Question

Determine the mass of helium required to fill a \(5.0-\mathrm{L}\) balloon to a pressure of \(1.1 \mathrm{~atm}\) at \(25^{\circ} \mathrm{C}\).

Short answer

0.9 g

Step-by-step solution

Identify the Given Information

We need to fill a balloon with helium. The given parameters are:- Volume \( V = 5.0 \, \text{L} \)- Pressure \( P = 1.1 \, \text{atm} \)- Temperature \( T = 25^{\circ} \text{C} \)- Gas constant \( R = 0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \)- Molar mass of helium \( 4.00 \, \text{g/mol} \)

Convert Temperature to Kelvin

Temperature needs to be in kelvins for calculations using the ideal gas law. Convert the given temperature from Celsius to Kelvin: \[ T = 25^{\circ} \text{C} + 273.15 = 298.15 \, \text{K} \]

Use the Ideal Gas Law

Apply the ideal gas law, \( PV = nRT \), to find the number of moles \( n \) of helium:\[ n = \frac{PV}{RT} \]Substitute the known values:\[ n = \frac{1.1 \, \text{atm} \times 5.0 \, \text{L}}{0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \times 298.15 \, \text{K}} \]

Calculate the Number of Moles

Calculate \( n \):\[ n = \frac{5.5 \, \text{L} \cdot \text{atm}}{24.4476 \, \text{L} \cdot \text{atm}/ \text{mol}} \approx 0.225 \, \text{mol} \]

Determine the Mass of Helium

Find the mass \( m \) using the molar mass of helium:\[ m = n \times \text{Molar Mass} \]\[ m = 0.225 \, \text{mol} \times 4.00 \, \text{g/mol} = 0.9 \, \text{g} \]

Conclusion Step: State the Result

The mass of helium required to fill the balloon is \( 0.9 \, \text{g} \).

Key concepts

Helium Gas

Helium gas, found in the periodic table as a noble gas, is a colorless, odorless, and tasteless element. Helium is the second lightest and second most abundant element in the observable universe. Due to being light and chemically inert, this gas is widely used in balloons, airships, and cooling systems.
In the context of the ideal gas law, helium behaves ideally under many conditions, making it a great real-world example. Its light molecular weight means it reacts swiftly for applications requiring rapid equilibrium, like scientific experiments and industrial applications.

Molecular Weight Calculation

Molecular weight, or molar mass, is the mass of one mole of a substance, usually measured in grams per mole. This value is crucial in the ideal gas law equation to relate the amount of gas to its mass. For helium, the molar mass is 4.00 g/mol.
This concept helps in translating the amount of substance from moles, calculated using the ideal gas equation, to actual mass. To compute the mass for a given pressure, volume, and temperature, you multiply the number of moles determined by the ideal gas law by the molar mass of helium. The formula is:

• Mass = Moles × Molar Mass

This conversion is fundamental in practical applications like filling balloons, where exact mass measurements are necessary.

Gas Volume and Pressure Calculations

Gas volume and pressure calculations are pivotal elements of the ideal gas law, expressed as \(PV = nRT\), where:
\(P\) = Pressure
\(V\) = Volume
\(n\) = Number of moles
\(R\) = Ideal gas constant
\(T\) = Temperature in Kelvin
These calculations help predict how changes in pressure, volume, or temperature of a gas will affect the others. Given volume, pressure, and temperature, you can determine the amount of gas, vital for understanding how substances behave in different conditions.
For practical scenarios, such as inflating balloons or assessing gas behavior in various pressure systems, understanding this relationship is key. Each variable's modification can reveal insights into the physical state and future reactions of the gas.