Textbook Question
What is the total rotational kinetic energy of 1.0 mol of nitrogen gas at 300 K?
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Ch 20: The Micro/Macro Connection
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 20, Problem 59
A gas of 1.0 x 1020 atoms or molecules has 1.0 J of thermal energy. Its molar specific heat at constant pressure is 20.8 J/ mol K. What is the temperature of the gas?
Verified step by step guidance1
Step 1: Start by identifying the given values in the problem. The number of atoms or molecules is \( N = 1.0 \times 10^{20} \), the thermal energy is \( Q = 1.0 \ \text{J} \), and the molar specific heat at constant pressure is \( C_p = 20.8 \ \text{J/mol·K} \).
Step 2: Recall the relationship between the number of moles \( n \) and the number of particles \( N \). Use the formula \( n = \frac{N}{N_A} \), where \( N_A \) is Avogadro's number \( 6.022 \times 10^{23} \ \text{particles/mol} \).
Step 3: Substitute the given value of \( N \) into the formula for \( n \) to calculate the number of moles of the gas. This will give you \( n \), the amount of substance in moles.
Step 4: Use the formula for thermal energy at constant pressure, \( Q = n C_p \Delta T \), where \( \Delta T \) is the change in temperature. Rearrange this equation to solve for \( \Delta T \): \( \Delta T = \frac{Q}{n C_p} \).
Step 5: Substitute the values of \( Q \), \( n \), and \( C_p \) into the equation for \( \Delta T \) to calculate the temperature change. This will give you the temperature of the gas in Kelvin.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Thermal Energy
Thermal energy refers to the total kinetic energy of the particles in a substance due to their motion. In the context of gases, this energy is related to the temperature and the number of particles present. The relationship between thermal energy, temperature, and the number of particles is crucial for understanding how energy is distributed among the gas molecules.
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Molar Specific Heat
Molar specific heat at constant pressure is the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin while maintaining constant pressure. It is a critical property that helps relate the thermal energy of a gas to its temperature change. In this problem, the specific heat value allows us to calculate the temperature change based on the given thermal energy.
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Ideal Gas Law
The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. Although not directly used in this question, understanding this law provides context for how temperature and energy are interrelated in gases. It helps in comprehending the behavior of gases under various conditions, which is essential for solving thermal energy problems.
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Related Practice
Textbook Question
A nitrogen molecule consists of two nitrogen atoms separated by 0.11 nm, the bond length. Treat the molecule as a rotating dumbbell and find the rms angular velocity at this temperature of a nitrogen molecule around the z-axis, as shown in Figure 20.10.
Textbook Question
2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K. How much heat energy is transferred, and in which direction?
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Textbook Question
A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. What is the final thermal energy of each gas?
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Textbook Question
A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. How much heat energy is transferred, and in which direction?
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Textbook Question
FIGURE P20.57 shows the thermal energy of 0.14 mol of gas as a function of temperature. What is Cᵥ for this gas?