Question

A balloon for long-distance flying contains \(1.2 \times 10^{7} \mathrm{L}\) of helium. If the helium pressure is \(737 \mathrm{mm}\) Hg at \(25^{\circ} \mathrm{C},\) what mass of helium (in grams) does the balloon contain? (See Study Question \(14 .\) )

Short answer

The balloon contains 1,903,090 grams of helium.

Step-by-step solution

Convert Volume from Liters to Cubic Meters

The volume given is in liters and needs to be converted to cubic meters for standard SI units in the gas laws. 1 m³ = 1000 L, so the volume in cubic meters is:\[V = \frac{1.2 \times 10^7}{1000} = 1.2 \times 10^4 \text{ m³}\]

Convert Pressure from mm Hg to Pascals

Pressure should be in Pascals for SI units. The conversion factor is 133.322 Pa/mm Hg. Thus, the pressure in Pascals is:\[P = 737 \text{ mm Hg} \times 133.322 \frac{\text{Pa}}{\text{mm Hg}} \approx 98208.434 \text{ Pa}\]

Convert Temperature from Celsius to Kelvin

Temperature should be in Kelvin for gas law calculations. Add 273.15 to convert Celsius to Kelvin:\[T = 25 + 273.15 = 298.15 \text{ K}\]

Apply the Ideal Gas Law to Calculate Moles

Use the ideal gas equation: \( PV = nRT \), where \( R = 8.314 \text{ J/(mol K)} \) is the ideal gas constant.Rearrange to find \( n \):\[n = \frac{PV}{RT}\]Substitute the known values:\[n = \frac{98208.434 \times 1.2 \times 10^4}{8.314 \times 298.15} \approx 475772.5 \text{ moles}\]

Calculate the Mass of Helium

The molar mass of helium is 4.00 g/mol. Multiply the moles by the molar mass to find the mass in grams:\[m = 475772.5 \text{ moles} \times 4.00 \text{ g/mol} = 1903090 \text{ grams}\]

Key concepts

Gas Laws

Gas laws are fundamental principles that describe how gases behave under different conditions of temperature, pressure, and volume. A key component is the Ideal Gas Law, which combines several simpler gas laws into one comprehensive equation: \( PV = nRT \).

• **Pressure (P):** The force exerted by gas particles colliding with the walls of their container. Typically measured in atmospheres (atm), mm of mercury (mm Hg), or Pascals (Pa).
• **Volume (V):** The space that the gas occupies, usually in liters (L) or cubic meters (m³).
• **Temperature (T):** The measure of the average kinetic energy of gas particles. For gas equations, it's always in Kelvin (K).
• **Number of Moles (n):** Represents the amount of gas. One mole contains approximately \(6.022 \times 10^{23}\) molecules.
• **Ideal Gas Constant (R):** A constant value that makes the equation work, typically \(8.314 \text{ J/(mol K)}\).

The Ideal Gas Law is a handy tool for predicting how a gas will react to changing conditions. It assumes the gas particles are in constant, random motion and do not interact with each other. This is an approximation, but it works well for many gases under a wide range of conditions.

Helium

Helium is a noble gas known for its chemical inertness and low atomic mass. It is the second element in the periodic table and has an atomic number of 2. Because of its properties: - **Inertness:** Helium does not react with other elements, making it perfect for applications requiring a non-reactive environment. - **Lightweight:** Helium's low density makes it ideal for filling balloons and airships, as it is lighter than air. - **Boiling Point:** It has an extremely low boiling point, remaining a gas at very low temperatures. In the context of the Ideal Gas Law, helium's role is determined by its atomic properties such as its molar mass (4.00 g/mol). This makes it easy to calculate the mass of helium when the number of moles is known. Helium's other properties ensure that calculations using the Ideal Gas Law tend to remain accurate, assuming conditions aren't too extreme.

Mole Calculations

Moles are a central unit in chemistry, used to quantify the amount of a substance. In the Ideal Gas Law, knowing the number of moles allows for calculating other properties of the gas. The process follows:1. **Identify the Required Values:** Gather the gas's pressure, volume, and temperature.2. **Rearrange the Ideal Gas Equation:** Solve the equation for moles: \( n = \frac{PV}{RT} \).3. **Calculate Moles:** Substitute the pressure (in Pascals), volume (in cubic meters), and temperature (in Kelvin) into the formula. Use the ideal gas constant \( R = 8.314 \text{ J/(mol K)} \). Once you find the number of moles (), you can relate this to the substance in question—like helium—to find the mass. Multiply the moles by the molar mass (in g/mol). This gives you the total mass of your substance.

Unit Conversions

Unit conversions play a critical role in solving problems using the Ideal Gas Law. Converting to the correct units ensures that calculations are accurate and follow standard conventions: - **Volume:** Often provided in liters but should be converted to cubic meters for the Ideal Gas Law. Since 1 m³ = 1000 L, divide the volume in liters by 1000. - **Pressure:** Many times, pressure is given in mm Hg. To use the Ideal Gas Law, convert pressure to Pascals using the conversion factor: 133.322 Pa/mm Hg. Multiply the given mm Hg value by this factor. - **Temperature:** Always convert Celsius to Kelvin by adding 273.15 to the Celsius temperature. Kelvin is the SI unit required for gas law calculations. Making these conversions ensures that every value is compatible with the other variables in the Ideal Gas Law, leading to a correct and reliable solution for your gas calculations.