Question

A 15.0-L tank is filled with helium gas at a pressure of \(1.00 \times 10^{2}\). How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(1.00 \mathrm{~atm}\), assuming that the temperature remains constant and that the tank cannot be emptied below \(1.00 \mathrm{~atm}\) ?

Short answer

Using Boyle's Law, we can calculate that 7 balloons can be inflated to a pressure of \(1.00 \mathrm{~atm}\) using the given quantity of helium gas in the 15.0-L tank at a pressure of \(1.00 \times 10^{2} \mathrm{~atm}\), without depleting the tank below \(1.00 \mathrm{~atm}\).

Step-by-step solution

1. Initial conditions

We have an initial volume, \(V_1=15.0 \mathrm{~L}\), and initial pressure, \(P_1= 1.00 \times 10^2 \mathrm{~atm}\). The maximum allowable final pressure is \(P_2=1.00 \mathrm{~atm}\).

2. Calculate final volume

As per Boyle's Law, \(P_1V_1=P_2V_2\). We will use this equation to find the final volume, \(V_2\). \(V_2 = \frac{P_1V_1}{P_2}\) Plug in the given values and calculate the final volume: \(V_2 = \frac{(1.00 \times 10^2)(15.0)}{1.00}\) \(V_2 = 1500 \mathrm{~L}\)

3. Calculate the volume of used gas

Next, we need to calculate the volume of gas used to inflate the balloons. This is the difference between the initial and final volume. \(\Delta V = V_1 - V_2 = 15.0 \mathrm{~L} - 1.00 \mathrm{~atm} = 14.0 \mathrm{~L}\)

4. Calculate the number of balloons

Now we will calculate the number of 2.00-L balloons that can be filled with 14.0 L of gas. Number of balloons = \(\frac{\Delta V}{Volume~of~each~balloon}\) Number of balloons = \(\frac{14.0 \mathrm{~L}}{2.00 \mathrm{~L}} = 7\)

Answer

Thus, 7 balloons can be inflated to a pressure of \(1.00 \mathrm{~atm}\) using the given quantity of helium gas.

Key concepts

Helium Gas

Helium is a fascinating element and plays a crucial role in various applications due to its unique properties. It is a noble gas with an atomic number of 2. Helium is colorless, odorless, and non-toxic, making it safe to use in many applications, such as in balloons, airships, and cooling systems. One of its most well-known features is its lighter-than-air property, which is why balloons filled with helium float. Its inertness means that helium does not react with other substances, maintaining stability in different environments. These properties make helium an excellent choice for inflating balloons, as shown in the exercise where it fills up a tank initially holding a specific volume at high pressure.

Pressure

Pressure is a fundamental concept in gas laws and is defined as the force exerted by the gas particles per unit area of the container they are in. It's typically measured in units like atmospheres (atm), pascals (Pa), or torrs. In the original exercise, pressure plays a key role as it helps determine how much helium can be used from the tank to inflate the balloons. The tank starts at a pressure of \(1.00 \times 10^2 \text{ atm}\), indicating how compressed the helium is initially. Understanding how to manage and calculate pressure differences using Boyle's Law allows us to predict how gases behave under different conditions without changing the temperature, a vital aspect in gas-related calculations like those for inflating balloons.

Volume

Volume refers to the space that a substance, like a gas, occupies. In gas laws, volume changes depending on the pressure exerted on the gas. The exercise illustrates this by starting with a given volume of helium in a tank, which is later used to calculate how much space or volume the helium will occupy after being used to fill balloons. According to Boyle's Law, if the temperature remains constant, increasing the pressure on a gas decreases its volume, and vice versa. For the given problem, the volume of helium that can be used is determined by calculating the initial and final volumes when helium is released at a reduced pressure, showing how pressure and volume interplay influences the ability to inflate a certain number of balloons.

Gas Laws

Gas laws are essential principles that describe the behavior of gases in various conditions. Boyle's Law is one of the fundamental gas laws, primarily used in this exercise. It states that for a given amount of gas at a constant temperature, the pressure and volume are inversely proportional. Mathematically, it is represented as \( P_1V_1 = P_2V_2 \).

In the exercise, Boyle's Law is used to determine the volume of helium available for inflating balloons when the pressure changes. By knowing the initial and final pressures and using Boyle's Law, we can calculate how the volume will change. Understanding these gas laws is crucial for solving problems involving gas compression or expansion, ensuring accurate and efficient results.