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What is the radius of a hydrogen atom whose electron moves at 7.3×105 m/s?
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Ch 38: Quantization
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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TOC.tableOfContents
- 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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Capitolo 38, Problema 34
Find the radius of the electron’s orbit, the electron’s speed, and the energy of the atom for the first three stationary states of He+.
Guida verificata passo dopo passo1
Understand the problem: The He+ ion is a hydrogen-like atom with a single electron orbiting a nucleus of charge +2e. The Bohr model can be used to calculate the radius of the electron's orbit, its speed, and the energy of the atom for the first three stationary states (n = 1, 2, 3). The formulas for these quantities depend on the principal quantum number (n) and the nuclear charge (Z).
Step 1: Write the formula for the radius of the electron's orbit in the Bohr model: , where is the Bohr radius (approximately 5.29 × 10⁻¹¹ m), n is the principal quantum number, and Z is the atomic number (Z = 2 for He+). Substitute the values of n = 1, 2, 3 to calculate the radii for the first three stationary states.
Step 2: Write the formula for the speed of the electron in the Bohr model:
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Bohr Model of the Atom
The Bohr model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. It quantizes the angular momentum of the electrons, leading to specific energy levels. This model is particularly useful for hydrogen-like atoms, such as He+, where the electron's orbit radius and energy can be calculated using simple formulas.
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01:06
Charge of Atom
Quantization of Energy Levels
In the context of the Bohr model, energy levels are quantized, meaning that electrons can only occupy certain discrete energy states. For He+, the energy levels can be calculated using the formula E_n = -Z² * 13.6 eV / n², where Z is the atomic number and n is the principal quantum number. This quantization explains why electrons do not radiate energy continuously and instead exist in stable orbits.
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04:10Intro to Energy & Types of Energy
Centripetal Force and Electron Speed
The speed of an electron in its orbit can be derived from the balance of centripetal force and electrostatic force. The centripetal force required to keep the electron in circular motion is provided by the electrostatic attraction between the positively charged nucleus and the negatively charged electron. This relationship allows for the calculation of the electron's speed in each stationary state.
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06:48Intro to Centripetal Forces
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