Textbook Question
2.0 mol of gas are at 30 °C and a pressure of 1.5 atm. How much work must be done on the gas to compress it to one third of its initial volume at constant pressure?
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Ch 19: Work, Heat, and the First Law of Thermodynamics
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
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Knight Calc 5th EditionCh 19: Work, Heat, and the First Law of ThermodynamicsProblem 52b
Knight Calc 5th EditionCh 19: Work, Heat, and the First Law of ThermodynamicsProblem 52bChapter 19, Problem 52b
An ideal-gas process is described by p=cV1/2, where c is a constant. 0.033 mol of gas at an initial temperature of 150°C is compressed, using this process, from 300 cm3 to 200 cm3. How much work is done on the gas?
Verified step by step guidance1
Step 1: Start by understanding the relationship given in the problem. The pressure is related to the volume by the equation \( p = cV^{\gamma} \), where \( \gamma = \frac{1}{2} \) and \( c \) is a constant. This indicates a polytropic process, where the work done can be calculated using the formula \( W = \frac{p_2V_2 - p_1V_1}{1 - \gamma} \).
Step 2: Convert the given initial temperature from Celsius to Kelvin. Use the formula \( T(K) = T(°C) + 273.15 \). For the initial temperature of 150°C, calculate \( T_1 \) in Kelvin.
Step 3: Use the ideal gas law \( pV = nRT \) to find the initial pressure \( p_1 \). Here, \( n = 0.033 \) mol, \( R = 8.314 \, \text{J/(mol·K)} \), \( T_1 \) is the initial temperature in Kelvin, and \( V_1 = 300 \, \text{cm}^3 = 0.0003 \, \text{m}^3 \). Solve for \( p_1 \).
Step 4: Use the relationship \( p = cV^{\gamma} \) to find the constant \( c \). Substitute \( p_1 \), \( V_1 \), and \( \gamma = \frac{1}{2} \) into the equation \( c = \frac{p_1}{V_1^{\gamma}} \). Then, use this constant to calculate \( p_2 \) at \( V_2 = 200 \, \text{cm}^3 = 0.0002 \, \text{m}^3 \) using \( p_2 = cV_2^{\gamma} \).
Step 5: Substitute \( p_1 \), \( V_1 \), \( p_2 \), \( V_2 \), and \( \gamma = \frac{1}{2} \) into the work formula \( W = \frac{p_2V_2 - p_1V_1}{1 - \gamma} \). Simplify the expression to find the work done on the gas.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ideal Gas Law
The Ideal Gas Law relates the pressure, volume, temperature, and amount of gas through the equation PV=nRT. It is essential for understanding the behavior of gases under various conditions. In this problem, the initial and final states of the gas are influenced by changes in volume and temperature, which can be analyzed using this law.
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Work Done on a Gas
In thermodynamics, the work done on a gas during a compression or expansion process is calculated using the formula W = ∫PdV, where P is the pressure and dV is the change in volume. For this specific process, the pressure is defined by the equation p=cV^2, which must be integrated over the volume change to find the total work done.
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Thermodynamic Processes
Thermodynamic processes describe how a system changes from one state to another, often involving heat transfer and work. In this case, the process is defined by a specific relationship between pressure and volume, indicating a non-isothermal compression. Understanding the type of process helps in applying the correct equations and principles to calculate work and other thermodynamic properties.
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Related Practice
Textbook Question
An ideal-gas process is described by p=cV1/2, where c is a constant. Find an expression for the work done on the gas in this process as the volume changes from V1 to V2.
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Textbook Question
0.25 mol of a gas are compressed at a constant pressure of 250 kPa from 6000 cm3 to 2000 cm3, then expanded at a constant temperature back to 6000 cm3. What is the net work done on the gas?
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Textbook Question
n moles of an ideal gas at temperature T1 and volume V1 expand isothermally until the volume has doubled. In terms of n, T1, and V1, what is the final temperature?
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Textbook Question
n moles of an ideal gas at temperature T1 and volume V1 expand isothermally until the volume has doubled. In terms of n, T1, and V1, what are the work done on the gas?
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Textbook Question
A 10-cm-diameter cylinder contains argon gas at 10 atm pressure and a temperature of 50°C. A piston can slide in and out of the cylinder. The cylinder's initial length is 20 cm. 2500 J of heat are transferred to the gas, causing the gas to expand at constant pressure. What are the final length of the cylinder?