Question

A 6.0-L tank is filled with helium gas at a pressure of 2 MPa. How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(101.3 \mathrm{kPa}\), assuming that the temperature remains constant and that the tank cannot be emptied below \(101.3 \mathrm{kPa}\) ?

Short answer

56 balloons can be inflated to a pressure of 101.3 kPa using the helium gas in the 6.0-L tank, without depleting the tank's pressure below 101.3 kPa.

Step-by-step solution

Write down the given information

We are given: - Initial volume of the gas in the tank (V1) = 6.0 L - Initial pressure of the gas in the tank (P1) = 2 MPa = 2000 kPa - Volume of each balloon (V2) = 2.00 L - Pressure of gas in each balloon (P2) = 101.3 kPa Since the temperature remains constant, we will use Boyle's Law, which states that the product of the initial pressure and volume of gas is equal to the final pressure and volume (P1 × V1 = P2 × V2).

Calculate the maximum number of balloons that can be inflated

To calculate the maximum number of balloons, we will first determine the number of balloons that can be inflated without depleting the tank's pressure to below 101.3 kPa. First, calculate the final volume of the gas remaining in the tank (Vf = V1 × (P1 / P2)): \(V_f = V_1 \times (\frac{P_1}{P_2})\) Plugging in the given values: \(V_f = 6.0\,\text{L} \times (\frac{2000\text{kPa}}{101.3\text{kPa}})\) \(V_f \approx 118.56\,\text{L}\) Since the tank cannot deplete below 101.3 kPa, we need to ensure that at least 6 L of gas remains in the tank. So, the available volume of gas that can be used to inflate the balloons is: \(V_\text{usable} = V_f - V_1 = 118.56\,\text{L} - 6.0\,\text{L} = 112.56\,\text{L}\) Now, divide the available volume of gas by the volume of each balloon to find the number of balloons that can be inflated: \(\text{Number of balloons} = \frac{V_\text{usable}}{V_2} = \frac{112.56\,\text{L}}{2.00\,\text{L}} \approx 56.28\)

Round down to the nearest whole number

Since we cannot inflate a fraction of a balloon, we round down to the nearest whole number: Number of balloons = 56 Hence, 56 balloons can be inflated to a pressure of 101.3 kPa using the helium gas in the 6.0-L tank, without depleting the tank's pressure below 101.3 kPa.

Key concepts

Understanding Gas Laws

Gas laws are fundamental principles that help us understand how gases behave under different conditions of pressure, volume, and temperature. One of the most important gas laws is Boyle's Law. It states that the pressure of a gas is inversely proportional to its volume when temperature is held constant. This means that if you increase the pressure exerted on a gas, its volume decreases, and vice versa, as long as the temperature doesn't change.
These concepts are essential for solving problems that involve gases being distributed or compressed, like inflating balloons. By applying these laws, you can predict how a gas will behave when moved from one container to another or when changing its pressure conditions.

Properties of Helium Gas

Helium is an interesting gas with unique properties. It is a noble gas, which means it is colorless, odorless, tasteless, non-toxic, and chemically inactive. This makes helium perfect for inflating balloons because it hardly reacts with anything and provides lift due to its low density compared to air.
Given these properties, helium is often used not just for balloons in festive settings but also for scientific purposes and medical equipment. It's important to consider these properties when working with helium because they directly affect how you manage and use the gas, such as storing it under pressure and ensuring it retains its purity during transfer from tanks to balloons.

Exploring Pressure Volume Relationship

The relationship between pressure and volume in gases is crucial for various practical applications. According to Boyle's Law, this relationship is an inverse one, expressed mathematically as \(P_1 \times V_1 = P_2 \times V_2\). This equation shows that if you know the pressure and volume of a gas initially, you can calculate what one of those parameters will be if the other changes, provided temperature stays constant.

• Initial pressure and volume refer to the starting conditions of the gas.
• Final pressure and volume denote the conditions the gas reaches after a change.

In our example of inflating balloons, understanding this relationship allows us to determine how much gas can be moved from a tank to the balloons without losing pressure below a set safe level.

Inflation of Balloons

Inflating balloons with helium involves understanding both the properties of helium and the gas laws that dictate how it behaves when compressed or expanded. The goal is to move gas from a dense storage tank to an expanded balloon at a lower pressure. This process requires determining how many balloons can be filled without emptying the tank below a critical pressure threshold, crucial for safety and efficiency.
Generally, when inflating balloons, you must account for:

• The volume of the balloon, which dictates how much gas is needed for adequate inflation.
• The desired pressure in the balloon, which influences its buoyancy and strength.
• Ensuring that enough gas remains in the tank to maintain pressure and prevent damage.

By understanding and applying Boyle's Law, one can appropriately manage the inflation process, ensuring all balloons are safely filled to their capacity without unnecessarily exhausting the tank.