Lesson 2: Exponential Decay and Atom Structure
Calculate the energy of a photon of wavelength 662 nm NOTE: h= 6.626 x 10^-34 J*s nano (n) = 10^-9 E= hf = hc/λ
3 x 10^-19
If an electron emits 3 eV of energy, what is the corresponding wavelength of the emitted photon? (NOTE: 1 eV = 1.60 x 10^-19 J, h = 6.626 x 10^-34 J*s) nano (n) = 10^-9 E= hf = hc/λ
400 x 10^-9 or 400 nm
Planck's constant
6.626 x 10⁻³⁴ J*s
The highest peak revealed on a mass spectra is a called ____ ____?
base peak *NOTE: The base peak may or may not be the parent peak **NOTE: On the spectra graph, the base peak is always made equal to 100% relative abundance (y-axis) & every other peak is compared to this base peak.
Once we know the frequency, we can easily find the wavelength (λ) according to the equation: ____
c = fλ
The energy that an electron has at some orbit is known as energy level or ____ _____ of an electron
energy state
If shining light on an electron does not excite that electron to a new energy state, then increasing intensity (amount) of light will have no effect. => b/c amount of energy is constant for any given frequency. However, increasing ____ of light will eventually allow it to excite the electron.
frequency (ƒ)
The orbit closest to the nucleus has the lowest energy level and an electron in this orbit is said to be in the ____ ____. Electrons higher up are said to be in an ____ _____.
ground state excited state
Einstein proposed that light behaved like a stream of particles called photons with an energy of ___{EQN}___
E = hν or E = hf *NOTE: Frequency symbol is both greek symbol "nu" (or "v") and "f".
Photoelectric effect equation
E photon = KE electron + Φ
The energy of each photon is proportional to the frequency of the light, given by: _______
E=hf E= energy h= planck's constant (6.626 x 10^-34) f= frequency of the light
The energy of the incident photon must be equal to the sum of the metal's work function and the photoelectron kinetic energy:
Ephoton = KEelectron + Φ
Ephoton = ?
Ephoton = planck's constant * frequency = hf NOTE: Sometimes, frequency is symbolized by "nu" which looks like a v. Do not confuse with velocity. So, Ephoton = hf or hv
Ground state
Generally, the state of lowest energy, in which all electrons are in the lowest possible orbitals. NOTE: All systems tend toward minimal energy; thus on the MCAT, atoms of any element will generally exist in the ground state unless subjected to extremely high temperatures or irradiation.
The peak of the parent ion is called ____ ____?
"parent peak" designated as "p" (aka "molecular ion peak" sometimes designated as "M")
If a photon of wavelength 525 nm hits metallic cesium (work function 3.43 x 10⁻¹⁹), what is the velocity of the photoelectron produced? NOTE: Ephoton = KE electron + Φ Ephoton = hv --> h= 6.626 x 10⁻³⁴ c = (wavelength)(frequency) = (λ)(v) --> c= 2.998 x 10⁸ *mass of electron = (9.11 x 10⁻³¹)
*mass of electron = (9.11 x 10⁻³¹) *Ephoton = KE electron + Φ --> KE = Ephoton - Φ --> Φ= 3.43 * 10⁻¹⁹ --> KE = ½mv² *Ephoton = hv --> h= 6.626 x 10⁻³⁴ c = (wavelength)(frequency) = (λ)(v) c= speed of light = 2.998 * 10⁸ λ= wavelength v= frequency c = (wavelength)(frequency) = (λ)(v) → v = c/λ → λ = 525nm = 525 * 10⁻⁹ Ephoton = hc/λ = [(6.626 * 10⁻³⁴)(2.998 * 10⁸)}/ (525 * 10⁻⁹) = 3.78 * 10⁻¹⁹ then, KEe = Ephoton - Φ = (3.78 * 10⁻¹⁹)-(3.43 * 10⁻¹⁹) = .35 * 10⁻¹⁹ = 3.5 * 10⁻²⁰ PAUSE!: This is enough energy to emit an electron with a photon because it's greater than the work function. then, KE = ½mv² so, ½mv² = 3.5 * 10⁻²⁰ J => √v² = = √(3.5 * 10⁻²⁰)/[½(9.11 * 10⁻³¹)] => v = 2.8 * 10⁵ m/s *NOTE: Recall that work function is the minimum energy required to emit an electron from a sheet of metal with a beam of light (photon). The value changes depending on the type of metal.
The energy level of an electron when infinitely far away from the attractive pull of the nucleus = ____
0
The Rutherford planetary model of the atom did not describe the structure of the atom very accurately because of what 2 reasons:
1.) Did not include the quantum theory of energy- did not incorporate the fact that energy is quantasized. 2.) Could not explain the line spectra; failed to explain why atoms emitted electromagnetic radiation with specific frequencies.
Explain in 2 statements the core principles of the Bohr model as proposed by Niels Bohr: Consider: 1.) Electrons movement in a single orbit 2.) Energy levels electrons as the move across orbitals.
1.) Electrons move about the nucleus in a circular orbit and each orbit corresponds to a discrete quantity of energy. As the electron moves around (in an orbit), it does not radiate any energy. 2.) Each possible orbit is called a stationary state and electrons EMIT (LOSE) ENERGY, in the form of electromagnetic radiation or a photon of light, only when an electron moves from a higher orbit to a lower orbit. Conversely, an electrons GAIN (ABSORB) ENERGY when they move from a lower to a higher energy level.
Find the wavelength of light that corresponds to an electron moving from n=3 to n=2 in a Hydrogen atom. E₃ = 1.5 eV E₂ = -3.4 eV 1.602 x 10⁻¹⁹ J = 1 eV E = hf c = fλ ; f = c/λ
1.) Find energy difference E₃ - E₂ = (1.5 eV) - (-3.4eV) = +1.9eV 2.) Convert electron volts (eV) to joules (J) (1.602 x 10⁻¹⁹ J/eV)(1.9 eV) = 3.04 x 10⁻¹⁹ 3.) Find the wavelength (must convert frequency to wavelength) E = hf = hc/λ => λ = hc/E = 654 nm Conclusion: 654 nm is the wavelength of the photon of the light emitted when the electron moves from the excited state in n=3 to n=2.
Though not completely accurate, give 2 reasons why the Bohr model is a useful model:
1.) Incorporates quantum theory which states that each electron moves at discrete energy levels. As electrons move down energy levels, they release photons with a wavelength (λ) that correspond to the energy needed to make the transition. 2.) It can be used to explain the occurrence of the discrete line spectra produced by individual atoms. e.g. Hydrogen Atom At room temperature, the hydrogen atom is in the ground state (n=1). If we increase the temperature, an electron in the ground state may gain enough energy to jump to a higher orbital (e.g. n = 2). Once in the excited state, the electron can now jump back down to a lower energy level, thereby releasing a photon. These photons correspond to the observed spectral lines.
Explain in more detail the simple mechanism (below) of a mass spectrometer? 1.) Ionization 2.) Acceleration ion over potential difference. 3.) Magnetic field chamber
1.) Ionization: e.g. removing an electron of a particle and given a charge of a positive ion. 2.) Accelerate ion over a potential difference (∆V). Accelerating voltage is applied to the charge-achieving a final velocity (v) 3.) Particle moves into a magnetic field chamber which exerts magnetic force on the particle giving it a trajectory of centripetal motion. The radius of the centripedal motion is function of the particle's mass. (m α r
What is mass spectrometry?
A field in chemistry that studies and analyzes molecules by using their molecular mass.
mass spectra
A graph that charts the collected data from the mass spectrometer. It can be analyzed by plotting the relative abundance versus mass-to-charge ratio on the xy-plane. NOTE: If the charge on the ion is equal to +1, the mass-to-charge ratio is simply equal to the mass of that ion.
Mnemonic: As electrons go from a lower energy level to a higher energy level, they get AHED: What does AHED signify?
A- Absorb light H- Higher potential E- Excited D- Distant (from the nucleus)
CONCEPT CHECK: As the wavelength of a photon increases, what happens to the photon's energy?
According to Planck's equation, the energy of a photon is proportional to the light frequency, "ν": E = hν The light frequency (ν) is inversely proportional to wavelength (λ): c = λν => λ = c/ν or ν = c/λ where c is the speed of light. That means that increasing the wavelength decreases the light's frequency. Therefore, as the wavelength of a photon increases, its energy decreases.
E = - RH/n^2 In the above equation, what is the proportional relationship between E and n?
At first glance, it may not be clear that the energy (E) is directly proportional to the principal quantum number (n). However, take notice of the negative sign, which causes the values to approach zero from a more negative value as n increases (thereby increasing the energy). Negative signs are as important as a variable's location in a fraction when it comes to determining proportionality. NOTE: This is an important point: while the magnitude of the fraction is getting smaller, the actual value it represents is getting larger (becoming less negative).
Because each element can have its electrons excited to a different set of distinct energy levels, each possesses a unique ___ ___ spectrum, which can be used as fingerprint for the element.
Atomic Emission NOTE: One particular application of atomic emission spectroscopy is in the analysis of stars and planets: while a physical sample may be impossible to procure, the light from a star can be resolved into its component wavelengths, which are then matched to the known line spectra.
If the blue light of frequency 6.00 x 10^14 Hz is incident on rubidium (W = 2.26 eV), will there be photoejection of electrons? If so, what is the maximum kinetic energy that an ejected electron will carry away? (Note: h= 6.626 x 10^-34 J*s = 4.14 x 10^-15 eV*s)
If the photons have a frequency of 6.00 x 10^14 Hz, each photon has an energy of: E = hf = (4.14 x 10^-15 eV*s)(6.00 x 10^14 Hz) = 2.48 eV Clearly then, any given photon has more than enough energy to allow an electron in the metal to overcome the 2.26 eV barrier. In fact, the maximum excess kinetic energy carried away by the electron turns out to be: K = hf - W = 2.48 - 2.26 = 0.22 eV
Excited state
In general, when at least one electron has moved to a subshell of higher than normal energy.
The reason that we only discuss electrons being ejected from metals (and not protons or neutrons) is because of the weak hold that metals have on their valence electrons due to their low _____ ______.
Ionization energy
The photoelectric effect is simply another example of energy transfer in which light energy causes an increase in electrical potential energy in the atom--enough to allow the electron to escape. If any energy is "left over," it cannot be destroyed. Rather it is transferred into ____ ____ in the ejected electron.
Kinetic Energy
If the frequency of a photon of light incident on a metal is at the threshold frequency for the metal, the electron barely escapes from the metal. However, if the frequency of an incident photon is above the threshold frequency of the metal, the photon will have more than enough energy to eject a single electron, and the excess energy will be converted to kinetic energy in the ejected electron. We can calculate the maximum kinetic energy of the ejected electron with the formula:
Kmax= hf - W W= work function
On a mass spectra, what axis is the mass (m):charge (q or z) ratio and what does it designate?
Mass to charge ratio sometimes written as: m/q or m/z is on the X-axis and designates the mass:charge ratio. m= molecular mass q or z = charge of ion (typically +1)
Bohr then related the permitted angular momentum values to the energy of an electron to obtain:
NOTE: Rydberg unit of energy (RH) = 2.18 x 10⁻¹⁸ J/electron
What is a fragmentation pattern in mass spectra?
Notice that the mass spectra graphs do not simply consist of "p" and "p + 1" peaks. There are many other peaks formed that correspond to lower weight molecules. This is known as the fragmentation pattern. It turns out that as you input energy and ionize the molecule, it can undergo other reactions in which it fragments into lower mass molecules and these can appear on the mass spectra.
The base peak equals the parent peak. (T/F)
Trick question! Could be True or False. The base peak may or may not be the parent peak
What is the "p + 1 peak"?
Recall that certain elements have isotopes that result in heavier masses (if the isotope has more neutrons). The "p + 1 peak" is the same molecule represented in the "parent peak", given from the "parent ion", but it corresponds to original molecule containing heavier isotopes (e.g. hydrogen and carbon, which are involved in many molecular structures, have various different isotopes)
What is a mass spectrometer?
TL;DR A mass spectrometer ionizes atoms and molecules with a high-energy electron beam and then deflects the ions through a magnetic field based on the mass-to-charge ratio of the ion, m/z. A mass spectrometer is a device that is used in analytical chemistry to determine the molecular weight (mass) of some unknown molecule. It uses an electric current to ionize the unknown sample. These ions are then accelerated using electric and magnetic fields that can be finely tuned to make sure the right molecules pass through the slits at the end. These unknown ions can then be detected at the ion detector.
What is the relationship between the Bohr model and Line Spectra?
The Bohr Model can be used to explain the occurrence of the discrete line spectra produced by individual atoms.
What electrical phenomenon results from the application of the photoelectric effect?
The accumulation of moving electrons creates a current during the photoelectric effect.
Balmer series
The group of hydrogen emission lines corresponding to transitions form energy levels n≥3 to n=2
Paschen Series
The group of hydrogen emission lines corresponding to transitions form energy levels n≥4 to n=3
Lyman Series
The group of hydrogen emission lines corresponding to transitions from energy levels n≥2 to n=1
Usually the charge of an ion in mass spectrometry is +1. In this case, what is typically the mass value?
The mass value would be equal to whatever m is because the x-axis is indicated by the m:q (aka m:z) ratio. If m/+1 , then the value of m is whatever m equals.
Who discovered the photoelectric effect?
The photoelectric effect was first observed by German physicist Heinrich Hertz in 1887. Hertz noticed that when certain frequencies of light were shone on a metal, the metal would sometimes exhibit a spark. Later, J.J. Thomson identified these sparks as light-excited electrons leaving the surface of the metal.
relative abundance
The relative abundance of an isotope is the fraction of a single element that exists on Earth with a specific atomic mass. The relative abundances of each isotope can be determined using mass spectrometry.
What does the threshold frequency depend upon?
The threshold frequency depends on the chemical composition of a material (that is, the identity of the metal).
How does the work function relate to the energy necessary to emit an electron from a metal?
The work function describes the minimum amount of energy necessary to emit an electron. Any additional energy from a photon will be converted to excess kinetic energy during the photoelectric effect.
work function
The work function, Φ, is the minimum amount of energy required to induce photoemission of electrons from a metal surface, and the value of Φ depends on the metal.
What is the purpose of a mass spectrometer?
To determine the mass of a particle. Often used in determining the mass of different isotopes.
On the spectra graph, the base peak is always made equal to 100% relative abundance (y-axis) & every other peak is compared to this base peak. (T/F)
True! All other peaks are in relative abundance to the base peak which corresponds to 100%
∆E is the same for absorption or emission between any two energy levels according to the conservation? (T/F)
True, due to to the conservation of energy. This is also the same as the energy of the photon of light absorbed or emitted.
Relate the following variables: 1.) energy of photon 2.) work function 3.) energy of an electron
We can analyze the frequency relationship using the law of conservation of energy. The total energy of the incoming photon, E photon, must be equal to the kinetic energy of the ejected electron, KE electron, plus the energy required to eject the electron from the metal. The energy required to free the electron from a particular metal is also called the metal's work function, which is represented by the symbol Φ (in units of J): E photon = KE electron + Φ Like the threshold frequency ν₀, the value of Φ also changes depending on the metal. We can now write the energy of the photon in terms of the light frequency using Planck's equation: E photon = hν = KE electron + Φ Rearranging this equation in terms of the electron's kinetic energy, we get: KE electron = hν − Φ We can see that kinetic energy of the photoelectron increases linearly with ν as long as the photon energy is greater than the work function Φ, which is exactly the relationship shown in graph (a) above. We can also use this equation to find the photoelectron velocity, v, which is related to KE electron as follows: KE electron = hν − Φ = ½mv² where "m" is the rest mass of an electron, 9.1094×10⁻³¹ kg.
photon emission
When an electron moves from a higher energy level to a lower energy level, it emits electromagnetic radiation in the form of light (aka photon) that has a direct relationship with the intensity of frequency.
What is the photoelectric effect?
When light shines on a metal, electrons can be ejected from the surface of the metal in a phenomenon known as the photoelectric effect. This process is also often referred to as photoemission, and the electrons that are ejected from the metal are called photoelectrons. In terms of their behavior and their properties, photoelectrons are no different from other electrons. The prefix, "photo-", simply tells us that the electrons have been ejected from a metal surface by incident light.
On a mass spectra, what axis is the relative abundance and what does it identify?
Y-axis Depends on the intensity of the ion molecule that is colliding with ion detector.
Electrons liberated from the metal by the photoelectric effect will produce a net charge flow per unit time, or _____. Provided that the light beam's frequency is above the threshold frequency of the metal, light beams of greater intensity produce larger _____ in this way. The higher the intensity of the light beam, the greater the number of photons per unit time that fall on an electrode, producing a greater number of electrons per unit time liberated from the metal When the light's frequency is above the threshold frequency, the magnitude of the resulting current is directly proportional to the intensity (and amplitude) of the light beam.
current
According to the equations: c = fλ and E = hf waves with _____ frequency have ____ wavelengths and ____ energy (toward the blue and ultraviolet end of the spectrum); waves with ____ frequency have _____ wavelengths and _____ energy (toward the red and infrared end of the spectrum).
higher frequency--> shorter wavelengths --> higher energy (toward the blue and ultraviolet end of the spectrum) lower frequency --> longer wavelengths --> lower energy (toward the red and infrared end of the spectrum)
To move an electron further from the nucleus, energy must be input; hence, energy becomes less negative as the electron is moved away. The minimum energy required to move an electron infinitely far from the attractive pull of atom's nucleus is the ____ ____.
ionization energy
Angular momentum
l = linear momentum p = mass x velocity (of electron) r = position of the particle from some defined point that we can assign a variable (e.g. "Q")
The photoelectric effect is for all intents and purposes, an "all-or-nothing" response: if the frequency of the incident photon is ____ than the threshold frequency (f __< or >___ fT), then no electron will be ejected because the photons do not have sufficient energy to dislodge the electron from its atom. But if the frequency of the incident photon is _____ than the threshold frequency (f __< or >___ fT), then an electron will be ejected and the maximum kinetic energy of the ejected electron will equal to the difference between hf and hfT (AKA the work function).
less, f < fT greater/more, f > fT
In nuclear physics, wavelength is commonly measured in ___choose prefix___meters and _____
nano-meters (1 nm = 10^-9 m) angstroms (1A = 10^-10 m)
An electron found in the ground state is assigned a __+/-__ energy.
negative
As an electron moves closer to the nucleus, the energy level becomes more ____ --decreasing in energy. As it moves away from the nucleus, the energy level becomes less ____--increasing in energy.
negative, negative
The ion that results from the parent molecule that is placed in a mass spectrometer is called the ____ ____?
parent ion (aka molecular ion)
When light is sufficiently high frequency (typically, blue to ultraviolet light) is incident on a metal in a vacuum, the metal atoms emit electrons. This phenomenon, discovered by Heinrich Hertz in 1887, is called the ____ ____
photoelectric effect
When light shines on a metal, electrons can be ejected from the surface of the metal in a phenomenon known as the ____ ____ .
photoelectric effect (aka photoemission)
electrons that are ejected from the metal are called _____.
photoelectrons *NOTE: The prefix, "photo-", simply tells us that the electrons have been ejected from a metal surface by incident light.
The light beam consists of an integral number of light quanta called ____. The energy of the each photon is ____ to the frequency of the light: What is the equation? _____
photons, proportional, Ephoton = hf = (planck's constant)(frequency)
Quantum mechanics also states that energy of electrons is _____ , moving energy levels in discrete units.
quantized.
The minimum frequency of light that causes ejection of electrons is known as the ____ ____. It depends on the type of metal being exposed to the radiation.
threshold frequency
The ____ ____ is the minimum energy required to eject an electron and is related to the threshold frequency (fT) of that metal by: W=h(fT)
work function *NOTE: Think of the work function like activation energy, in the sense that it must be matched or exceeded to cause the reaction (escape of an electron) to occur.
How do we relate wavelength to frequency?
λ = c/ƒ => ƒ= c/λ λ= wavelength ƒ = frequency
Quantum theory demonstrates that electromagnetic energy is carried in quantasized/discrete units given by what equation:
∆E = hƒ = (Planck's constant)(frequency) h= 6.626 x 10⁻³⁴ ƒ = frequency NOTE: Recall how we get direct proportionality between frequency and energy of a photon: Based on the wave model of light, physicists predicted that increasing light amplitude would increase the kinetic energy of emitted photoelectrons, while increasing the frequency would increase measured current. Contrary to the predictions, experiments showed that increasing the light frequency increased the kinetic energy of the photoelectrons, and increasing the light amplitude increased the current.
The valence electron in a lithium atom jumps from energy level n=2 to n=4. What is the energy of this transition in joules? In eV? Use the equation: -RH [1/ni^2 - 1/nf^2] NOTE: RH = 2.18 x 10^-18 J/electron = 13.6 eV/electron
≈ 4 x 10^-19 J ≈ 2.5 eV
