Chapter 40: Q. 22 (page 1175)

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a $C = O$ carbon-oxygen double bond.

Short Answer

Expert verified

Energy of $n = 1$ state = $0.107 e V$

Energy of $n = 2 s t a t e = 0.321 e V$

Energy of $n = 3 s t a t e = 0.535 e V$

Step by step solution

Step 1. Given information

Energy level of harmonic quantum oscillator,

$E_{n} = (n - \frac{1}{2}) \frac{h ω}{2 π}$

Here,

n denotes the state,

h = Planck's constant and

$ω =$ classical angular frequency.

$E_{n} = (n - \frac{1}{2}) \frac{h (2 π f)}{2 π}$

$= (n - \frac{1}{2}) h f$

$= (n - \frac{1}{2}) h (\frac{c}{λ})$

Step 2. Energy of n=1 state

$E_{1} = (1 - \frac{1}{2}) h (\frac{c}{λ})$

$= \frac{h c}{2 λ}$

$E_{1} = \frac{(6.63 \times 10^{- 34} J \cdot s) (3.0 \times 10^{8} m / s)}{2 (5.8 μ m)}$

$= \frac{(6.63 \times 10^{- 34} J \cdot s) (3.0 \times 10^{8} m / s)}{2 (5.8 μ m) (\frac{10^{- 6} m}{1.0 μ m})} (\frac{1.0 eV}{1.6 \times 10^{- 19} J})$

$= 0.107 eV$

Step 2. Energy of n=2 state

$E_{2} = (2 - \frac{1}{2}) h (\frac{c}{λ})$

$= \frac{3 h c}{2 λ}$

$E_{2} = \frac{3 (6.63 \times 10^{- 34} J \cdot s) (3.0 \times 10^{8} m / s)}{2 (5.8 μ m)}$

$= \frac{3 (6.63 \times 10^{- 34} J \cdot s) (3.0 \times 10^{8} m / s)}{2 (5.8 μ m) (\frac{10^{- 6} m}{1.0 μ m})} (\frac{1.0 eV}{1.6 \times 10^{- 19} J})$

$= 0.321 eV$

Step 3. Energy of n=3 state

$E_{3} = (3 - \frac{1}{2}) h (\frac{c}{λ})$

$= \frac{5 h c}{2 λ}$

E_{3} = \frac{5 (6.63 \times 10^{- 34} J \cdot s) (3.0 \times 10^{8} m / s)}{2 (5.8 μ m)}

$= \frac{5 (6.63 \times 10^{- 34} J \cdot s) (3.0 \times 10^{8} m / s)}{2 (5.8 μ m) (\frac{10^{- 6} m}{1.0 μ m})} (\frac{1.0 eV}{1.6 \times 10^{- 19} J})$

$= 0.535 eV$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a $C = O$ carbon-oxygen double bond.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Energy of n=1 state

Step 2. Energy of n=2 state

Step 3. Energy of n=3 state

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Classical Mechanics

Fields in Physics

Dynamics

Fluids

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a C=Ocarbon-oxygen double bond.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Energy of n=1 state

Step 2. Energy of n=2 state

Step 3. Energy of n=3 state

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Classical Mechanics

Fields in Physics

Dynamics

Fluids

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a $C = O$ carbon-oxygen double bond.