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

Answer: The volume of the tank is approximately 1.494 cubic feet.

Step-by-step solution

Convert pressures to pascal (Pa)

First, we need to convert both initial pressure in psi and atmospheric pressure to pascals (Pa). 1 psi ≈ 6894.76 Pa Initial pressure (\(P_1\)) = 2400 psi = 2400 * 6894.76 = 16,547,424 Pa Atmospheric pressure (\(P_2\)), assuming sea level, is approximately 101,325 Pa.

Use Boyle's Law equation to find the volume of the tank

Now, we can plug the values into the Boyle's Law equation: \(P_1V_1 = P_2V_2\) \(16,547,424V_1 = 101,325 * 244\) To find the volume of the tank (\(V_1\)), we just need to divide both sides by the initial pressure (\(P_1=16,547,424\)): \(V_1 = \frac{101,325 * 244}{16,547,424}\)

Calculate the volume of the tank

Now, we can calculate the volume of the tank: \(V_1 \approx \frac{24,723,300}{16,547,424} \approx 1.494 \,\text{ft}^{3}\) The volume of the tank is approximately 1.494 cubic feet.

Key concepts

Boyle's Law in Physics

Understanding Boyle's Law is crucial when dealing with problems related to gas behavior under pressure. This fundamental law dictates that for a given mass of gas, at a constant temperature, the pressure and volume of the gas are inversely proportional. In simpler terms, if you increase the pressure on a gas, its volume will decrease, and if you decrease the pressure, the volume will increase, as long as the temperature remains unchanged.

Boyle's Law is mathematically expressed as: \[ P_1V_1 = P_2V_2 \
\]where \(P_1\) and \(V_1\) represent the initial pressure and volume, and \(P_2\) and \(V_2\) represent the final pressure and volume. This relationship is foundational not only in physics but also in chemistry and engineering disciplines, effectively predicting how gases will behave when compressed or allowed to expand.

Gas Laws

Gas laws are a set of scientifically established principles that describe the behavior of gases and how variables such as temperature, volume, and pressure are interrelated. Aside from Boyle's Law, other important gas laws include Charles's Law, which deals with volume and temperature, and Gay-Lussac's Law, which connects pressure and temperature. Additionally, the Combined Gas Law integrates Boyle's, Charles's, and Gay-Lussac’s laws into a single formula to make calculations involving simultaneous changes in pressure, volume, and temperature more manageable.

For comprehensive understanding, the Ideal Gas Law further combines these individual laws, adding the amount of gas into consideration with the gas constant. This law is essential in predicting the behavior of gases under various conditions and is expressed by the equation \[ PV = nRT \
\]where \(n\) is the number of moles of gas and \(R\) is the universal gas constant. These laws are cornerstone concepts for students of physical sciences and are critical for practical applications like HVAC system design, automotive engineering, and even space exploration.

Pressure-Volume Relationship

At the heart of Boyle's Law lies the pressure-volume relationship. This relationship implies that if you alter the pressure exerted on a confined gas, its volume will respond accordingly, provided the temperature doesn’t change. In the example of the helium tank, as the gas is moved from a high-pressure tank to the lower-pressure atmosphere, its volume increases substantially.

To further illustrate, imagine squeezing a balloon. The decrease in volume leads to an increase in pressure inside the balloon. When you stop squeezing, the volume increases and the pressure decreases. Understanding this relationship not only helps in solving textbook problems but also offers practical insight into how everyday objects around us work, such as air pumps, syringes, and even our own lungs as we breathe. As an educational tip, always make sure to account for temperature as a constant when applying Boyle’s Law; any variation could shift the dynamics of the pressure-volume relationship and make Boyle's Law inapplicable.