Textbook Question
Two moles of an ideal gas occupy a volume . The gas expands isothermally and reversibly to a volume . Is the velocity distribution changed by the isothermal expansion? Explain.
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Ch 20: The Second Law of Thermodynamics
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 20, Problem 31
A lonely party balloon with a volume of L and containing mol of air is left behind to drift in the temporarily uninhabited and depressurized International Space Station. Sunlight coming through a porthole heats and explodes the balloon, causing the air in it to undergo a free expansion into the empty station, whose total volume is m3. Calculate the entropy change of the air during the expansion.
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Understand the concept of entropy change during free expansion: In a free expansion, the gas expands into a vacuum without doing work and without heat exchange. The entropy change is determined by the change in the number of accessible microstates.
Identify the initial and final states of the system: Initially, the air is confined to a volume of 2.40 L. After expansion, it occupies the entire volume of the space station, which is 425 m³. Convert the initial volume to cubic meters for consistency: 2.40 L = 0.00240 m³.
Use the formula for entropy change in free expansion: The entropy change \( \Delta S \) can be calculated using the formula \( \Delta S = nR \ln \left( \frac{V_f}{V_i} \right) \), where \( n \) is the number of moles, \( R \) is the ideal gas constant (8.314 J/mol·K), \( V_f \) is the final volume, and \( V_i \) is the initial volume.
Substitute the known values into the formula: \( n = 0.100 \) mol, \( V_i = 0.00240 \) m³, \( V_f = 425 \) m³, and \( R = 8.314 \) J/mol·K. Calculate the natural logarithm of the volume ratio \( \ln \left( \frac{425}{0.00240} \right) \).
Calculate the entropy change \( \Delta S \) using the substituted values: \( \Delta S = 0.100 \times 8.314 \times \ln \left( \frac{425}{0.00240} \right) \). This will give you the entropy change in joules per kelvin (J/K).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Entropy
Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the amount of energy in a system that is unavailable to do work. During a free expansion, such as the one described in the question, the entropy of the system increases because the gas molecules spread out into a larger volume, increasing disorder.
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Intro to Entropy
Free Expansion
Free expansion is a process where a gas expands into an evacuated space without doing work on the surroundings and without heat exchange. It is an irreversible process, and in an ideal gas, it results in an increase in entropy. The internal energy remains constant because no work is done and no heat is transferred.
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Ideal Gas Law
The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. It describes the behavior of an ideal gas and is used to relate the initial and final states of the gas in processes like expansion, helping to calculate changes in properties such as entropy.
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Related Practice
Textbook Question
You make tea with kg of °C water and let it cool to room temperature (°C). Calculate the entropy change of the water while it cools.
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Textbook Question
A box is separated by a partition into two parts of equal volume. The left side of the box contains molecules of nitrogen gas; the right side contains molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature. On average, how many molecules of each type will there be in either half of the box?
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