How to Calculate the Volume of a Helium-Filled Balloon Using the Ideal Gas Law

When dealing with gases at different pressures and temperatures, you might need to determine the volume of a gas-filled balloon. For instance, consider a balloon that holds 26 kg of helium with a pressure of 1.15 ATM and a temperature of 23 degrees Celsius. In this article, we will walk you through the process of calculating the volume of the balloon using the ideal gas law.

Understanding the Ideal Gas Law

The ideal gas law describes the behavior of an ideal gas and is represented by the equation:

[ PV nRT ]

where:

P is the pressure of the gas, V is the volume of the gas, n is the number of moles of the gas, R is the Universal Gas Constant, T is the temperature of the gas in Kelvin.

Step-by-Step Calculation

Let's apply the ideal gas law to calculate the volume of the balloon holding 26 kg of helium with the given conditions.

Given Data:

P 1.15 ATM mHe (mass of helium) 26 kg 26,000 g (1 kg 1,000 g) MHe (molar mass of helium) 4.0026 g/mol R 0.082057 L·ATM/(K·mol) T 23°C 23 273.15 296.15 K

Unknown:

(V)

First, let's convert the mass of helium to moles:

[ n frac{m}{M} frac{26,000 text{ g}}{4.0026 text{ g/mol}} 6,500 text{ moles} ]

Now, use the ideal gas law formula to solve for (V>:

[ V frac{nRT}{P} ]

Substitute the known values:

[ V frac{6,500 text{ moles} times 0.082057 text{ L·ATM/(K·mol)} times 296.15 text{ K}}{1.15 text{ ATM}} ]

Perform the multiplication and division:

[ V frac{157,330 text{ L·ATM·K}}{1.15 text{ ATM}} 136,813 text{ L} approx 137,000 text{ L} ]

Therefore, the volume of the balloon is approximately 137,000 liters or 137.367 m3.

Why Use This Calculation?

Knowing how to calculate the volume of a gas-filled balloon using the ideal gas law is crucial in various applications, such as balloon design, scientific experiments, and even in understanding the behavior of gases. This method is particularly useful for ensuring that a balloon can hold the desired volume of gas at the specified pressure and temperature.

Conclusion

By applying the ideal gas law, we can determine the volume of a helium-filled balloon accurately. This article has provided a step-by-step guide to performing the calculation, making it a valuable resource for students, scientists, and enthusiasts who want to understand and use the ideal gas law in practical scenarios.