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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 121.

Step by step solution

01

Understand the Problem

We need to find out the maximum number of balloons that can be filled with helium from a cylinder. We know the volume and pressure of helium in each balloon and the volume and pressure of helium in the cylinder.
02

Apply Boyle's Law for Gases

Use Boyle's Law, which states that for a constant temperature, \(P_1V_1 = P_2V_2\), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume. Here, \(P_1 = 1.6 \times 10^7\, \text{Pa}\), \(V_1 = 0.0031\, \text{m}^3\); \(P_2 = 1.2 \times 10^5\, \text{Pa}\).
03

Calculate Total Volume of Helium at Balloon Pressure

Rearrange Boyle's Law to find \(V_2\): \[V_2 = \frac{P_1 \times V_1}{P_2}\] \[V_2 = \frac{1.6 \times 10^7 \times 0.0031}{1.2 \times 10^5}\] \[V_2 \] is the total volume of helium available to fill balloons at the pressure of \(1.2 \times 10^5\, \text{Pa}\).
04

Perform the Calculation

Calculate \(V_2\): \[V_2 = \frac{1.6 \times 10^7 \times 0.0031}{1.2 \times 10^5} = \frac{4.96 \times 10^4}{1.2 \times 10^5} \approx 4.13\, \text{m}^3\]
05

Calculate Maximum Number of Balloons

Divide the total volume \(V_2\) by the volume of one balloon: \[\text{Number of balloons} = \frac{4.13}{0.034} \approx 121\]
06

Conclude the Solution

The maximum number of balloons that can be filled is approximately 121.

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Key Concepts

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

Ideal Gas Law Applications
The ideal gas law is a pivotal concept when dealing with gases. It incorporates key properties such as pressure, volume, and temperature to describe a gas's state. Although the exercise uses Boyle's Law, which is a variant applicable under constant temperature, understanding the ideal gas law helps know why conditions are kept constant in some scenarios. The ideal gas law is expressed as:
\[ PV = nRT \]
where:
  • \( P \) is the pressure
  • \( V \) is the volume
  • \( n \) is the amount of substance of gas (measured in moles)
  • \( R \) is the ideal gas constant
  • \( T \) is the temperature
Applications of the ideal gas law enable us to predict how a gas will behave under different pressures and volumes, granting us insights into various everyday and scientific tasks. This application is crucial in fields ranging from meteorology to engineering. It simplifies calculating gas behaviors when the temperature is a variable we can keep stable.
Pressure and Volume Relationship
Boyle's Law highlights the intrinsic relationship between pressure and volume in a gas kept at constant temperature. It is a specific case of the more general ideal gas law, where the variables of interest are pressure and volume. Boyle's Law can be mathematically represented as:
\[ P_1V_1 = P_2V_2 \]
According to Boyle's Law:
  • If the volume of a gas increases, its pressure decreases.
  • Conversely, if the volume decreases, the pressure increases.
This inverse relationship between pressure and volume is crucial when compressing and expanding gases, as seen in the helium cylinder and balloon exercise. Understanding this concept allows us to predict changes in gas behaviors in industries like automotive (in shock absorbers) and even in balloons, where helium's volume changes based on surrounding pressure.
Gas Laws in Physics
Gas laws like Boyle's Law and the ideal gas law are foundational in physics for describing the behavior of gases under varying conditions. These laws facilitate a deeper understanding of how gases interact with their surroundings and can be utilized in many scientific and engineering applications. Some key laws include:
  • **Boyle's Law:** Pressure is inversely proportional to volume at a constant temperature.
  • **Charles's Law:** Volume is directly proportional to temperature at constant pressure.
  • **Avogadro's Law:** Volume is directly proportional to the number of gas moles at constant temperature and pressure.
These gas laws provide simplified models for complex gas behaviors and are indispensable tools for scientists. They offer explanations for various phenomena, such as why hot air balloons rise or why soda cans might burst when left in a hot environment. Understanding these basic principles allows predictions about gas reactions to pressure changes, temperature shifts, and more, offering clear and concise guidelines for anticipating a gas's behavior in closed systems.

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Most popular questions from this chapter

The preparation of homeopathic "remedies" involves the repeated dilution of solutions containing an active ingredient such as arsenic trioxide \(\left(\mathrm{As}_{2} \mathrm{O}_{3}\right) .\) Suppose one begins with \(18.0 \mathrm{g}\) of arsenic trioxide dissolved in water, and repeatedly dilutes the solution with pure water, each dilution reducing the amount of arsenic trioxide remaining in the solution by a factor of \(100 .\) Assuming perfect mixing at each dilution, what is the maximum number of dilutions one may perform so that at least one molecule of arsenic trioxide remains in the diluted solution? For comparison, homeopathic "remedies" are commonly diluted 15 or even 30 times.

You and your team have been given the task of preparing a gas piston device that is to be deployed to the bottom of the Southern Sea, beneath the Brunt Ice Shelf off the coast of Antarctica. The inner volume of the cylinder has a diameter of \(9.00 \mathrm{cm}\) and a length of \(25.0 \mathrm{cm},\) and is fitted with a movable piston that compresses the argon (Ar) gas that it contains. Before deployment, the piston is located and held at the very top of the cylinder, providing the maximum volume for the gas. When the device is submerged, however, the piston will move inward due to the water pressure created by the depth, and the change in temperature of the gas. The device is to be taken to a depth of \(350 \mathrm{m}\) in salt water (density \(\rho=1.03\) \(\left.\mathrm{g} / \mathrm{cm}^{3}\right)\) at a temperature of \(33.0^{\circ} \mathrm{F} .\) If the maximum allowable compression distance of the piston is \(13.0 \mathrm{cm},\) what is the minimum gas pressure that must be attained when the device is filled at the surface (at atmospheric pressure and \(T=59.0^{\circ} \mathrm{F}\) ) to prevent the maximum displacement of the piston from exceeding its maximum ( \(13.0 \mathrm{cm}\) ) when submerged to a depth of \(350 \mathrm{m} ?\)

When a gas is diffusing through air in a diffusion channel, the diffusion rate is the number of gas atoms per second diffusing from one end of the channel to the other end. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. The atomic mass of ideal gas \(A\) is \(1.0 \mathrm{u},\) and that of ideal gas \(\mathrm{B}\) is 2.0 u. For diffusion through the same channel under the same conditions, find the ratio of the diffusion rate of gas \(A\) to the diffusion rate of gas \(B\).

Near the surface of Venus, the rms speed of carbon dioxide molecules \(\left(\mathrm{CO}_{2}\right)\) is \(650 \mathrm{m} / \mathrm{s} .\) What is the temperature (in kelvins) of the atmosphere at that point?

The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is \(A=2.50 \times 10^{-3} \mathrm{m}^{2}\) The initial pressure, volume, and temperature of the gas are, respectively, \(P_{0}, V_{0}=6.00 \times 10^{-4} \mathrm{m}^{3},\) and \(T_{0}=273 \mathrm{K}\) and the spring is initially stretched by an amount \(x_{0}=\) \(0.0800 \mathrm{m}\) with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are \(P_{\mathrm{f}}, V_{\mathrm{f}},\) and \(T_{\mathrm{f}},\) and the spring is stretched by an amount \(x_{\mathrm{f}}=0.1000 \mathrm{m}\) with respect to its unstrained length. What is the final temperature of the gas?
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