Chemistry: The Molecular Nature of Matter and Change
Chemistry: The Molecular Nature of Matter and Change
8th Edition
ISBN: 9781259631757
Author: Martin Silberberg Dr., Patricia Amateis Professor
Publisher: McGraw-Hill Education
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Chapter 7, Problem 7.81P

(a)

Interpretation Introduction

Interpretation:

The highest frequency and minimum wavelength of radiation emitted when the electron makes a transition from n=3 is to be determined.

Concept introduction:

Electromagnetic waves are radiations that are formed by oscillating electric and magnetic fields. The electric and magnetic field components of an electromagnetic wave are perpendicular to each other.

The difference between the energies of two energy levels is the amount of energy in a photon emitted or absorbed when an electron makes a transition. The energy of a photon emitted or absorbed by an electron is,

ΔE=hν (1)

Here,

ΔE is the energy.

h is the Plank’s constant.

ν is the frequency

(a)

Expert Solution
Check Mark

Answer to Problem 7.81P

The highest frequency of radiation emitted when the electron makes a transition from n=3 is 8×1014 s1. The wavelength of this radiation is 4×107 m.

Explanation of Solution

The energies of the energy levels 3 and 1 are 15×1019 J and 20×1019 J respectively.

The formula to calculate the difference in the energies of energy level 1 and 2 is,

ΔE=E3E1 (2)

Substitute 15×1019 J for E3 and 20×1019 J for E1 in equation (2).

ΔE=(15×1019 J)(20×1019 J)=15×1019 J+20×1019 J=5×10-19 J

Substitute 5×10-19 J for ΔE and 6.63×1034 Js for h in equation (1).

5×10-19 J=(6.63×1034 Js)ν

Rearrange the above equation to calculate the value for ν as follows:

ν=5×10-19 J6.63×1034 Js=7.546×1014 s1=8×1014 s1

The equation to relate the frequency and wavelength of radiation is as follows:

ν=cλ (3)

Substitute 3×108 m/s for c and 8×1014 s1 for ν in equation (3).

8×1014 s1=3×108 m/sλ

Rearrange the above equation to calculate the value of λ as follows:

λ=3×108 m/s8×1014 s1=3.75×107 m=4×10-7 m

Conclusion

The highest frequency of radiation emitted when the electron makes a transition from n=3 is 8×1014 s1. The wavelength of this radiation is 4×107 m.

(b)

Interpretation Introduction

Interpretation:

The ionization energy of the atom in its ground state in kJ/mol is to be determined.

Concept introduction:

Ionization energy is defined as the amount of energy required to remove an electron from an isolated gaseous atom. The energy required to remove an electron from an atom depends on the position of the electron in the atom. The closer the electron is to the nucleus in the atom, the harder it is to pull it out of the atom. As the distance of an electron from the nucleus increases, the magnitude of the forces of attraction between the electron and the nucleus decreases. Thus it becomes easier to remove it from the atom.

(b)

Expert Solution
Check Mark

Answer to Problem 7.81P

The ionization energy of the atom in its ground state in kJ/mol is 1.2×103.

Explanation of Solution

The total number of atoms in one mole of a compound is 6.023×1023.

The ionization energy required to remove an electron from the ground state is the same as the energy of the state.

Thus for one atom, the ionization energy for the ground state is 20×1019 J.

For 1 mole (6.023×1023 atoms), the ionization energy is calculated as,

Ionization energy=(20×1019 J/atom)(1 kJ103 J)(6.023×1023 atoms/mol)=1204.4 kJ/mol=1.2×103 kJ/mol

Conclusion

The ionization energy of the atom in its ground state in kJ/mol is 1.2×103.

(c)

Interpretation Introduction

Interpretation:

The shortest wavelength of radiation that could be absorbed by the electron in the n=4 level without causing ionization is to be determined.

Concept introduction:

Electromagnetic waves are radiations that are formed by oscillating electric and magnetic fields. The electric and magnetic field components of an electromagnetic wave are perpendicular to each other.

The difference between the energies of two energy levels is the amount of energy in a photon emitted or absorbed when an electron makes a transition. The energy of a photon emitted or absorbed by an electron is,

ΔE=hν (1)

Here,

ΔE is the energy.

h is the Plank’s constant.

ν is the frequency

The equation to relate the frequency and wavelength of radiation is as follows:

ν=cλ

The above relation can be modified as follows:

ΔE=hcλ (4)

(c)

Expert Solution
Check Mark

Answer to Problem 7.81P

The shortest wavelength of radiation that could be absorbed by the electron in the n=4 level without causing ionization is 2×102 nm.

Explanation of Solution

Since the absorption of radiation occurs, hence the electron from n=4 will move to a level of higher energy.

The shortest wavelength of radiation will be absorbed for a transition from n=4 to n=6.

The energy of the energy level 4 and 6 are 11×1019 J and 2×1019 J respectively.

The formula to calculate the difference in the energies of energy level 4 and 6 is,

ΔE=E6E4 (5)

Substitute 2×1019 J for E6 and 11×1019 J for E4 in equation (5).

ΔE=(2×1019 J)(11×1019 J)=2×1019 J+11×1019 J=9×10-19 J

Substitute 9×10-19 J for ΔE, 3×108 m/s for c and 6.63×1034 Js for h in equation (4).

9×10-19 J=(6.63×1034 Js)(3×108 m/s)λ

Rearrange the above equation to calculate the value of λ as follows:

λ=(6.63×1034 Js)(3×108 m/s)9×10-19 J(1 nm109 m)=2.21 nm=2×102 nm

Conclusion

The shortest wavelength of radiation that could be absorbed by the electron in the n=4 level without causing ionization is 2×102 nm.

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Chapter 7 Solutions

Chemistry: The Molecular Nature of Matter and Change

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