/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 How does the energy of a princip... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 38

How does the energy of a principal energy level depend on the value of \(n\) ? Does a higher value of \(n\) mean a higher or lower energy?

Short Answer

Expert verified
The energy of an electron in a principal energy level $n$ of a hydrogen-like atom is given by the formula \(E = -\dfrac{13.6 \, eV}{n^2}\). As $n$ increases, the energy E becomes less negative (closer to zero), indicating a higher energy. Therefore, a higher value of $n$ corresponds to a higher energy, while a lower value of $n$ corresponds to a lower energy.

Step by step solution

01

Energy formula for an electron in a hydrogen-like atom

The energy of an electron in a principal energy level n of a hydrogen-like atom can be calculated using the formula: \[ E = -\dfrac{13.6 \, eV}{n^2} \] Where E is the energy of the electron, and n is the principal quantum number (an integer n = 1, 2, 3, ...).
02

Analyzing the relationship between energy and n

Looking at the formula, we can notice that as n increases, the denominator \(n^2\) also increases, which makes the entire fraction smaller. Since it's a negative fraction, when the fraction becomes smaller, the overall energy E becomes less negative or closer to zero. When n is small (n = 1, for example), the energy is more negative, and as n gets larger, the energy becomes less negative.
03

Conclusion: Higher or lower energy for larger n?

From the analysis, we can conclude that a higher value of n means a higher energy as the energy E becomes closer to zero (less negative). Conversely, a lower value of n means lower energy (more negative) in the energy level.

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

Most popular questions from this chapter

In each of the following sets of elements, indicate which element has the smallest atomic size. a. \(\mathrm{Ba}, \mathrm{Ca}, \mathrm{Ra}\) b. \(\mathrm{P}, \mathrm{Si}, \mathrm{Al}\) c. \(\mathrm{Rb}, \mathrm{Cs}, \mathrm{K}\)

Using the symbol of the previous noble gas to indicate the core electrons, write the valence shell electron configuration for each of the following elements. a. phosphorus, \(Z=15\) b. chlorine, \(Z=17\) c. magnesium, \(Z=12\) d. \(\operatorname{zinc}, Z=30\)

Arrange the following sets of elements in order of increasing atomic size. a. \(\mathrm{Sn}, \mathrm{Xe}, \mathrm{Rb}, \mathrm{Sr}\) b. \(\mathrm{Rn}, \mathrm{He}, \mathrm{Xe}, \mathrm{Kr}\) c. \(\mathrm{Pb}, \mathrm{Ba}, \mathrm{Cs}, \mathrm{At}\)

The number of sublevels in a principal energy level (increases/decreases) as \(n\) increases.

In the modern theory of the atom, a(n) _______ resents a region of space in which there is a high probability of finding an electron.
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Reactions

Read Explanation

Organic Chemistry

Read Explanation

Physical Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Kinetics

Read Explanation

The Earths Atmosphere

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.