Question

Sodium atoms emit a spectral line with a wavelength in the yellow, \(589.6 \mathrm{~nm}\). What is the approximate difference in energy between the two energy levels involved in the emission of this spectral line? (a) \(3.37 \times 10^{-19} \mathrm{~J}\) (b) \(2.1 \mathrm{eV}\) (c) \(48.35 \mathrm{kcal} / \mathrm{mol}\) (d) all of these

Short answer

The approximate difference in energy between the two energy levels involved is indeed represented by all provided options: (a) \(3.37 \times 10^{-19} \mathrm{J}\), (b) \(2.1 \mathrm{eV}\), and (c) \(48.35 \mathrm{kcal/mol}\).

Step-by-step solution

Calculate the Energy of the Photon

To find the difference in energy between the two energy levels involved in the emission, use the formula for the energy of a photon, which is given by the equation: \(E = \frac{hc}{\lambda}\).Here, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{J\cdot s}\)), \(c\) is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)), and \(\lambda\) is the wavelength of the emitted light. First, convert the wavelength from nanometers to meters by dividing by \(10^9\).

Perform the Calculation for Energy

Substitute the given wavelength (\(589.6 \mathrm{nm}\)), the value for \(h\), and \(c\) into the equation to calculate the energy of the emitted photon. \(E = \frac{(6.626 \times 10^{-34} \mathrm{J\cdot s})(3.00 \times 10^8 \mathrm{m/s})}{589.6 \times 10^{-9} \mathrm{m}} = 3.37 \times 10^{-19} \mathrm{J}\).

Convert Energy to Electron Volts

To convert the energy from joules to electron volts (eV), use the conversion factor \(1 \mathrm{eV} = 1.602 \times 10^{-19} \mathrm{J}\). \(E = \frac{3.37 \times 10^{-19} \mathrm{J}}{1.602 \times 10^{-19} \mathrm{J/eV}} = 2.1 \mathrm{eV}\).

Convert Energy to kcal/mol

To convert the energy from joules to kilocalories per mole (kcal/mol), use two conversion factors: \(1 \mathrm{J} = 0.239 \mathrm{cal}\) and Avogadro's number, \(6.022 \times 10^{23} \mathrm{mol}^{-1}\). \(E = \frac{3.37 \times 10^{-19} \mathrm{J}}{0.239 \mathrm{cal/J}} = E_{\mathrm{cal}}\).Then, \(E_{\mathrm{kcal/mol}} = \frac{E_{\mathrm{cal}}}{1000} \times 6.022 \times 10^{23}\).This yields approximately \(48.35 \mathrm{kcal}/\mathrm{mol}\).

Compare the Calculated Values with the Options

From the calculated values in Joules, eV, and kcal/mol, check against the provided answer choices. If the calculations are correct, they should correspond to the answer choices given in the question, and thus all of these would be correct.

Key concepts

Energy Level Transition

When we talk about spectral lines and their appearance, particularly as colors like the yellow observed in sodium atoms, we are referring to energy level transitions. These transitions occur when an electron in an atom moves between quantized energy levels. Each atom has a unique set of energy levels, and the difference in energy between these levels is what determines the properties of the emitted light.

The process involves an electron dropping from a higher energy level to a lower one. This loss of energy is not simply lost; instead, it gets emitted as a photon of light, which is why we see a spectral line. The color or wavelength of the light corresponds to the energy difference between the two levels. Specifically for sodium, the emission of a yellow spectral line implies a transition between two specific energy states of its electrons.

Planck's Constant

The foundation of the calculation required for determining the energy of a spectral line lies in a fundamental constant of nature known as Planck's constant. Denoted by the symbol 'h', Planck's constant is a key quantity in quantum mechanics, with a value of approximately \(6.626 \times 10^{-34} \text{ J}\text{•s} \).

This constant relates the frequency of a photon to its energy, reflecting the inherently quantized nature of energy in the atomic and subatomic domains. It's a pivotal factor in the formula \(E = \frac{hc}{\lambda}\), which is used to find the energy associated with a photon emitted during an energy level transition. Here, 'c' is the speed of light and '\(\text{λ}\)' is the wavelength of the emitted photon. Planck's constant serves as a bridge connecting the energy of microscopic particles with characteristics of the wave-like phenomena they exhibit.

Photon Energy

A photon is a quantum of electromagnetic energy, and its energy is directly related to the frequency of the radiation it represents. The energy of a photon can be calculated using the formula \(E = \frac{hc}{\lambda}\), tying together the Planck constant (h), the speed of light (c), and the wavelength (\(\text{λ}\)) of the emitted light.

In the sodium atom example, when an electron transitions between two energy levels, it emits a photon with a specific energy, which we calculated as \(3.37 \times 10^{-19} \text{ J}\). It is this energy that is exactly the difference between the two energy levels the electron transitioned between. This number is not arbitrary; it corresponds with the color (wavelength) of light we see, in this case, yellow from a sodium lamp.

Wavelength to Energy Conversion

Relating a photon's wavelength to its energy is a key aspect of understanding electromagnetic radiation and spectral lines. It demonstrates the inversely proportional relationship between wavelength and energy. To convert the wavelength of light into energy, we utilize the formula \(E = \frac{hc}{\lambda}\).

Conversion to different units, such as electron volts (eV) or kilocalories per mole (kcal/mol), is frequently required in different scientific contexts. For example, electron volts are a common unit in physics for quantifying particle energies, while kilocalories per mole are often used in chemistry to express energies per number of molecules. These conversions make use of known constants and conversion factors, enabling scientists and students alike to communicate and compare energy values across different disciplines and applications.