Question

A tank of compressed helium for inflating balloons is advertised as containing helium at a pressure of 2400 . psi, which, when allowed to expand at atmospheric pressure, will occupy a volume of \(244 \mathrm{ft}^{3}\). Assuming no temperature change takes place during the expansion, what is the volume of the tank in cubic feet? \(?\) ?

Short answer

Based on the given information and using Boyle's Law, the volume of the tank containing compressed helium is approximately 1.49 cubic feet.

Step-by-step solution

Convert pressure units

First, we need to convert the given pressure of the compressed helium from psi (pounds per square inch) to atm (atmosphere) for consistency. We know that 1 atm = 14.7 psi. Therefore, \(P_1 = \frac{2400 \ \text{psi}}{14.7 \ \frac{\text{psi}}{\text{atm}}} = 163.3 \ \text{atm}\) We also know that atmospheric pressure, \(P_2\), is 1 atm.

Apply Boyle's Law

Applying Boyle's Law, we have: \(P_1V_1 = P_2V_2\) Substitute the values of \(P_1\), \(P_2\), and \(V_2\): \(163.3 \ \text{atm} \cdot V_1 = 1 \ \text{atm} \cdot 244 \ \mathrm{ft}^{3}\)

Solve for the volume of the tank (\(V_1\))

Now, we'll isolate \(V_1\) by dividing both sides of the equation by 163.3 atm: \(V_1 = \frac{1 \ \text{atm} \cdot 244 \ \mathrm{ft}^{3}}{163.3 \ \text{atm}}\) Calculating this, we get: \(V_1 = 1.49 \ \mathrm{ft}^{3}\) The volume of the tank is approximately 1.49 cubic feet.

Key concepts

Pressure Unit Conversion

Understanding pressure unit conversion is essential when working with Boyle's Law, as it often requires consistent units to correctly apply the law. In the realm of physics and engineering, pressure can be measured in various units including pounds per square inch (psi), atmospheres (atm), pascals (Pa), and bars, among others. For our purposes, the conversion between psi and atm is crucial.
When solving the problem regarding a tank of helium, we start by converting the pressure from psi to atm. Given that 1 atm is equal to 14.7 psi, you can convert psi to atm by dividing the pressure in psi by 14.7. This step standardizes pressure units, enabling us to apply Boyle's Law accurately. Accurate conversions are important because they ensure the reliability of the law's application and the correctness of the results.

Boyle's Law Application

Boyle's Law is a fundamental concept in the gas laws of physics, expressing the inverse relationship between the pressure and volume of a gas at constant temperature. This means that as pressure increases, volume decreases, and vice versa, assuming the amount of gas and temperature remains unchanged.
In practical application, let's consider a tank of helium used for inflating balloons. The law is used to find the volume of the gas inside the tank at a different pressure. If we know the volume the gas will occupy at atmospheric pressure, and the pressure at which it's initially stored, we can rearrange Boyle's Law to solve for the unknown volume.
This is precisely what was done in the solution provided—it demonstrates the computation of the original volume of the compressed helium tank using Boyle's Law after the necessary pressure unit conversion. Such practical applications of Boyle's Law are abundant in various fields, including respiratory medicine, scuba diving, and the design of pressure vessels.

Gas Laws in Physics

Gas laws in physics are empirically derived principles that describe how gases behave under varying conditions of temperature, pressure, and volume. Boyle's Law is just one piece of this larger puzzle. Alongside Charles's Law, Gay-Lussac's Law, and Avogadro's Law, it forms the combined gas law, which is a cornerstone principle in chemical and physical calculations.
In the case of the helium tank, we primarily focus on Boyle's Law, but understanding how it fits within the full scope of gas laws is beneficial for comprehending the behavior of gases. These laws are crucial not just in academic exercises, but also in real-world applications such as determining the lift capacity of balloons, the expansion of air in tires, and even in the calculation of the correct gas mixtures for safe scuba diving.
Becoming familiar with these gas laws, their interrelations, and their applications allow for a deeper understanding of physical phenomena and how matter behaves under various environmental conditions. It also emphasizes the importance of mastering pressure unit conversions and their role in the application of gas laws like Boyle's Law.