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(a) For n = 4, what are the possible values of l?
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Ch.6 - Electronic Structure of Atoms
Brown14th EditionChemistry: The Central ScienceISBN: 9780134414232
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TOC.tableOfContents
- Ch.1 - Introduction: Matter, Energy, and Measurement177
- Ch.2 - Atoms, Molecules, and Ions231
- Ch.3 - Chemical Reactions and Reaction Stoichiometry216
- Ch.4 - Reactions in Aqueous Solution189
- Ch.5 - Thermochemistry175
- Ch.6 - Electronic Structure of Atoms168
- Ch.7 - Periodic Properties of the Elements150
- Ch.8 - Basic Concepts of Chemical Bonding177
- Ch.9 - Molecular Geometry and Bonding Theories200
- Ch.10 - Gases179
- Ch.11 - Liquids and Intermolecular Forces113
- Ch.12 - Solids and Modern Materials154
- Ch.13 - Properties of Solutions157
- Ch.14 - Chemical Kinetics199
- Ch.15 - Chemical Equilibrium128
- Ch.16 - Acid-Base Equilibria164
- Ch.17 - Additional Aspects of Aqueous Equilibria163
- Ch.18 - Chemistry of the Environment105
- Ch.19 - Chemical Thermodynamics164
- Ch.20 - Electrochemistry171
- Ch.21 - Nuclear Chemistry122
- Ch.22 - Chemistry of the Nonmetals82
- Ch.23 - Transition Metals and Coordination Chemistry79
- Ch.24 - The Chemistry of Life: Organic and Biological Chemistry16
Brown14th EditionChemistry: The Central ScienceISBN: 9780134414232Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 51a
Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50-mg mosquito moving at a speed of 1.40 m/s if the speed is known to within {0.01 m/s;
Guida verificata passo dopo passo1
Identify the given values: mass of the mosquito \( m = 1.50 \times 10^{-3} \text{ g} = 1.50 \times 10^{-6} \text{ kg} \), speed \( v = 1.40 \text{ m/s} \), and uncertainty in speed \( \Delta v = 0.01 \text{ m/s} \).
Recall Heisenberg's uncertainty principle formula: \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).
Calculate the uncertainty in momentum \( \Delta p \) using \( \Delta p = m \cdot \Delta v \).
Substitute \( \Delta p \) into the uncertainty principle formula to solve for \( \Delta x \): \( \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \).
Plug in the known values and solve for \( \Delta x \) to find the uncertainty in the position of the mosquito.
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Heisenberg's Uncertainty Principle
Heisenberg's uncertainty principle states that it is impossible to simultaneously know both the exact position and momentum of a particle. The more accurately we know one of these values, the less accurately we can know the other. This principle is fundamental in quantum mechanics and highlights the limitations of measurement at microscopic scales.
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01:09
Heisenberg Uncertainty Principle
Momentum
Momentum is defined as the product of an object's mass and its velocity (p = mv). In the context of the uncertainty principle, momentum is a key variable because it relates to the motion of the particle. For the mosquito in the question, its momentum can be calculated using its mass and the given speed, which is essential for determining the uncertainty in its position.
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00:40Angular Momentum Quantum Number
Uncertainty Calculation
To calculate the uncertainty in position (Δx) using the uncertainty principle, the formula Δx * Δp ≥ ħ/2 is used, where Δp is the uncertainty in momentum and ħ is the reduced Planck's constant. The uncertainty in momentum can be derived from the uncertainty in velocity (Δv) and the mass of the object. This calculation allows us to quantify the limits of our knowledge about the particle's position.
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02:05Uncertainty Principle Formula
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Neutron diffraction is an important technique for determining
the structures of molecules. Calculate the velocity of a
neutron needed to achieve a wavelength of 125 pm. (Refer to
the inside cover for the mass of the neutron.)
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Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (b) a proton moving at a speed of 15.00 { 0.012 * 104 m/s. (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)
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Domanda del libro di testo
Calculate the uncertainty in the position of (a) an electron moving at a speed of 13.00 ± 0.012 × 105 m/s (b) a neutron moving at this same speed. (The masses of an electron and a neutron are given in the table of fundamental constants in the inside cover of the text.)
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Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at 8.85 * 105 cm/s.
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Use the de Broglie relationship to determine the wavelengths of the following objects: (c) a lithium atom moving at 2.5 × 105 m/s (d) an ozone (O3) molecule in the upper atmosphere moving at 550 m/s.
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