Textbook Question
How much heat does it take to increase the temperature of mol of an ideal gas by K near room temperature if the gas is held at constant volume and is diatomic?
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Ch 18: Thermal Properties of Matter
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 18, Problem 41a
For diatomic carbon dioxide gas (CO2, molar mass g/mol) at K, calculate the most probable speed .
Verified step by step guidance1
First, understand that the most probable speed \( v_{mp} \) for a gas molecule is given by the formula \( v_{mp} = \sqrt{\frac{2kT}{m}} \), where \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \text{ J/K}) \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule.
Convert the molar mass of CO2 from grams per mole to kilograms per molecule. Since the molar mass is 44.0 g/mol, convert this to kg/mol by dividing by 1000, resulting in 0.044 kg/mol. Then, divide by Avogadro's number \( (6.022 \times 10^{23} \text{ molecules/mol}) \) to find the mass of a single molecule.
Substitute the values into the formula for \( v_{mp} \). Use \( T = 300 \text{ K} \) and the calculated mass of a single CO2 molecule.
Calculate the expression \( \frac{2kT}{m} \) using the values for \( k \), \( T \), and \( m \).
Finally, take the square root of the result from the previous step to find the most probable speed \( v_{mp} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Maxwell-Boltzmann Distribution
The Maxwell-Boltzmann distribution describes the distribution of speeds among particles in a gas. It is crucial for understanding how the most probable speed, average speed, and root-mean-square speed are derived. The distribution is dependent on temperature and particle mass, influencing the speed of gas molecules.
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Speed Distribution & Special Speeds of Ideal Gases
Most Probable Speed
The most probable speed (v_mp) is the speed at which the largest number of gas molecules are moving, according to the Maxwell-Boltzmann distribution. It is calculated using the formula v_mp = sqrt(2kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a molecule.
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Molar Mass and Molecular Mass
Molar mass is the mass of one mole of a substance, expressed in grams per mole, and is essential for converting between moles and grams. For gases, the molecular mass is used to calculate the mass of individual molecules, which is necessary for determining speeds using the Maxwell-Boltzmann distribution.
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Related Practice
Textbook Question
How much heat does it take to increase the temperature of mol of an ideal gas by K near room temperature if the gas is held at constant volume and is monatomic?
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Textbook Question
For diatomic carbon dioxide gas (CO2, molar mass g/mol) at K, calculate the average speed .
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Textbook Question
Compute the specific heat at constant volume of nitrogen (N2) gas, and compare it with the specific heat of liquid water. The molar mass of N2 is g/mol.
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Textbook Question
For a gas of nitrogen molecules (N2), what must the temperature be if of all the molecules have speeds less than m/s? Use Table . The molar mass of N2 is g/mol.
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Textbook Question
For diatomic carbon dioxide gas (CO2, molar mass g/mol) at K, calculate the root-mean-square speed .
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