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Chapter 14: Problem 17

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?

Short Answer

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

Step by step solution

01

Understand the Problem

We are given a helium cylinder and need to determine how many balloons can be filled with helium from it. The problem involves pressures and volumes, making it a gas laws problem. Since temperature is constant, we will apply the ideal gas law principle.
02

Calculate Total Moles of Helium in Cylinder

Using the ideal gas law, the number of moles of helium in the cylinder is calculated by \( PV = nRT \). Since temperature and gas constant \( R \) are constant and will cancel out when finding the ratio between the gas in the cylinder and the balloons, we can find the volume or pressure equivalent directly by rearranging the formula to \( n = \frac{PV}{RT} \) .
03

Determine Volume of Helium Transferable to Balloons

The tank pressure initially is \( P_1 = 1.6 \times 10^7 \, \mathrm{Pa} \) and volume \( V_1 = 0.0031 \, \mathrm{m^3} \). Using \( pV \) as an equivalent expression for moles (due to constant temperature), the equivalent volume at balloon pressure \( P_{balloon} = 1.2 \times 10^5 \, \mathrm{Pa} \) is \( V_{balloon} = \frac{P_1 \times V_1}{P_{balloon}} \). Plugging in, \( V_{balloon} = \frac{1.6 \times 10^7 \times 0.0031}{1.2 \times 10^5} \approx 0.4133 \mathrm{m^3} \).
04

Calculate Number of Balloons

Each balloon requires \( 0.034 \, \mathrm{m^3} \) at \( 1.2 \times 10^5 \, \mathrm{Pa} \). Divide the calculated total equivalent volume \( 0.4133 \, \mathrm{m^3} \) by the volume of one balloon: \(\text{Number of Balloons} = \frac{0.4133}{0.034} \approx 12.15\). Since only whole balloons can be filled, round down to 12 balloons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Helium
Helium is a lightweight, colorless, and odorless gas often used for inflating balloons.

It's part of the noble gases group, meaning it doesn't easily react with other elements.

Since helium is less dense than air, it allows balloons to float upward. This property is crucial in many entertaining and scientific applications.
  • Helium is non-flammable, making it safe for both fun and industrial purposes.
  • It has a very low boiling point, so it stays gaseous in normal conditions.
  • In this problem, helium helps us apply the ideal gas law, connecting pressure, volume, and temperature.
Understanding Pressure
Pressure is the force exerted on a surface per unit area. It plays a vital role in how gases behave.

In the context of balloons, pressure determines how much gas can be filled before something bursts.
  • The pressure inside the helium cylinder is very high: 1.6 脳 10鈦� Pa.
  • Each balloon is filled to a pressure of 1.2 脳 10鈦� Pa.
Using the ideal gas law, \( PV = nRT \), we can calculate how much helium transfers between the tank and balloons by comparing pressures. By maintaining constant temperature, we simplify calculations.
Understanding Volume
Volume refers to the amount of space that an object or substance occupies. For gases, volume changes with pressure.

In this context, we're concerned about the volume of helium in the cylinder and balloons.
  • The helium tank has a volume of 0.0031 m鲁.
  • Each balloon has a volume of 0.034 m鲁 when filled.
Using \( V_{balloon} = \frac{P_1 \times V_1}{P_{balloon}} \), we find how much of the helium cylinder's volume can be converted into balloon volume. This allows us to calculate the number of balloons.
Understanding Balloons
Balloons provide a practical application of gas laws, illustrating the relationship between pressure, volume, and temperature.

For a balloon to float and be festive, helium is filled to a specific pressure and volume.
  • We aim to maximize the number of balloons filled from a single cylinder.
  • Each balloon needs a volume of 0.034 m鲁 at a pressure of 1.2 脳 10鈦� Pa.
By calculating the total equivalent volume of the helium cylinder at balloon pressure, we determine how many balloons can be filled. Rounding down to whole numbers gives us the final number of 12 balloons.

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Most popular questions from this chapter

A cylindrical glass beaker of height \(1.520 \mathrm{m}\) rests on a table. The bottom half of the beaker is filled with a gas, and the top half is filled with liquid mercury that is exposed to the atmosphere. The gas and mercury do not mix because they are separated by a frictionless movable piston of negligible mass and thickness. The initial temperature is \(273 \mathrm{K}\). The temperature is increased until a value is reached when one-half of the mercury has spilled out. Ignore the thermal expansion of the glass and the mercury, and find this temperature.

A \(0.030-\mathrm{m}^{3}\) container is initially evacuated. Then, \(4.0 \mathrm{g}\) of water is placed in the container, and, after some time, all the water evaporates. If the temperature of the water vapor is \(388 \mathrm{K},\) what is its pressure?

One assumption of the ideal gas law is that the atoms or molecules themselves occupy a negligible volume. Verify that this assumption is reasonable by considering gaseous xenon (Xe). Xenon has an atomic radius of \(2.0 \times 10^{-10} \mathrm{m} .\) For STP conditions, calculate the percentage of the total volume occupied by the atoms.

A cylindrical glass of water \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) has a radius of \(4.50 \mathrm{cm}\) and a height of \(12.0 \mathrm{cm} .\) The density of water is \(1.00 \mathrm{g} / \mathrm{cm}^{3} .\) How many moles of water molecules are contained in the glass?

The pressure of sulfur dioxide \(\left(\mathrm{SO}_{2}\right)\) is \(2.12 \times 10^{4} \mathrm{Pa} .\) There are 421 moles of this gas in a volume of \(50.0 \mathrm{m}^{3} .\) Find the translational rms speed of the sulfur dioxide molecules.
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