Chapter 6: Q41E (page 1356)

For the sodium atom of Example 40.8, find (a) the ground-state energy; (b) the wavelength of a photon emitted when the n= 4 to n= 3 transition occurs; (c) the energy difference for any $∆ n = 1$ transition.

Short Answer

Expert verified

(a) The ground state energy is $5.89 \times 10^{- 3} e V$ .

(b) The wavelength of photon when it makes a transition from n= 4 to n= 3 is $106 μ m$ .

Step by step solution

(a) Determination of the ground state energy.

For harmonic oscillator, the energy levels expression is,

$E_{n} = (n + \frac{1}{2}) h \sqrt{\frac{k^{'}}{m}} = (n + \frac{1}{2}) h ω$

For ground state, n=0

So,

$E_{0} = \frac{1}{2} ħ ω = \frac{1}{2} h \sqrt{\frac{k^{'}}{m}} = \frac{1.055 \times 10^{- 34} J \times s}{2} \sqrt{\frac{12.2 N / m}{3.82 \times 10^{- 26} kg}} = 9.43 \times 10^{- 22} J = 5.89 \times 10^{- 3} eV$

Thus, the ground state energy is $5.89 \times 10^{- 3} e V$ .

(b) Determination of wavelength of photon when it makes a transition from n=4 to n=3.

For, n=4,

$E_{4} = (4 + \frac{1}{2}) ħ ω = \frac{9}{2} ħ ω$

For, n=3,

$E_{3} = (3 + \frac{1}{2}) ħ ω = \frac{7}{2} ħ ω$

Thus,

$E_{4} - E_{3} = \frac{9}{2} ħ ω - \frac{7}{2} ħ ω = ħ ω = 2 E_{0} = 0.0118 eV$

Now, solve for the wavelength using equation,

$λ = \frac{hc}{E} = \frac{6.63 \times 10^{- 34} J \cdot s) \times (3.00 \times 10^{8} m / s)}{1.88 \times 10^{- 21} J} = 106 μ m$

Thus the wavelength emitted in transition is $= 106 μ m$ .

(c) Determination of the energy difference for any ∆n=1 transition.

Like part (b), for $n = \pm 1, i . e . Δ n = 1$ transition, the difference in energy level is,

$E_{n + 1} - E_{n} = ħ ω = 2 E_{0} = 0.0118 eV$

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For the sodium atom of Example 40.8, find (a) the ground-state energy; (b) the wavelength of a photon emitted when the n= 4 to n= 3 transition occurs; (c) the energy difference for any $∆ n = 1$ transition.

Short Answer

Step by step solution

(a) Determination of the ground state energy.

(b) Determination of wavelength of photon when it makes a transition from n=4 to n=3.

(c) Determination of the energy difference for any ∆n=1 transition.

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For the sodium atom of Example 40.8, find (a) the ground-state energy; (b) the wavelength of a photon emitted when the n= 4 to n= 3 transition occurs; (c) the energy difference for any ∆n=1transition.

Short Answer

Step by step solution

(a) Determination of the ground state energy.

(b) Determination of wavelength of photon when it makes a transition from n=4 to n=3.

(c) Determination of the energy difference for any ∆n=1 transition.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Measurements

Electricity

Electricity and Magnetism

Nuclear Physics

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

For the sodium atom of Example 40.8, find (a) the ground-state energy; (b) the wavelength of a photon emitted when the n= 4 to n= 3 transition occurs; (c) the energy difference for any $∆ n = 1$ transition.