Problem 171 - Chemistry

Question

In 2012 , Felix Baumgartner jumped from a balloon roughly \(24 \mathrm{mi}\) above Earth, breaking the record for the highest skydive. He reached speeds of more than 700 miles per hour and became the first skydiver to exceed the speed of sound during free fall. The helium-filled plastic balloon used to carry Baumgartner to the edge of space was designed to expand to \(8.5 \times 10^{8} \mathrm{~L}\) in order to accommodate the low pressures at the altitude required to break the record. (a) Calculate the mass of helium in the balloon from the conditions at the time of the jump \((8.5 \times\) \(\left.10^{8} \mathrm{~L},-67.8^{\circ} \mathrm{C}, 0.027 \mathrm{mmHg}\right) .\) (b) Determine the volume of the helium in the balloon just before it was released, assuming a pressure of 1.0 atm and a temperature of \(23^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The mass of helium in the balloon at the time of the jump is approximately 0.011 kg, and the volume of the helium just before it was released is approximately \(2.44 x 10^5 m^3\).

Step 1: Convert the Given Values to SI Units

We first need to convert the given values to SI units. The volume \(V=8.5x10^8\) liters should be converted to cubic meters. As 1 liter = \(1x10^{-3} m^3\), so \(V=8.5x10^5 m^3\). The pressure P=0.027mmHg needs to be converted to pascals. As 1mmHg = 133.322 Pascals, so P = 3.6 Pascals. The temperature T=-67.8°C is to be converted to Kelvin. As 1°C = 273.15 Kelvin, so T = 205.34 Kelvin.

Step 2: Calculate the Number of Moles

The number of moles (n) of helium can be found using Ideal Gas Law equation \(PV=nRT\), by rearranging the equation to \(n = \frac{PV}{RT}\). Use the values from Step 1 and gas constant R = 8.31 J/(mol*K) to calculate n.

Step 3: Calculate the Mass of Helium

Once the number of moles is found, the mass of helium can be calculated by using the formula \(m = nM\), where m is the mass, n is number of moles from Step 2 and M is the molar mass of helium = 4 g/mol. Convert the molar mass to kg to match SI units (M = 0.004 kg/mol).

Step 4: Apply Ideal Gas Law Again

In the second part, we need to find the volume of helium before it was released. Assume a pressure of 1.0 atm = 101325 Pa and a temperature of 23°C = 296.15 K. We can rearrange the ideal gas law to \(V = \frac{nRT}{P}\). Use the number of moles n calculated in Step 2, R = 8.31 J/(mol*K), the given values of temperature and pressure to calculate the volume V.

Key Concepts

Helium Gas

Helium is a unique gas with fascinating properties that make it perfect for many scientific applications, including balloon flights into the stratosphere.
It is a noble gas, meaning it is less reactive because of its full electron shell. Due to its low density, it has lifting properties that make it ideal for balloons.
Helium is much lighter than air, which allows it to easily lift balloons high into the atmosphere.
Some key characteristics of helium include:

• Noble gas: Helium belongs to Group 18 of the periodic table, making it non-reactive.
• Lightweight: With a density of about 0.1786 grams per liter, it's much lighter than air.
• Inert: Helium does not form stable compounds with other elements under normal conditions.

These properties of helium are vital for tasks such as carrying large objects to high altitudes, similar to the balloon used by Felix Baumgartner in his record-breaking skydive.

Mole Calculations

Understanding mole calculations is essential when working with gases, as they relate the amount of gas to its molecular properties.
The mole is a basic unit in chemistry that helps describe a substance's amount. It links mass, number of particles, and volume in calculations.
In the exercise, the Ideal Gas Law is used: \(PV = nRT\).
This formula rearranges to \(n = \frac{PV}{RT}\), where:

• \(n\): Number of moles
• \(P\): Pressure of the gas
• \(V\): Volume of the gas
• \(R\): Ideal gas constant (8.31 \(J/(mol \, K)\))
• \(T\): Temperature in Kelvin

By substituting values from the problem, you can determine how many moles of helium were present at the time of the jump.

Pressure Conversions

Pressure conversions are crucial for scientific calculations when dealing with different units.
In many cases, you might find pressures given in units like mmHg or atmospheres. However, the SI unit for pressure, which is often used in formulas, is the pascal (Pa).
Here's how to convert:

• 1 mmHg = 133.322 Pa; thus, when converting from mmHg to Pa, multiply by 133.322.
• 1 atm = 101,325 Pa; when converting from atm to Pa, use this factor.

In the solution, the conversion from 0.027 mmHg to pascals results in about 3.6 Pa. Correct conversion ensures calculations using the Ideal Gas Law remain consistent and reliable.

Temperature Conversions

Temperature conversions are critical in applying the Ideal Gas Law, as the temperatures need to be in Kelvin.
Kelvin is the absolute temperature scale used in scientific calculations, providing a more direct scale for thermodynamic equations.
Converting Celsius to Kelvin is simple:

• Add 273.15 to the Celsius value to convert to Kelvin.

For example, changing -67.8°C to Kelvin:

• Add 273.15: -67.8 + 273.15 = 205.34 K.

Thus, using Kelvin ensures the calculation aligns with physical properties and provides accurate results for gas laws.