Textbook Question
Propane gas (C3H8) behaves like an ideal gas with . Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.
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Ch 19: The First 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 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
The engine of a Ferrari F355 F1 sports car takes in air at °C and atm and compresses it adiabatically to times the original volume. The air may be treated as an ideal gas with . Find the final temperature and pressure.
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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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