Q46P Calculate the probability that t... [FREE SOLUTION]

91影视

Chapter 39: Q46P (page 1217)

Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a , where a is the Bohr radius?

Short Answer

Expert verified

The required probability is P=0.439.

Step by step solution

Identification of the given data:

The given data is listed below.

Radii of the spherical shells are given as a and 2a .

Formula for finding the probability of electron:

The ground state wave function of hydrogen atom is given by,

$蠄_{100} (r, 胃, 蠒) = \frac{1}{\sqrt{蟺}} {(\frac{1}{a})}^{\frac{3}{2}} e^{- \frac{r}{a}}$

Here, the Bohr radius is $a = 5.292 脳 10^{- 11} m$ .

Determine the probability of the electron in the hydrogen atom in its ground state:

The probability of finding the electron found between spherical shells is,

$P = {鈭�}_{a}^{2 a} {鈭�}_{0}^{蟺} {鈭�}_{0}^{2 蟺} {|蠄_{100} (r, 胃, 蠒)|}^{2} r^{2} 诲谤蝉颈苍胃诲胃诲蠒 = {鈭�}_{a}^{2 a} {|\frac{1}{\sqrt{蟺}} {(\frac{1}{a})}^{\frac{3}{2}} e^{\frac{- r}{a}}|}^{2} r^{2} dr {鈭�}_{0}^{蟺} 诲蠒 = \frac{1}{蟺} {(\frac{1}{a})}^{3} {鈭�}_{a}^{2 a} {|e^{- \frac{r}{a}}|}^{2} r^{2} dr 脳 {(- 肠辞蝉胃)}_{0}^{蟺} {(胃)}_{0}^{2 蟺} = \frac{1}{蟺} {(\frac{1}{a})}^{3} {[- \frac{1}{4} {ae}^{\frac{- 2 r}{a}} (a^{2} + 2 ar + 2 r^{2})]}_{a}^{2 a} 脳 (- 肠辞蝉蟺 + \cos 0) (2 蟺 - 0)$

$P = \frac{1}{蟺} {(\frac{1}{a})}^{3} [- \frac{a}{4} \{e^{\frac{- 2 (2 a)}{a}} (a^{2} + 2 a (2 a) + 2 {(2 a)}^{2}) - e^{\frac{- 2 (a)}{a}} (a^{2} + 2 a^{2} + 2 a^{2})\}] 脳 (- (- 1) + 1) (2 蟺) = \frac{1}{蟺} {(\frac{1}{a})}^{3} [- \frac{a}{4} \{e^{- 4} (13 a^{2}) - e^{- 2} (5 a^{2})\}] 脳 (4 蟺) = 4 {(\frac{1}{a})}^{3} [- \frac{a}{4} \{0.0183 (13 a^{2}) - 0.1353 (5 a^{2})\}] P = 4 [- \frac{1}{4} \{0.0183 (13) - 0.1353 (5)\}] = - 4 [\frac{1}{4} \{0.2379 - 0.6765\}] = 0.439$

Hence, the probability of the electron in the hydrogen atom in its ground state is 0.439.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a , where a is the Bohr radius?

Short Answer

Step by step solution

Identification of the given data:

Formula for finding the probability of electron:

Determine the probability of the electron in the hydrogen atom in its ground state:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Turning Points in Physics

Conservation of Energy and Momentum

Radiation

Oscillations

Electrostatics

Study anywhere. Anytime. Across all devices.