Light, Energy, and the Hydrogen Atom a. Which has the...

Light, Energy, and the Hydrogen Atom a. Which has the greater wavelength, blue light or red light? b. How do the frequencies of blue light and red light compare? C. How does the energy of blue light compare with that of red light? d. Does blue light have a greater speed than red light? e. How does the energy of three photons from a blue light source compare with the energy of one photon of blue light from the same source? How does the energy of two photons corresponding to a wavelength of $451 \mathrm{~nm}$ (blue light) compare with the energy of three photons corresponding to a wavelength of $704 \mathrm{~nm}$ (red light)? f. A hydrogen atom with an electron in its ground state interacts with a photon of light with a wavelength of $1.22 \times$ $10^{-6} \mathrm{~m} .$ Could the electron make a transition from the ground state to a higher energy level? If it does make a transition, indicate which one. If no transition can occur, explain. g. If you have one mole of hydrogen atoms with their electrons in the $n=1$ level, what is the minimum number of photons you would need to interact with these atoms in order to have all of their electrons promoted to the $n=3$ level? What wavelength of light would you need to perform this experiment?

Light, Energy, and the Hydrogen Atom
a. Which has the greater wavelength, blue light or red light?
b. How do the frequencies of blue light and red light compare?
C. How does the energy of blue light compare with that of red light?
d. Does blue light have a greater speed than red light?
e. How does the energy of three photons from a blue light source compare with the energy of one photon of blue light from the same source? How does the energy of two photons corresponding to a wavelength of $451 \mathrm{~nm}$ (blue light) compare with the energy of three photons corresponding to a wavelength of $704 \mathrm{~nm}$ (red light)?
f. A hydrogen atom with an electron in its ground state interacts with a photon of light with a wavelength of $1.22 \times$ $10^{-6} \mathrm{~m} .$ Could the electron make a transition from the ground state to a higher energy level? If it does make a transition, indicate which one. If no transition can occur, explain.
g. If you have one mole of hydrogen atoms with their electrons in the $n=1$ level, what is the minimum number of photons you would need to interact with these atoms in order to have all of their electrons promoted to the $n=3$ level? What wavelength of light would you need to perform this experiment?

Key Concepts

Quantization and Photon Counting in Energy Absorption

Since energy is absorbed in discrete packets (photons), calculations involving multiple photons require summing these quantized energy units. When considering many atoms (e.g., one mole), the concept extends to counting the minimum number of photons needed to cause simultaneous transitions, integrating ideas of quantization with stoichiometry.

Photon Absorption in Electron Transitions

For an electron in an atom to move from one energy level to another, it must absorb a photon whose energy exactly matches the energy gap between the two levels. This principle is key to determining whether a particular photon (characterized by its wavelength) can induce a transition, such as promoting an electron from the ground state to an excited state.

Electromagnetic Spectrum Properties

The electromagnetic spectrum includes all types of light, characterized by wavelength and frequency. In this context, understanding that longer wavelengths correspond to lower frequencies and shorter wavelengths to higher frequencies is fundamental. For instance, red light has a longer wavelength and lower frequency than blue light, a relationship governed by the constant speed of light.

Photon Energy and Planck’s Equation

Photon energy is related to its frequency by Planck’s equation (E = h·f) and inversely to its wavelength (E = h·c/?). This means that photons with higher frequencies (or shorter wavelengths) have higher energy. This principle explains why blue light photons carry more energy than red light photons.

Hydrogen Atom Energy Levels

The hydrogen atom has discrete energy levels as described by the Bohr model. Electrons can only exist in these quantized states, and transitions between levels occur only when the electron absorbs or emits a photon with energy equal to the difference between these levels. This concept is crucial for understanding photon-induced electron transitions.

Transcript

00:01 We're looking at the introduction to the quantum mechanical model of the atom, and we're thinking about electrons as probabilistic matter waves.

00:08 So looking at the visible spectrum, we know that the red light has a greater wavelength, where wavelength is described as lambda, because it lies to the right of the visible region of the electromagnetic spectrum.

00:29 So in the next part, we need to look at the following equation where we have the frequency is equal to the speed of light divided by the, the wavelength, so the wavelength and frequency are inversely related, so the wavelength of the red is greater compared to the blue, so therefore the frequency of red will be less than blue.

00:54 So continuing on the next equation that we need to pay some attention to is e is equal to hb, so we've got energy, plants constant, and the frequency again.

01:03 So the red light has a lower frequency, so the energy of the red light will be less.

01:14 So both red and blue light will travel at the same speed in a vacuum, but in another medium red light will travel faster than the blue light.

01:46 So that's explained by the following formula.

01:48 Mue is equal to c over n, where mute is a speed of light in medium, where mute is the frequency, c is the speed of light, and n is the refractive index.

02:03 So the refractive index of blue light is greater than that of the bright.

02:06 Red light and therefore it travels with less speed.

02:10 So we know the energy of one photon that is equal to e equals h mu, and the energy of three photons equals to e equals 3h mu.

02:21 So the energy of three photons is a lot greater than the energy of one photon if energy is emitted from the same blue light source.

02:29 So the energy of the two photons, we can calculate on the next page.

02:34 So we have the energy is equal to two lot, of 6 .626 times 10 to the negative 34 joules per second, multiply by 2 .998 times 10 to the 8 meters per second.

02:52 Divide that by 451 times 10 to the minus 9 meters...

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