Waves - MCAT Chemical and Physical Foundations of Biological Systems
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If the first overtone of a sound wave in a pipe with one end open and one end closed has a frequency of 300Hz, what is the frequency of the second overtone?
If the first overtone of a sound wave in a pipe with one end open and one end closed has a frequency of 300Hz, what is the frequency of the second overtone?
In an open-closed pipe, we get only odd harmonics, since there must always be a displacement node at the closed end and a displacement antinode at the open end. So, the first overtone is the same as the third harmonic, which has 
. Similarly, the second overtone is the fifth harmonic, which has 
. Using the given value of 
in the first overtone to be 300Hz, we find 
and 
.
In an open-closed pipe, we get only odd harmonics, since there must always be a displacement node at the closed end and a displacement antinode at the open end. So, the first overtone is the same as the third harmonic, which has
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A string of length 2.5m is struck producing a harmonic. The wavelength produced is 1.25m. Which harmonic is the string vibrating at?
A string of length 2.5m is struck producing a harmonic. The wavelength produced is 1.25m. Which harmonic is the string vibrating at?
A harmonic is a standing wave that has certain points (nodes) which do not go through any displacement, and certain points (antinodes) that are moving through maximum displacement during the wave's resonation.
Wavelength is related to harmonic on a string fixed at both ends through the formula 
, where 
is the length of the string and 
is the number harmonic. By plugging in the given length and wavelength, we can solve for 
.
The string must be vibrating at the fourth harmonic.
A harmonic is a standing wave that has certain points (nodes) which do not go through any displacement, and certain points (antinodes) that are moving through maximum displacement during the wave's resonation.
Wavelength is related to harmonic on a string fixed at both ends through the formula
The string must be vibrating at the fourth harmonic.
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At a local concert, a speaker is set up to produce low-pitched base sounds with a frequency range of 20Hz to 200Hz, which can be modeled as sine waves. In a simplified model, the sound waves the speaker produces can be modeled as a cylindrical pipe with one end closed that travel through the air at a velocity of 
, where T is the temperature in °C.
How are the first three harmonics of the base speaker designated?
At a local concert, a speaker is set up to produce low-pitched base sounds with a frequency range of 20Hz to 200Hz, which can be modeled as sine waves. In a simplified model, the sound waves the speaker produces can be modeled as a cylindrical pipe with one end closed that travel through the air at a velocity of
How are the first three harmonics of the base speaker designated?
First, notice that the paragraph above tells us that the wave can be modeled as a pipe with one end closed. This is in contrast to the other possibility, where the wave is modeled as a pipe with two ends open. It is critical to recognize this difference, as the definition of sequential harmonics and the formula used to calculate them changes depending on whether both ends are open or not. In the situation where one end is closed, the harmonics are odd numbers, meaning that the first three harmonics are 1st, 3rd, and 5th. In the situation where both ends are open or both ends are closed, the harmonics are sequential, meaning that the first three harmonics are 1st, 2nd, and 3rd.
First, notice that the paragraph above tells us that the wave can be modeled as a pipe with one end closed. This is in contrast to the other possibility, where the wave is modeled as a pipe with two ends open. It is critical to recognize this difference, as the definition of sequential harmonics and the formula used to calculate them changes depending on whether both ends are open or not. In the situation where one end is closed, the harmonics are odd numbers, meaning that the first three harmonics are 1st, 3rd, and 5th. In the situation where both ends are open or both ends are closed, the harmonics are sequential, meaning that the first three harmonics are 1st, 2nd, and 3rd.
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At a local concert, a speaker is set up to produce low-pitched base sounds with a frequency range of 20Hz to 200Hz, which can be modeled as sine waves. In a simplified model, the sound waves the speaker produces can be modeled as a cylindrical pipe with one end closed that travel through the air at a velocity of 
, where T is the temperature in °C.
What is the closest distance a person could be standing away from the speaker in order to hear the loudest fundamental frequency of a 1.2m wavelength wave at the concert?
At a local concert, a speaker is set up to produce low-pitched base sounds with a frequency range of 20Hz to 200Hz, which can be modeled as sine waves. In a simplified model, the sound waves the speaker produces can be modeled as a cylindrical pipe with one end closed that travel through the air at a velocity of
What is the closest distance a person could be standing away from the speaker in order to hear the loudest fundamental frequency of a 1.2m wavelength wave at the concert?
This question asks us to incorporate information we learned in the pre-question text and new information in the question. First, we are told that we are looking to calculate a distance where maximal sound is heard, in other words, where the amplitude is at its maximum. This only occurs at an anti-node.
Additionally, we know that the speaker can be modeled as a one-end closed pipe, meaning that the wavelengths of the harmonics are calculated as 
.
In a one-end closed pipe model, each harmonic ends with maximal amplitude, so we can find L (the length of the pipe, i.e. where the person standing would hear maximal sound), as we know the fundamental harmonic has n = 1.
This question asks us to incorporate information we learned in the pre-question text and new information in the question. First, we are told that we are looking to calculate a distance where maximal sound is heard, in other words, where the amplitude is at its maximum. This only occurs at an anti-node.
Additionally, we know that the speaker can be modeled as a one-end closed pipe, meaning that the wavelengths of the harmonics are calculated as
In a one-end closed pipe model, each harmonic ends with maximal amplitude, so we can find L (the length of the pipe, i.e. where the person standing would hear maximal sound), as we know the fundamental harmonic has n = 1.
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You observe several stars in the distance with varying colors. Which of the following stars would have the highest surface temperature?
You observe several stars in the distance with varying colors. Which of the following stars would have the highest surface temperature?
The light portion of the electromagnetic spectrum, from lowest to highest frequency, is red, orange, yellow, green, blue, indigo, violet (ROYGBIV).
Frequency is proportional to temperature, and wavelength is inversely proportional to frequency. Since the energy level corresponds with the temperature, objects that emit a higher frequency and shorter wavelength photon will have higher energy. This corresponds with violet, as it is the highest frequency (shortest wavelength) of visible light
The light portion of the electromagnetic spectrum, from lowest to highest frequency, is red, orange, yellow, green, blue, indigo, violet (ROYGBIV).
Frequency is proportional to temperature, and wavelength is inversely proportional to frequency. Since the energy level corresponds with the temperature, objects that emit a higher frequency and shorter wavelength photon will have higher energy. This corresponds with violet, as it is the highest frequency (shortest wavelength) of visible light
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Which of the following waves carry the greatest amount of energy?
Which of the following waves carry the greatest amount of energy?
The energy of a wave increases with increasing frequency and decreasing wavelength. Considering these different waves, radiowaves possess the longest wavelengths and gamma rays the shortest wavelength, thus gamma rays carry the greatest amount of energy.
The energy of a wave increases with increasing frequency and decreasing wavelength. Considering these different waves, radiowaves possess the longest wavelengths and gamma rays the shortest wavelength, thus gamma rays carry the greatest amount of energy.
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An incandescent light bulb is shown through a glass prism. The certain wavelength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
What type of light is produced by the incandescent light bulb?
An incandescent light bulb is shown through a glass prism. The certain wavelength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
What type of light is produced by the incandescent light bulb?
Incandescent light bulbs produce visible light of all wavelengths. The mix of red, orange, yellow, green, blue, indigo, and violet (ROYGBIV) give the light its characteristic white appearance. For the MCAT, it is important to know the relative wavelengths of light for the visible spectrum (390 – 700nm) and where visiable wavelengths fit into the overall spectrum of electromagnetic radiation. From longest wavelength to shortest, the sequence of wavelengths is listed below.
Radio > Microwaves > Infrared > Visible > Ultraviolet > X-Rays > Gamma Rays
Incandescent light bulbs produce visible light of all wavelengths. The mix of red, orange, yellow, green, blue, indigo, and violet (ROYGBIV) give the light its characteristic white appearance. For the MCAT, it is important to know the relative wavelengths of light for the visible spectrum (390 – 700nm) and where visiable wavelengths fit into the overall spectrum of electromagnetic radiation. From longest wavelength to shortest, the sequence of wavelengths is listed below.
Radio > Microwaves > Infrared > Visible > Ultraviolet > X-Rays > Gamma Rays
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An incandescent light bulb is shown through a glass prism. The certain wavlength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
The prism is designed to select for the color of visible light that has the longest wavelength. What color is this, and what are the expected wavelength ranges?
An incandescent light bulb is shown through a glass prism. The certain wavlength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
The prism is designed to select for the color of visible light that has the longest wavelength. What color is this, and what are the expected wavelength ranges?
We know that the visible spectrum has wavelengths of 390nm to 700nm. The wavelengths of light around the 700nm range are red.
Using the ROYGBIV acronym, we know that red has the longest wavelength and violet the shortest. While this may seen to be a difficult question asking for the wavelengths of light that correspond to certain colors, this is helpful on the MCAT for estimating answers and may be worth the time learning. The expected range of wavelengths for the red portion of the visible spectrum is 650nm to 700 nm. Keep in mind that, because it has the longest wavelength, red light will also have the lowest frequency.
We know that the visible spectrum has wavelengths of 390nm to 700nm. The wavelengths of light around the 700nm range are red.
Using the ROYGBIV acronym, we know that red has the longest wavelength and violet the shortest. While this may seen to be a difficult question asking for the wavelengths of light that correspond to certain colors, this is helpful on the MCAT for estimating answers and may be worth the time learning. The expected range of wavelengths for the red portion of the visible spectrum is 650nm to 700 nm. Keep in mind that, because it has the longest wavelength, red light will also have the lowest frequency.
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Which color of the visible spectrum of light has the highest energy of the answer choices?
Which color of the visible spectrum of light has the highest energy of the answer choices?
The highest energy of any visible light belongs to violet. The greater the wavelength, the lower the energy of the light. The greater the frequency, the higher the energy of the light. This is why ultraviolet light ("ultra" meaning "beyond" violet) is so damaging to DNA. Out of the answer choices, blue light has the lowest wavelength and greatest frequency, making it the highest energy.
The highest energy of any visible light belongs to violet. The greater the wavelength, the lower the energy of the light. The greater the frequency, the higher the energy of the light. This is why ultraviolet light ("ultra" meaning "beyond" violet) is so damaging to DNA. Out of the answer choices, blue light has the lowest wavelength and greatest frequency, making it the highest energy.
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Which of the following is/are ionizing radiation?
- microwaves
- X-rays
- gamma rays
- all of the above
- 2 and 3, but not 1
Which of the following is/are ionizing radiation?
- microwaves
- X-rays
- gamma rays
- all of the above
- 2 and 3, but not 1
Choice 5 is correct. Highly energetic frequencies such as X-rays and gamma rays can displace electrons from materials upon which they impinge. Microwaves are a form of radio waves, which are long-wavelength, low frequency waves with little energy.
Mnemonic: Microwave ovens are fundamentally safe household items.
Choice 5 is correct. Highly energetic frequencies such as X-rays and gamma rays can displace electrons from materials upon which they impinge. Microwaves are a form of radio waves, which are long-wavelength, low frequency waves with little energy.
Mnemonic: Microwave ovens are fundamentally safe household items.
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An incandescent light bulb is shown through a glass prism. The certain wavlength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
Light is a .
An incandescent light bulb is shown through a glass prism. The certain wavlength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
Light is a .
As Einstein determined, light has properties of both a particle and a wave. In the particle sense, it has mass, velocity, and momentum. The wave property of light allows for diffraction, and constructive and destructive interference. For the MCAT, it is important to know that the photon (the particle of light) has both particle and wave properties. In fact, all objects have both particle and wave properties; however, their wave property becomes less obvious with increasing mass.
As Einstein determined, light has properties of both a particle and a wave. In the particle sense, it has mass, velocity, and momentum. The wave property of light allows for diffraction, and constructive and destructive interference. For the MCAT, it is important to know that the photon (the particle of light) has both particle and wave properties. In fact, all objects have both particle and wave properties; however, their wave property becomes less obvious with increasing mass.
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Light is passed through a prism to separate the wavelengths, and red, green and blue portions of the spectrum are sequentially projected onto a photo-electric surface connected to an electrical circuit. Which of these observations is correct?
- Red light, but not green or blue light, will generate an electric current.
- Red light and blue light, but not green light, will generate an electric current.
- Green light, but not red or blue light, will generate an electric current.
- Blue light, but not red or green light, will generate an electric current.
- All the frequencies will generate an electric current if the intensity is adequate.
Light is passed through a prism to separate the wavelengths, and red, green and blue portions of the spectrum are sequentially projected onto a photo-electric surface connected to an electrical circuit. Which of these observations is correct?
- Red light, but not green or blue light, will generate an electric current.
- Red light and blue light, but not green light, will generate an electric current.
- Green light, but not red or blue light, will generate an electric current.
- Blue light, but not red or green light, will generate an electric current.
- All the frequencies will generate an electric current if the intensity is adequate.
Choice 4 is correct because blue light is much more energetic than red light. The most intense low-frequency light will not generate any current at all, because the wavelength is not sufficiently energetic to displace electrons from the photo-electric surface; however, once it is established that a certain frequency is capable of generating current, then the amount of current is dependent upon intensity.
Mnemonic: “Blue is better.”
Choice 4 is correct because blue light is much more energetic than red light. The most intense low-frequency light will not generate any current at all, because the wavelength is not sufficiently energetic to displace electrons from the photo-electric surface; however, once it is established that a certain frequency is capable of generating current, then the amount of current is dependent upon intensity.
Mnemonic: “Blue is better.”
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Which of the following electron transitions in a hydrogen atom would emit a photon of the lowest frequency?
Which of the following electron transitions in a hydrogen atom would emit a photon of the lowest frequency?
A photon is emitted if the electron goes from a higher to lower energy level, so we need a choice where the energy level, n, decreases. Also, we need to look for the transition that has the smallest energy difference, since frequency is proportional to energy (f = E/h, where h is Planck's constant). Higher energy levels are closer together, so the highest pair of levels has the smallest difference in energy and the lowest frequency of emitted photons.
A photon is emitted if the electron goes from a higher to lower energy level, so we need a choice where the energy level, n, decreases. Also, we need to look for the transition that has the smallest energy difference, since frequency is proportional to energy (f = E/h, where h is Planck's constant). Higher energy levels are closer together, so the highest pair of levels has the smallest difference in energy and the lowest frequency of emitted photons.
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The first four energy levels of a hydrogen atom have the energies given above. If a hydrogen atom is initially in the n = 2 state, photons of which of the following energies could be absorbed?
The first four energy levels of a hydrogen atom have the energies given above. If a hydrogen atom is initially in the n = 2 state, photons of which of the following energies could be absorbed?
Absorbing a photon would have the effect of pushing the atom into a higher energy state, in this case n = 3 or n = 4. Photons with an energy equal to the difference betweeen E2 and E3 or between E2 and E4, could be absorbed.
E3 – E2 = –1.51 – (–3.40) = 1.89eV
E4 – E2 = –0.85 – (–3.40) = 2.55eV
Absorbing a photon would have the effect of pushing the atom into a higher energy state, in this case n = 3 or n = 4. Photons with an energy equal to the difference betweeen E2 and E3 or between E2 and E4, could be absorbed.
E3 – E2 = –1.51 – (–3.40) = 1.89eV
E4 – E2 = –0.85 – (–3.40) = 2.55eV
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An incandescent light bulb is shown through a glass prism. The certain wavlength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
How much energy does a photon of red light with a wavelength of 690nm traveling through the solution contain?
An incandescent light bulb is shown through a glass prism. The certain wavlength of the light is then directed into a glass cuvette containing an unknown concentration of protein. Commonly, this process is called spectroscopy and is used to determine the concentrations of DNA, RNA, and proteins in solutions. The indices of reflection of air, glass, and the solution are 1, 1.5, and 1.3, respectively.
How much energy does a photon of red light with a wavelength of 690nm traveling through the solution contain?
This question asks us about the particle nature of light and how much energy a photon would contain. From our light equations, we know that 
, where E is the energy of a single photon, h is Plank’s constant, and f is the frequency of the photon.
From the information in the problem, we need to determine the frequency.
We need to determine the velocity of the light in the solution. We can use the definition of index of refraction to determine this value, along with the speed of light and the index of refraction of the solution.
Now we can compute the frequency.
Substituting the frequency we found, along with Plank’s constant, we can find the energy.
This question asks us about the particle nature of light and how much energy a photon would contain. From our light equations, we know that
From the information in the problem, we need to determine the frequency.
We need to determine the velocity of the light in the solution. We can use the definition of index of refraction to determine this value, along with the speed of light and the index of refraction of the solution.
Now we can compute the frequency.
Substituting the frequency we found, along with Plank’s constant, we can find the energy.
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In the photoelectric effect experiment, incoming photons with enough energy cause the ejection of an electron from a metal plate, proving that light carries energy. Which color of visible light would you expect to produce the fastest ejected electrons?
In the photoelectric effect experiment, incoming photons with enough energy cause the ejection of an electron from a metal plate, proving that light carries energy. Which color of visible light would you expect to produce the fastest ejected electrons?
This question can be approached in two ways. The first way is to have a general understanding of the photoelectric effect, and its equation: 
, where the kinetic energy of an ejected electron is equal to difference between the energy of the incoming photon, and the work function of the metal plate. If the energy of the photon is greater than that of the work function, an electron will be emitted with a speed that is proportional that difference, thus the greater the energy of the photon, the greater the total kinetic energy, and the faster the speed of the outgoing electron.
Since wavelength is inversely proportional to energy, and because we know that blue light has a wavelength around 400nm, and red light approximately 700nm, we would expect blue light to carry the most energy and thus result in the fastest ejected electron.
A second approach to this question is to use critical reasoning. Using the concept of energy conservation, we can predict the energy of the incoming photon will be transferred to the outgoing electron. Because we know that energy is inverse to wavelength, the lowest wavelength photon will have the most energy. Applying conservation of energy principles and the fact that energy is directly proportional to velocity, it is a good assumption to reason that the lowest wavelength photon will create the highest velocity electron. This leads to the answer of blue light.
This question can be approached in two ways. The first way is to have a general understanding of the photoelectric effect, and its equation:
Since wavelength is inversely proportional to energy, and because we know that blue light has a wavelength around 400nm, and red light approximately 700nm, we would expect blue light to carry the most energy and thus result in the fastest ejected electron.
A second approach to this question is to use critical reasoning. Using the concept of energy conservation, we can predict the energy of the incoming photon will be transferred to the outgoing electron. Because we know that energy is inverse to wavelength, the lowest wavelength photon will have the most energy. Applying conservation of energy principles and the fact that energy is directly proportional to velocity, it is a good assumption to reason that the lowest wavelength photon will create the highest velocity electron. This leads to the answer of blue light.
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A student wishes to run an experiment that requires photons with energy between 12.5eV and 13.0eV per photon. The light source is a hydrogen lamp. Which hydrogen electron transition would produce an appropriate photon?
A student wishes to run an experiment that requires photons with energy between 12.5eV and 13.0eV per photon. The light source is a hydrogen lamp. Which hydrogen electron transition would produce an appropriate photon?
To calculate photon energy from an electron transition, we use the following equation.
In the formula, 
is the initial energy level and 
is the final energy level. 
is a constant for the given compound. Our first step will to find the difference described in the formula using the constant given for hydrogen and an estimate for the energy produced.
We can use guess and check to estimate the discrete values that can be used for the electron energy levels.
We find that if the initial energy level is 4 and the final energy level is 1, the value of the difference is approximately -0.94.
An electron transition from energy level 4 to energy level 1 would produce a photon in the appropriate range.
To calculate photon energy from an electron transition, we use the following equation.
In the formula,
We can use guess and check to estimate the discrete values that can be used for the electron energy levels.
We find that if the initial energy level is 4 and the final energy level is 1, the value of the difference is approximately -0.94.
An electron transition from energy level 4 to energy level 1 would produce a photon in the appropriate range.
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What color of visible light contains enough energy to eject an electron off of a metal with a work function of 
?
What color of visible light contains enough energy to eject an electron off of a metal with a work function of
Work function is the minimum amount of energy, in electron-volts, required to remove an electron from a metal surface. To calculate the energy of a single photon of light, one must use the equation:
In this formula, 
is energy in Joules, 
is Planck's Constant, and 
is the speed of light.
To convert from Joules to eV, one must use the given conversion factor.
Use this value in the energy formula to find the wavelength of the light.
The visible spectrum spans from approximately 
to 
, with smaller wavelengths corresponding to violet and blue and larger wavelengths corresponding to red. Even without knowing the exact wavelength correlations in the spectrum, we know that our wavelength is very small and will be found in the violet end of the spectrum.
Work function is the minimum amount of energy, in electron-volts, required to remove an electron from a metal surface. To calculate the energy of a single photon of light, one must use the equation:
In this formula,
To convert from Joules to eV, one must use the given conversion factor.
Use this value in the energy formula to find the wavelength of the light.
The visible spectrum spans from approximately
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The electron shell energies of a particular atom are given below:
Which of the following colors can this atom emit when an electron falls from a higher to lower energy shell?
The electron shell energies of a particular atom are given below:
Which of the following colors can this atom emit when an electron falls from a higher to lower energy shell?
We can see that eV differences between the shells for this atom range anywhere from 
(between n2 and n1) to 
(between n4 and n1). We can use these two values to find the range of wavelengths that can be produced by photons from this atom. To find the wavelength, we will use the formula:
In this equation, 
is the energy in Joules, 
is Planck's constant, 
is the speed of light, and 
is the wavelength.
First, convert the energy changes to Joules using the given conversion factor.
Use these energies in the first equation to solve for the range of possible wavelengths.
The visible spectrum spans from approximately 
to 
, with smaller wavelengths corresponding to violet and blue and larger wavelengths corresponding to red. Since the range of possible wavelengths only overlaps with a small portion of the visible spectrum, we can see that the only color that can be generated when an electron falls from a higher level to a lower level is violet. The other transitions will generate photons in the ultraviolet range.
We can see that eV differences between the shells for this atom range anywhere from
In this equation,
First, convert the energy changes to Joules using the given conversion factor.
Use these energies in the first equation to solve for the range of possible wavelengths.
The visible spectrum spans from approximately
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A sodium lamp emits a yellow light with a frequency of 
. How much energy is contained in 
of photons?
A sodium lamp emits a yellow light with a frequency of
The energy of a single photon is given by the equation:
We are given the frequency and the value of the constant, allowing us to solve.
The above gives the energy contained in one photon. Next, solve for the energy contained in 
using Avogadro's number:
The energy of a single photon is given by the equation:
We are given the frequency and the value of the constant, allowing us to solve.
The above gives the energy contained in one photon. Next, solve for the energy contained in
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