Questions: The energy E of the electron in a hydrogen atom can be calculated from the Bohr formula: E = -Ry / n^2 In this equation Ry stands for the Rydberg energy, and n stands for the principal quantum number of the orbital that holds the electron. (You can find the value of the Rydberg energy using the Data button on the ALEKS toolbar.) Calculate the wavelength of the line in the absorption line spectrum of hydrogen caused by the transition of the electron from an orbital with n=7 to an orbital with n=12. Round your answer to 3 significant digits. m

The energy E of the electron in a hydrogen atom can be calculated from the Bohr formula:
E = -Ry / n^2

In this equation Ry stands for the Rydberg energy, and n stands for the principal quantum number of the orbital that holds the electron. (You can find the value of the Rydberg energy using the Data button on the ALEKS toolbar.)

Calculate the wavelength of the line in the absorption line spectrum of hydrogen caused by the transition of the electron from an orbital with n=7 to an orbital with n=12. Round your answer to 3 significant digits.
m

Transcript text: The energy $E$ of the electron in a hydrogen atom can be calculated from the Bohr formula: \[ E=-\frac{R_{y}}{n^{2}} \] In this equation $R_{y}$ stands for the Rydberg energy, and $n$ stands for the principal quantum number of the orbital that holds the electron. (You can find the value of the Rydberg energy using the Data button on the ALEKS toolbar.) Calculate the wavelength of the line in the absorption line spectrum of hydrogen caused by the transition of the electron from an orbital with $n=7$ to an orbital with $n=12$. Round your answer to 3 significant digits. $\square$ $\square \mathrm{m}$

Solution

Solution Steps

Step 1: Calculate the Energy Levels

The energy of an electron in a hydrogen atom is given by the Bohr formula:

\[ E_n = -\frac{R_y}{n^2} \]

where $ R_y $ is the Rydberg energy, approximately $ 13.6057 \, \text{eV} $.

For $ n = 7 $:

\[ E_7 = -\frac{13.6057}{7^2} = -\frac{13.6057}{49} \approx -0.2777 \, \text{eV} \]

For $ n = 12 $:

\[ E_{12} = -\frac{13.6057}{12^2} = -\frac{13.6057}{144} \approx -0.0945 \, \text{eV} \]

Step 2: Calculate the Energy Difference

The energy difference $\Delta E$ between the two levels is:

\[ \Delta E = E_{12} - E_7 = -0.0945 - (-0.2777) = 0.1832 \, \text{eV} \]

Step 3: Convert Energy Difference to Wavelength

The energy of a photon is related to its wavelength $\lambda$ by the equation:

\[ E = \frac{hc}{\lambda} \]

where $ h = 4.1357 \times 10^{-15} \, \text{eV} \cdot \text{s} $ is Planck's constant and $ c = 3.00 \times 10^8 \, \text{m/s} $ is the speed of light.

Rearranging for $\lambda$:

\[ \lambda = \frac{hc}{\Delta E} \]

Substituting the values:

\[ \lambda = \frac{(4.1357 \times 10^{-15} \, \text{eV} \cdot \text{s})(3.00 \times 10^8 \, \text{m/s})}{0.1832 \, \text{eV}} \]

\[ \lambda \approx \frac{1.2407 \times 10^{-6} \, \text{m} \cdot \text{eV}}{0.1832 \, \text{eV}} \approx 6.772 \times 10^{-6} \, \text{m} \]