Textbook Question
An experimenter adds J of heat to mol of an ideal gas to heat it from °C to °C at constant pressure. The gas does J of work during the expansion. Calculate for the gas.
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Ch 19: The First Law of Thermodynamics
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors37
- Ch 02: Motion Along a Straight Line57
- Ch 03: Motion in Two or Three Dimensions45
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws54
- Ch 06: Work & Kinetic Energy11
- Ch 07: Potential Energy & Conservation6
- Ch 08: Momentum, Impulse, and Collisions15
- Ch 09: Rotation of Rigid Bodies37
- Ch 10: Dynamics of Rotational Motion44
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics27
- Ch 13: Gravitation34
- Ch 14: Periodic Motion26
- Ch 15: Mechanical Waves22
- Ch 16: Sound & Hearing28
- Ch 17: Temperature and Heat28
- Ch 18: Thermal Properties of Matter35
- Ch 19: The First Law of Thermodynamics29
- Ch 20: The Second Law of Thermodynamics18
- Ch 21: Electric Charge and Electric Field28
- Ch 22: Gauss' Law21
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF24
- Ch 26: Direct-Current Circuits29
- Ch 27: Magnetic Field and Magnetic Forces16
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction22
- Ch 30: Inductance14
- Ch 31: Alternating Current20
- Ch 33: The Nature and Propagation of Light17
- Ch 34: Geometric Optics29
- Ch 35: Interference13
- Ch 36: Diffraction19
- Ch 37: Special Relativity19
- Ch 38: Photons: Light Waves Behaving as Particles12
- Ch 39: Particles Behaving as Waves25
- Ch 40: Quantum Mechanics I: Wave Functions20
- Ch 41: Quantum Mechanics II: Atomic Structure21
- Ch 42: Molecules and Condensed Matter17
- Ch 43: Nuclear Physics13
- Ch 44: Particle Physics and Cosmology17
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
Chapter 19, Problem 27a
A monatomic ideal gas that is initially at Pa and has a volume of m3 is compressed adiabatically to a volume of m3. What is the final pressure?
Verified step by step guidance1
Understand that the process is adiabatic, meaning no heat is exchanged with the surroundings. For an adiabatic process involving an ideal gas, the relation between pressure and volume is given by the equation: \( P_1 V_1^\gamma = P_2 V_2^\gamma \), where \( \gamma \) is the adiabatic index (ratio of specific heats, \( C_p/C_v \)). For a monatomic ideal gas, \( \gamma = \frac{5}{3} \).
Identify the known values: initial pressure \( P_1 = 1.50 \times 10^5 \) Pa, initial volume \( V_1 = 0.0800 \) m\(^3\), and final volume \( V_2 = 0.0400 \) m\(^3\). The final pressure \( P_2 \) is what we need to find.
Substitute the known values into the adiabatic equation: \( 1.50 \times 10^5 \times (0.0800)^{\frac{5}{3}} = P_2 \times (0.0400)^{\frac{5}{3}} \).
Solve for \( P_2 \) by isolating it on one side of the equation: \( P_2 = \frac{1.50 \times 10^5 \times (0.0800)^{\frac{5}{3}}}{(0.0400)^{\frac{5}{3}}} \).
Calculate the expression to find the final pressure \( P_2 \). Ensure to handle the powers and division correctly to arrive at the numerical value for \( P_2 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Adiabatic Process
An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. For an ideal gas, this means that any change in internal energy is due solely to work done on or by the system. In an adiabatic compression, the gas's temperature and pressure increase as the volume decreases.
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Entropy & Ideal Gas Processes
Ideal Gas Law
The ideal gas law is a fundamental equation that relates the pressure, volume, and temperature of an ideal gas through the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps in understanding how gases behave under different conditions, although it assumes no interactions between gas molecules.
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Adiabatic Equation for Ideal Gases
For an adiabatic process involving an ideal gas, the relationship between pressure and volume is given by the equation PV^γ = constant, where γ (gamma) is the heat capacity ratio (Cp/Cv). This equation allows us to calculate the final pressure of the gas after adiabatic compression or expansion, given the initial conditions and the final volume.
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Related Practice
Textbook Question
A monatomic ideal gas that is initially at Pa and has a volume of m3 is compressed adiabatically to a volume of m3. What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?
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Textbook Question
Five moles of monatomic ideal gas have initial pressure Pa and initial volume m3. While undergoing an adiabatic expansion, the gas does J of work. What is the final pressure of the gas after the expansion?
Textbook Question
A player bounces a basketball on the floor, compressing it to of its original volume. The air (assume it is essentially N2 gas) inside the ball is originally at °C and atm. The ball's inside diameter is cm. What temperature does the air in the ball reach at its maximum compression? Assume the compression is adiabatic and treat the gas as ideal.
Textbook Question
The temperature of mol of an ideal gas is held constant at °C while its volume is reduced to of its initial volume. The initial pressure of the gas is atm. Does the gas exchange heat with its surroundings? If so, how much? Does the gas absorb or liberate heat?
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Textbook Question
Heat flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?
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