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Chapter 6: Problem 76

If you put 120 volts of electricity through a pickle, the pickle will smoke and start glowing orange-yellow. The light is emitted because sodium ions in the pickle become excited; their return to the ground state results in light emission. (a) The wavelength of this emitted light is \(589 \mathrm{nm} .\) Calculate its frequency. (b) What is the energy of 0.10 mole of these photons? (c) Calculate the energy gap between the excited and ground states for the sodium ion. (d) If you soaked the pickle for a long time in a different salt solution, such as strontium chloride, would you still observe \(589-\mathrm{nm}\) light emission? Why or why not?

Short Answer

Expert verified
The frequency of the emitted light is \(5.09 \times 10^{14} Hz\). The energy of 0.10 mole of these photons is \(4.01 \times 10^{-17} J\). The energy gap between the excited and ground states for the sodium ion is \(4.01 \times 10^{-17} J\). If the pickle was soaked in another salt solution like strontium chloride, we would not observe the 589 nm light emission because the resulting light emission would depend on the properties of the ions present in the new salt solution.

Step by step solution

01

(a) Calculate the frequency of the emitted light

To calculate the frequency (ν) of the emitted light, we will use the equation: \(c = λν\) where c is the speed of light \( (3 \times 10^8 m/s) \) and λ is the wavelength of the light (589 nm). Remember to convert the wavelength from nm to meters before using the formula: \(λ = 589 nm = 589 \times 10^{-9} m\) Now, we can calculate the frequency: \( ν = \frac{c}{λ} = \frac{3 \times 10^8 m/s}{589 \times 10^{-9} m} = 5.09 \times 10^{14} Hz \) The frequency of the emitted light is \(5.09 \times 10^{14} Hz\).
02

(b) Calculate the energy of 0.10 mole of photons

To find the energy of 0.1 mole of photons, we will use the equation: \(E = nhν\) where E is the energy, n is the number of moles, h is the Planck's constant \((6.626 \times 10^{-34} Js)\), and ν is the frequency ( \(5.09 \times 10^{14} Hz\)) calculated in part (a). Calculate the energy: \(E = (0.10 \,\text{mol}) (6.022 \times 10^{23}\, \text{photons/mol}) (6.626 \times 10^{-34} Js) (5.09 \times 10^{14} Hz) = 4.01 \times 10^{-17} J\) The energy of 0.10 mole of these photons is \(4.01 \times 10^{-17} J\).
03

(c) Calculate the energy gap between the excited and ground states for the sodium ion

Since we have already calculated the energy of one photon in part (b), we can use the same result to find the energy gap between the excited and ground states for the sodium ion: Energy gap = Energy of one photon \(ΔE = 4.01 \times 10^{-17} J\) The energy gap between the excited and ground states for the sodium ion is \(4.01 \times 10^{-17} J\).
04

(d) Light emission for a pickle soaked in strontium chloride

If the pickle was soaked in another salt solution like strontium chloride, we would not observe the 589 nm light emission. This is because the excited sodium ions in the initial pickle are responsible for this particular wavelength. When soaked in another salt solution, the other ions would become excited and produce different wavelengths of light. The resulting light emission would depend on the properties of the ions present in the new salt solution, which would most likely not result in a 589 nm light emission.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Electromagnetic Spectrum
The electromagnetic spectrum is a comprehensive range of all possible frequencies of electromagnetic radiation. It includes, in order of increasing frequency and decreasing wavelength, radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Visible light, which is just a small part of the spectrum, is what humans can see with the naked eye. The light emitted from a glowing pickle due to excited sodium ions falls into this visible range, specifically characterized by a certain wavelength—measured in nanometers (nm)—which determines its color, in this case, orange-yellow.
Planck's Constant and Its Role in Emission Spectroscopy
Planck's constant (symbolized as h) is a fundamental constant in quantum mechanics, with a value of approximately 6.626 x 10^-34 joule-seconds. This constant is key to understanding emission spectroscopy, as it relates the energy of a photon to its frequency through the equation E = hν. When solving for the energy of a photon associated with the 589 nm light from the pickle exercise, Planck's constant is an essential component. Using this constant, one can calculate the energy of multiple photons (in moles) and understand the energy transitions in atoms, like those in the sodium ions of the pickle.
Excited States: The Origin of Light Emission
When an electron in an atom absorbs energy, it moves from a lower-energy ground state to a higher-energy excited state. This energy can come from various sources, such as electrical currents, which was the case with the sodium ions in the pickle. When the electron returns to its ground state, it releases the absorbed energy in the form of light, a process which is the essence of emission spectroscopy. The wavelength (and thus the color) of the emitted light is determined by the energy difference between the excited and ground states. This energy gap is crucial to understanding the color change when the pickle is soaked in different solutions.
The Ground State: Sodium Ions' Starting Point
The ground state of an atom or ion is its lowest energy state. It is the starting point for any excitation process. In the context of the glowing pickle, sodium ions initially exist in their ground state. Once excited by electrical energy, they transition to higher energy levels. When these excited ions return to their ground state, they emit light of a specific wavelength which corresponds to the energy difference between these states. This phenomenon is evidenced by the distinctive orange-yellow glow of the sodium emissions. In emission spectroscopy, understanding the ground state helps predict how an atom or ion will behave when interacting with energy.

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Most popular questions from this chapter

(a) Using Equation \(6.5,\) calculate the energy of an electron in the hydrogen atom when \(n=2\) and when \(n=6 .\) Calculate the wavelength of the radiation released when an electron moves from \(n=6\) to \(n=2 .\) (b) Is this line in the visible region of the electromagnetic spectrum? If so, what color is it?

It is possible to convert radiant energy into electrical energy using photovoltaic cells. Assuming equal efficiency of conversion, would infrared or ultraviolet radiation yield more electrical energy on a per-photon basis?

Suppose that the spin quantum number, \(m_{s}\), could have three allowed values instead of two. How would this affect the number of elements in the first four rows of the periodic table?

Bohr's model can be used for hydrogen-like ions -ions that have only one electron, such as \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+}\). (a) Why is the Bohr model applicable to \(\mathrm{He}^{+}\) ions but not to neutral He atoms? (b) The ground-state energies of \(\mathrm{H}, \mathrm{He}^{+},\) and \(\mathrm{Li}^{2+}\) are tabulated as follows: By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z\). (c) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{C}^{5+}\) ion.

Sodium metal requires a photon with a minimum energy of \(4.41 \times 10^{-19} \mathrm{~J}\) to emit electrons. (a) What is the minimum frequency of light necessary to emit electrons from sodium via the photoelectric effect? (b) What is the wavelength of this light? (c) If sodium is irradiated with light of \(405 \mathrm{nm},\) what is the maximum possible kinetic energy of the emitted electrons? (d) What is the maximum number of electrons that can be freed by a burst of light whose total energy is \(1.00 \mu \mathrm{J}\) ?
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