Problem 14 - Physics

Question

A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains \(0.034 \mathrm{~m}^{3}\) of helium at an absolute pressure of \(1.2 \times 10^{5} \mathrm{~Pa}\). The cylinder contains helium at an absolute pressure of \(1.6 \times 10^{7} \mathrm{~Pa}\) and has a volume of \(0.0031 \mathrm{~m}^{3}\). The temperature of the helium in the tank and in the balloons is the same and remains constant. What is the maximum number of balloons that can be filled?

Step-by-Step Solution

Answer

The maximum number of balloons that can be filled is 12.

Step 1: Understand the Problem

We need to find out how many balloons can be filled using the helium in the cylinder. We know the volume and pressure of the helium in both the cylinder and each balloon.

Step 2: Use Ideal Gas Law

The problem assumes constant temperature and ideal gas behavior. Therefore, we can use the relationship \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. Since \(R\) and \(T\) are constant, we can use \(PV\) without directly computing \(n\).

Step 3: Calculate Total Helium in Cylinder

Compute the total pressure-volume product for the gas in the cylinder: \[PV_{\text{cylinder}} = (1.6 \times 10^7 \text{ Pa}) \times (0.0031 \text{ m}^3) = 49600 \text{ Pa} \cdot \text{m}^3\]

Step 4: Calculate Helium per Balloon

Compute the pressure-volume product for one balloon: \[PV_{\text{balloon}} = (1.2 \times 10^5 \text{ Pa}) \times (0.034 \text{ m}^3) = 4080 \text{ Pa} \cdot \text{m}^3.\]

Step 5: Determine Maximum Number of Balloons

Divide the total \(PV_{\text{cylinder}}\) by \(PV_{\text{balloon}}\) to find the number of balloons: \[\text{Number of Balloons} = \frac{49600}{4080} \approx 12.16.\] Since the number of whole balloons is needed, round down to get 12.

Key Concepts

Helium Balloon Calculations

When calculating how many balloons can be filled using helium from a cylinder, an understanding of volumes and pressures is crucial. Helium balloons are filled with a specific volume of gas at a certain pressure. In this exercise, each balloon holds \(0.034 \text{ m}^3\) of helium at an absolute pressure of \(1.2 \times 10^5 \text{ Pa}\).

To find out how many such balloons can be filled from a cylinder, we first need to know how much helium is available in the cylinder. The cylinder in this scenario contains helium at an absolute pressure of \(1.6 \times 10^7 \text{ Pa}\) and has a volume of \(0.0031 \text{ m}^3\). With these quantities, we can calculate the product of pressure and volume (PV), which will help us understand the total amount of helium present.

Exploring these calculations, we begin by determining the PV product for the cylinder:

• PVi.e., \(PV_{\text{cylinder}} = 1.6 \times 10^7 \text{ Pa} \times 0.0031 \text{ m}^3 \)
• This equates to \(49600 \text{ Pa} \cdot \text{m}^3\).

This calculation is key to understanding the total helium available in the cylinder, setting the stage for filling multiple balloons.

Pressure-Volume Relationship

The pressure-volume relationship of a gas is an essential concept when working with gases such as helium. This relationship is rooted in the Ideal Gas Law, represented by the equation \(PV = nRT\). In practical applications like filling balloons, we focus on the product of pressure (P) and volume (V), since temperature (T) and the number of moles (n) are constant.

In our balloon exercise, the constant temperature assumption implies that the total amount of gas (measured in PV) shifts consistently from the cylinder to the balloons.

Let's look at PV for individual balloons:

• Each balloon's PV is computed as: \(PV_{\text{balloon}} = 1.2 \times 10^5 \text{ Pa} \times 0.034 \text{ m}^3\).
• This equates to \(4080 \text{ Pa} \cdot \text{m}^3\).

This indicates that each balloon uses up a fixed PV of 4080. By having the PV values of both the cylinder and a single balloon, we can determine the number of balloons filled by dividing the cylinder's PV by that of one balloon.

Gas Cylinder Problem Solving

Solving problems involving gas cylinders focuses on utilizing the Ideal Gas Law in practical scenarios. In the case of a helium cylinder used to fill balloons, the aim is to maximize the number of balloons from the available gas.

Once we have the pressure-volume values for both the cylinder and individual balloons, finding how many balloons can be filled is straightforward. This is achieved by dividing the cylinder's total PV by the PV of a single balloon:

• Total PV for the cylinder: \(49600 \text{ Pa} \cdot \text{m}^3\)
• PV for one balloon: \(4080 \text{ Pa} \cdot \text{m}^3\)
• The number of balloons is given by \(\frac{49600}{4080} \approx 12.16\)

Since we can't fill a fraction of a balloon, we round down to fill 12 whole balloons.

This process of using PV values is a critical part of gas cylinder problem-solving, making it easier to work out how much gas can be distributed into smaller units or containers like balloons.