QuestionApril 19, 2025

What is the energy in joules, and the wavelength, in meters, of the line spectrum of hydrogen that represents e-movement from n=4 to n=6' ? What is the energy for a mole of photons?Is this absorption or emission of energy? What kind of radiation is this? Sketch the levels and draw an arrow representing the transition

What is the energy in joules, and the wavelength, in meters, of the line spectrum of hydrogen that represents e-movement from n=4 to n=6' ? What is the energy for a mole of photons?Is this absorption or emission of energy? What kind of radiation is this? Sketch the levels and draw an arrow representing the transition
What is the energy in joules, and the wavelength, in meters, of the line spectrum of hydrogen that represents e-movement from n=4 to n=6' ? What is the energy for a mole of photons?Is this absorption or emission of energy? What kind of radiation is this? Sketch the levels and draw an arrow representing the transition

Solution
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Answer

Energy difference: \( \Delta E = 4.84 \times 10^{-20} \, \text{J} \) ### Wavelength: \( \lambda = 4.05 \times 10^{-6} \, \text{m} \) ### Energy for a mole of photons: \( E_{\text{mole}} = 2.92 \times 10^{4} \, \text{J/mol} \) ### This is absorption of energy. ### The radiation is infrared. Explanation 1. Determine the energy difference between levels The energy difference for a hydrogen atom transition is given by the formula: \[ \Delta E = -R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \] where \( R_H = 2.18 \times 10^{-18} \, \text{J} \) is the Rydberg constant, \( n_i = 4 \), and \( n_f = 6 \). Calculate ( ( \(\Delta E\) ) ). 2. Calculate the energy in joules Substitute \( n_i = 4 \) and \( n_f = 6 \) into the formula: \[ \Delta E = -2.18 \times 10^{-18} \left( \frac{1}{6^2} - \frac{1}{4^2} \right) \] Calculate the value of ( ( \(\Delta E\) ) ). 3. Convert energy to wavelength Use the relation between energy and wavelength: \[ E = \frac{hc}{\lambda} \] where \( h = 6.626 \times 10^{-34} \, \text{Js} \) and \( c = 3.00 \times 10^8 \, \text{m/s} \). Solve for ( ( \(\lambda\) ) ): \[ \lambda = \frac{hc}{|\Delta E|} \] 4. Calculate energy for a mole of photons Multiply the energy per photon by Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)): \[ E_{\text{mole}} = |\Delta E| \times N_A \] 5. Determine if it is absorption or emission Since the electron moves from a lower energy level (( n=4 )) to a higher one (( n=6 )), this process requires energy input, indicating absorption. 6. Identify the type of radiation Calculate the wavelength and compare it with known ranges of electromagnetic spectrum to determine the type of radiation. 7. Sketch the energy levels and transition Draw energy levels for ( n=4 ) and ( n=6 ) and an arrow pointing upwards to represent the transition.

Explanation

1. Determine the energy difference between levels<br /> The energy difference for a hydrogen atom transition is given by the formula:<br />\[<br />\Delta E = -R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)<br />\]<br />where \( R_H = 2.18 \times 10^{-18} \, \text{J} \) is the Rydberg constant, \( n_i = 4 \), and \( n_f = 6 \). Calculate ( \(\Delta E\) ).<br /><br />2. Calculate the energy in joules<br /> Substitute \( n_i = 4 \) and \( n_f = 6 \) into the formula:<br />\[<br />\Delta E = -2.18 \times 10^{-18} \left( \frac{1}{6^2} - \frac{1}{4^2} \right)<br />\]<br />Calculate the value of ( \(\Delta E\) ).<br /><br />3. Convert energy to wavelength<br /> Use the relation between energy and wavelength:<br />\[<br />E = \frac{hc}{\lambda}<br />\]<br />where \( h = 6.626 \times 10^{-34} \, \text{Js} \) and \( c = 3.00 \times 10^8 \, \text{m/s} \). Solve for ( \(\lambda\) ):<br />\[<br />\lambda = \frac{hc}{|\Delta E|}<br />\]<br /><br />4. Calculate energy for a mole of photons<br /> Multiply the energy per photon by Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)):<br />\[<br />E_{\text{mole}} = |\Delta E| \times N_A<br />\]<br /><br />5. Determine if it is absorption or emission<br /> Since the electron moves from a lower energy level (( n=4 )) to a higher one (( n=6 )), this process requires energy input, indicating absorption.<br /><br />6. Identify the type of radiation<br /> Calculate the wavelength and compare it with known ranges of electromagnetic spectrum to determine the type of radiation.<br /><br />7. Sketch the energy levels and transition<br /> Draw energy levels for ( n=4 ) and ( n=6 ) and an arrow pointing upwards to represent the transition.
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