A hydrogen atom consists of a proton and an electron. According to the Bohr theory, the electron revolves about the proton in a circle of radius a (a = 5 · 10−9cm for the ground state). According to quantum mechanics, the electron may be at any distance r (from 0 to ∞) from the proton; for the ground state, the probability that the electron is in a volume element dV , at a distance r to r + dr from the proton, is proportional to e−2r/adV , where a is the Bohr radius. Write dV in spherical coordinates (see Chapter 5, Section 4) and find the density function f(r) so that f(r) dr is the probability that the electron is at a distance between r and r + drfrom the proton. (Remember that the probability for the electron to be somewhere must be 1.) Computer plot f(r) and show that its maximum value is at r = a; we then say that the most probable value of r is a. Also show that the average value of r−1 is a−1.

General Chemistry - Standalone book (MindTap Course List)
11th Edition
ISBN:9781305580343
Author:Steven D. Gammon, Ebbing, Darrell Ebbing, Steven D., Darrell; Gammon, Darrell Ebbing; Steven D. Gammon, Darrell D.; Gammon, Ebbing; Steven D. Gammon; Darrell
Publisher:Steven D. Gammon, Ebbing, Darrell Ebbing, Steven D., Darrell; Gammon, Darrell Ebbing; Steven D. Gammon, Darrell D.; Gammon, Ebbing; Steven D. Gammon; Darrell
Chapter7: Quantum Theory Of The Atom
Section: Chapter Questions
Problem 7.24QP: Investigating Energy Levels Consider the hypothetical atom X that has one electron like the H atom...
icon
Related questions
Question

A hydrogen atom consists of a proton and an electron. According to the Bohr theory, the electron revolves about the proton in a circle of radius a (a = 5 · 10−9cm for the ground state). According to quantum mechanics, the electron may be at any distance r (from 0 to ∞) from the proton; for the ground state, the probability that the electron is in a volume element dV , at a distance r to r + dr from the proton, is proportional to e−2r/adV , where a is the Bohr radius. Write dV in spherical coordinates (see Chapter 5, Section 4) and find the density function f(r) so that f(r) dr is the probability that the electron is at a distance between r and r + dr
from the proton. (Remember that the probability for the electron to be somewhere must be 1.) Computer plot f(r) and show that its maximum value is at r = a; we then say that the most probable value of r is a. Also show that the average value of r−1 is a−1.

Expert Solution
steps

Step by step

Solved in 2 steps with 12 images

Blurred answer
Knowledge Booster
Quantum Mechanical Model of Atom
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, chemistry and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
General Chemistry - Standalone book (MindTap Cour…
General Chemistry - Standalone book (MindTap Cour…
Chemistry
ISBN:
9781305580343
Author:
Steven D. Gammon, Ebbing, Darrell Ebbing, Steven D., Darrell; Gammon, Darrell Ebbing; Steven D. Gammon, Darrell D.; Gammon, Ebbing; Steven D. Gammon; Darrell
Publisher:
Cengage Learning
Chemistry: The Molecular Science
Chemistry: The Molecular Science
Chemistry
ISBN:
9781285199047
Author:
John W. Moore, Conrad L. Stanitski
Publisher:
Cengage Learning
Chemistry for Engineering Students
Chemistry for Engineering Students
Chemistry
ISBN:
9781337398909
Author:
Lawrence S. Brown, Tom Holme
Publisher:
Cengage Learning
Chemistry: Principles and Practice
Chemistry: Principles and Practice
Chemistry
ISBN:
9780534420123
Author:
Daniel L. Reger, Scott R. Goode, David W. Ball, Edward Mercer
Publisher:
Cengage Learning
Chemistry: Principles and Reactions
Chemistry: Principles and Reactions
Chemistry
ISBN:
9781305079373
Author:
William L. Masterton, Cecile N. Hurley
Publisher:
Cengage Learning
Chemistry: An Atoms First Approach
Chemistry: An Atoms First Approach
Chemistry
ISBN:
9781305079243
Author:
Steven S. Zumdahl, Susan A. Zumdahl
Publisher:
Cengage Learning