance, expressed in grams per mole (g/mol). In our example, we're dealing with helium, which has a molar mass of approximately 4 g/mol.

Understanding molar mass enables us to determine how much mass corresponds to a specific number of moles. In calculations involving gases, such as helium in a blimp, once we find the number of moles using the Ideal Gas Law, we can easily calculate the mass by multiplying the moles by the molar mass.

Temperature Conversion

Temperature conversion is an important step when working with gas laws, as these laws generally require the temperature to be in Kelvin. In the exercise given, the initial temperature is provided in degrees Celsius, which needs conversion.

To convert Celsius to Kelvin, we add 273.15 to the Celsius temperature using the formula: \[T(K) = T(^\circ C) + 273.15\]

Therefore, for a temperature of \(23 ^\circ C\), the conversion to Kelvin would result in \(296.15 K\). Working in Kelvin ensures that calculations align with absolute temperature measures used in scientific computations.

Pressure Conversion

When dealing with gases, pressure is often provided in various units such as atmospheres, kilopascals, or pascals. For consistency in calculations, particularly with the Ideal Gas Law, conversion to a common unit like Pascals is necessary.

In this exercise, the pressure in kilopascals (kPa) is converted to Pascals (Pa). We use the conversion factor that 1 kPa equals 1000 Pa:

• Given pressure: \(101.33 \text{ kPa}\)
• Conversion to Pa: \(101.33 \times 1000 = 101330 \text{ Pa}\)

This conversion ensures accuracy and consistency, especially necessary when utilizing the Ideal Gas Law formula \(PV = nRT\), where pressure is often expressed in Pascals.

Volume Calculations

Understanding how to perform volume calculations is essential in determining the amount of gas present in a container like a blimp. Volume in the Ideal Gas Law is typically expressed in cubic meters (\(m^3\)).

In the example, the volume was already given in \(4955 \text{ m}^3\), matching the required unit for the Ideal Gas Law equation \(PV = nRT\).

Correct volume measurements are critical for accurate computation of moles and, subsequently, the mass of a gas using its molar mass. Accurate measurement and consistency in unit management allow correct and meaningful interpretation when calculating the amount of gas present based on conditions like temperature and pressure.