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)}

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

$= 0.535 eV$

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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

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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

Fundamentals of Physics

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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.