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Chapter 7: Problem 138

The bright yellow light emitted by a sodium vapor lamp consists of two emission lines at 589.0 and 589.6 nm. What are the frequency and the energy of a photon of light at each of these wavelengths? What are the energies in kJ/mol?

Short Answer

Expert verified
The frequency and energy of a photon of light at \(589.0 nm\) are \(5.093 \times 10^{14} Hz\) and \(203.0 kJ/mol\), respectively. The frequency and energy of a photon of light at \(589.6 nm\) are \(5.088 \times 10^{14} Hz\) and \(202.9 kJ/mol\), respectively.

Step by step solution

01

Convert Wavelengths to Meters

First, we need to convert the wavelengths from nanometers (nm) to meters (m) by multiplying by \(10^{-9}\). This is because the speed of light is in meters/second. \(589.0 nm = 589.0 \times 10^{-9}m = 5.89 \times 10^{-7}m\) \(589.6 nm = 589.6 \times 10^{-9}m = 5.896 \times 10^{-7}m\)
02

Calculate Frequency

Now, we will calculate the frequency of each wavelength using the formula: \(v = \frac{c}{\lambda}\) For wavelength \(589.0 nm\): \(v = \frac{3.0 \times 10^8 m/s}{5.89 \times 10^{-7} m} = 5.093 \times 10^{14} Hz\) For wavelength \(589.6 nm\): \(v = \frac{3.0 \times 10^8 m/s}{5.896 \times 10^{-7} m} = 5.088 \times 10^{14} Hz\)
03

Calculate Energy

Using the frequencies we just found, we can now calculate the energy of a photon of light at each wavelength using the formula \(E = h \times v\): For wavelength \(589.0 nm\): \(E = (6.626 \times 10^{-34} Js)(5.093 \times 10^{14} Hz) = 3.372 \times 10^{-19} J\) For wavelength \(589.6 nm\): \(E = (6.626 \times 10^{-34} Js)(5.088 \times 10^{14} Hz) = 3.370 \times 10^{-19} J\)
04

Convert Energies to kJ/mol

Finally, we need to convert the energies from joules to kJ/mol using Avogadro's number: For wavelength \(589.0 nm\): \(\frac{3.372 \times 10^{-19} J}{1 photon} \times \frac{10^3 kJ}{1 J} \times \frac{6.022 \times 10^{23} photons}{1 mol} = 203.0 kJ/mol\) For wavelength \(589.6 nm\): \(\frac{3.370 \times 10^{-19} J}{1 photon} \times \frac{10^3 kJ}{1 J} \times \frac{6.022 \times 10^{23} photons}{1 mol} = 202.9 kJ/mol\) So, the frequency and energy of a photon of light at \(589.0 nm\) are \(5.093 \times 10^{14} Hz\) and \(203.0 kJ/mol\), respectively. The frequency and energy of a photon of light at \(589.6 nm\) are \(5.088 \times 10^{14} Hz\) and \(202.9 kJ/mol\), respectively.

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Most popular questions from this chapter

What are the possible values for the quantum numbers \(n, \ell,\) and $m_{\ell} ?$

Calculate the wavelength of light emitted when each of the following transitions occur in the hydrogen atom. What type of electromagnetic radiation is emitted in each transition? a. \(n=3 \rightarrow n=2\) b. \(n=4 \rightarrow n=2\) c. \(n=2 \rightarrow n=1\)

An electron is excited from the \(n=1\) ground state to the \(n=\) 3 state in a hydrogen atom. Which of the following statements are true? Correct the false statements to make them true. a. It takes more energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be shorter than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1 .\) d. The wavelength of light emitted when the electron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\) e. For \(n=3,\) the electron is in the first excited state.

Calculate the maximum wavelength of light capable of removing an electron for a hydrogen atom from the energy state characterized by \(n=1,\) by \(n=2\)

Which of the following statements is (are) true? a. The 2\(s\) orbital in the hydrogen atom is larger than the 3s orbital also in the hydrogen atom. b. The Bohr model of the hydrogen atom has been found to be incorrect. c. The hydrogen atom has quantized energy levels. d. An orbital is the same as a Bohr orbit. e. The third energy level has three sublevels, the s,p, and d sublevels.
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