Textbook Question
A gas at 100°C fills volume V₀. If the pressure is held constant, what is the volume if the Kelvin temperature is doubled?
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Ch 18: A Macroscopic Description of Matter
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 18, Problem 25c
A 20-cm-diameter cylinder that is 40 cm long contains 50 g of oxygen gas at 20°C. What is the number density of the oxygen?
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Step 1: Start by understanding the concept of number density. Number density is defined as the number of particles (in this case, molecules of oxygen) per unit volume. Mathematically, it is given by \( n = \frac{N}{V} \), where \( n \) is the number density, \( N \) is the total number of molecules, and \( V \) is the volume of the container.
Step 2: Calculate the volume of the cylinder. The formula for the volume of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length) of the cylinder. Convert the diameter (20 cm) to radius (10 cm) and the height (40 cm) to meters before substituting into the formula.
Step 3: Use the molar mass of oxygen to determine the number of moles of oxygen gas. The molar mass of oxygen (O₂) is approximately 32 g/mol. The number of moles \( n_{\text{moles}} \) is given by \( n_{\text{moles}} = \frac{m}{M} \), where \( m \) is the mass of the gas (50 g) and \( M \) is the molar mass.
Step 4: Use Avogadro's number to find the total number of molecules \( N \). Avogadro's number is \( 6.022 \times 10^{23} \) molecules/mol. Multiply the number of moles \( n_{\text{moles}} \) by Avogadro's number to get \( N \).
Step 5: Finally, calculate the number density \( n \) by dividing the total number of molecules \( N \) by the volume \( V \) of the cylinder. Ensure that the units are consistent (e.g., volume in cubic meters) to obtain the correct number density in molecules per cubic meter.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Number Density
Number density is defined as the number of particles (atoms, molecules, etc.) per unit volume. It is typically expressed in units such as particles per cubic meter (m³). In this context, calculating the number density of oxygen gas involves determining how many molecules are present in the given volume of the cylinder.
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Ideal Gas Law
The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding the behavior of gases under various conditions and can be used to find the number of moles of oxygen in the cylinder, which is necessary for calculating number density.
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Volume of a Cylinder
The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height. In this problem, the diameter of the cylinder is given, allowing us to find the radius and subsequently the volume, which is crucial for determining the number density of the oxygen gas contained within.
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Related Practice
Textbook Question
A 20-cm-diameter cylinder that is 40 cm long contains 50 g of oxygen gas at 20°C. What is the reading of a pressure gauge attached to the tank?
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Textbook Question
The total lung capacity of a typical adult is 5.0 L. Approximately 20% of the air is oxygen. At sea level and at a body temperature of 37°C, how many oxygen molecules do the lungs contain at the end of a strong inhalation?
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Textbook Question
A gas with initial state variables p1, V1, and T1 is cooled in an isochoric process until p2 = (1/3)p1. What are V2?
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Textbook Question
A gas with initial state variables p1, V1, and T1 expands isothermally until V2 = 2V1. What are T1?
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Textbook Question
The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about 2×106 K. The gas pressure in the corona is about 0.03 Pa. Estimate the number density of particles in the solar corona.
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