Ch.8 - The Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 5th Edition

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Tro 5th Edition

Ch.8 - The Quantum-Mechanical Model of the Atom

Problema 55

Capitolo 8, Problema 55

An electron has an uncertainty in its position of 552 pm. What is the uncertainty in its velocity?

Guida verificata passo dopo passo

Step 1: Understand the problem. This problem is based on Heisenberg's Uncertainty Principle, which states that it is impossible to simultaneously measure the exact position and momentum (or velocity) of a particle. The principle can be mathematically expressed as Δx * Δp ≥ h/4π, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, h is Planck's constant, and π is a mathematical constant.

Step 2: Convert the given uncertainty in position from picometers (pm) to meters (m) because the SI unit for position is meter. 1 pm = 1e-12 m.

Step 3: Rearrange the Heisenberg's Uncertainty Principle equation to solve for the uncertainty in momentum (Δp). Δp = h/(4π*Δx).

Step 4: Calculate the uncertainty in momentum using the converted uncertainty in position and the value of Planck's constant (h = 6.62607015 × 10^-34 m^2 kg / s).

Step 5: Finally, calculate the uncertainty in velocity (Δv) using the relation Δp = m*Δv, where m is the mass of the electron (9.10938356 × 10^-31 kilograms). Rearrange the equation to solve for Δv: Δv = Δp/m.

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Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum (which includes velocity) of a particle. This principle highlights a fundamental limit to measurement at the quantum level, indicating that the more precisely one property is measured, the less precisely the other can be known.

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Heisenberg Uncertainty Principle

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces concepts such as wave-particle duality and quantization of energy, which are essential for understanding the behavior of electrons and other particles in terms of probabilities rather than certainties.

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Introduction to Quantum Mechanics

Uncertainty in Measurements

Uncertainty in measurements refers to the inherent limitations in measuring physical quantities. In the context of the Heisenberg Uncertainty Principle, the uncertainty in position and momentum (or velocity) can be quantitatively related, allowing for calculations that predict the degree of uncertainty in one variable based on the known uncertainty in the other.

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Uncertainty Principle Formula

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