College Physics

# Section Summary

College PhysicsSection Summary

### 30.1Discovery of the Atom

• Atoms are the smallest unit of elements; atoms combine to form molecules, the smallest unit of compounds.
• The first direct observation of atoms was in Brownian motion.
• Analysis of Brownian motion gave accurate sizes for atoms ($10 −10 m 10 −10 m$ on average) and a precise value for Avogadro’s number.

### 30.2Discovery of the Parts of the Atom: Electrons and Nuclei

• Atoms are composed of negatively charged electrons, first proved to exist in cathode-ray-tube experiments, and a positively charged nucleus.
• All electrons are identical and have a charge-to-mass ratio of
$q e m e = − 1.76 × 10 11 C/kg. q e m e = − 1.76 × 10 11 C/kg. size 12{ { {q rSub { size 8{e} } } over {m rSub { size 8{e} } } } = - 1 "." "76" times "10" rSup { size 8{"11"} } " C/kg" "." } {}$
• The positive charge in the nuclei is carried by particles called protons, which have a charge-to-mass ratio of
$q p m p = 9 . 57 × 10 7 C/kg . q p m p = 9 . 57 × 10 7 C/kg . size 12{ { {q rSub { size 8{p} } } over {m rSub { size 8{p} } } } =9 "." "57" times "10" rSup { size 8{7} } " C/kg" "." } {}$
• Mass of electron,
$m e = 9 . 11 × 10 − 31 kg . m e = 9 . 11 × 10 − 31 kg . size 12{m rSub { size 8{e} } =9 "." "11" times "10" rSup { size 8{ - "31"} } " kg" "." } {}$
• Mass of proton,
$m p = 1 . 67 × 10 − 27 kg. m p = 1 . 67 × 10 − 27 kg. size 12{m rSub { size 8{p} } =1 "." "67" times "10" rSup { size 8{ - "27"} } " kg"} {}$
• The planetary model of the atom pictures electrons orbiting the nucleus in the same way that planets orbit the sun.

### 30.3Bohr’s Theory of the Hydrogen Atom

• The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula
$1λ=R1nf2−1ni2,1λ=R1nf2−1ni2, size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right )} {}$
where $λλ size 12{λ} {}$ is the wavelength of the emitted EM radiation and $RR size 12{R} {}$ is the Rydberg constant, which has the value
$R = 1.097 × 10 7 m −1 . R = 1.097 × 10 7 m −1 .$
• The constants $nini size 12{n rSub { size 8{i} } } {}$ and $nfnf size 12{n rSub { size 8{f} } } {}$ are positive integers, and $nini$ must be greater than $nfnf size 12{n rSub { size 8{f} } } {}$.
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
$ΔE=hf=Ei−Ef,ΔE=hf=Ei−Ef, size 12{ΔE= ital "hf"=E rSub { size 8{i} } - E rSub { size 8{f} } } {}$
where $ΔEΔE size 12{ΔE} {}$ is the change in energy between the initial and final orbits and $hfhf size 12{ ital "hf"} {}$ is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
• Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by
$L=mevrn=nh2πn=1, 2, 3 …,L=mevrn=nh2πn=1, 2, 3 …,$
where $LL size 12{L} {}$ is the angular momentum, $rnrn size 12{r rSub { size 8{n} } } {}$ is the radius of the $nthnth size 12{n"th"} {}$ orbit, and $hh size 12{h} {}$ is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by
$rn=n2ZaB (allowed orbits n=1, 2, 3, ...),rn=n2ZaB (allowed orbits n=1, 2, 3, ...),$
$ZZ size 12{Z} {}$ is the atomic number of an element (the number of electrons is has when neutral) and $aBaB size 12{a rSub { size 8{B} } } {}$ is defined to be the Bohr radius, which is
$a B = h 2 4π 2 m e kq e 2 = 0.529 × 10 − 10 m . a B = h 2 4π 2 m e kq e 2 = 0.529 × 10 − 10 m . size 12{a rSub { size 8{B} } = { {h rSup { size 8{2} } } over {4π rSup { size 8{2} } m rSub { size 8{e} } ital "kq" rSub { size 8{e} } rSup { size 8{2} } } } =0 "." "529" times "10" rSup { size 8{ - "10"} } " m" "." } {}$
• Furthermore, the energies of hydrogen-like atoms are given by
$En=−Z2n2E0n=1, 2, 3 ...,En=−Z2n2E0n=1, 2, 3 ..., size 12{ left (n=1, 2, 3 "." "." "." right )} {}$
where $E0E0 size 12{E rSub { size 8{0} } } {}$ is the ground-state energy and is given by
$E 0 = 2π 2 q e 4 m e k 2 h 2 = 13.6 eV. E 0 = 2π 2 q e 4 m e k 2 h 2 = 13.6 eV. size 12{E rSub { size 8{0} } = { {2π rSup { size 8{2} } q rSub { size 8{e} } rSup { size 8{4} } m rSub { size 8{e} } k rSup { size 8{2} } } over {h rSup { size 8{2} } } } ="13" "." 6" eV"} {}$
Thus, for hydrogen,
$E n = − 13.6 eV n 2 n = 1, 2, 3 ... . E n = − 13.6 eV n 2 size 12{E rSub { size 8{n} } = - { {"13" "." 6" eV"} over {n rSup { size 8{2} } } } } {} n = 1, 2, 3 ... . size 12{ left (n=1, 2, 3 "." "." "." right ) "." } {}$
• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

### 30.4X Rays: Atomic Origins and Applications

• X rays are relatively high-frequency EM radiation. They are produced by transitions between inner-shell electron levels, which produce x rays characteristic of the atomic element, or by decelerating electrons.
• X rays have many uses, including medical diagnostics and x-ray diffraction.

### 30.5Applications of Atomic Excitations and De-Excitations

• An important atomic process is fluorescence, defined to be any process in which an atom or molecule is excited by absorbing a photon of a given energy and de-excited by emitting a photon of a lower energy.
• Some states live much longer than others and are termed metastable.
• Phosphorescence is the de-excitation of a metastable state.
• Lasers produce coherent single-wavelength EM radiation by stimulated emission, in which a metastable state is stimulated to decay.
• Lasing requires a population inversion, in which a majority of the atoms or molecules are in their metastable state.

### 30.6The Wave Nature of Matter Causes Quantization

• Quantization of orbital energy is caused by the wave nature of matter. Allowed orbits in atoms occur for constructive interference of electrons in the orbit, requiring an integral number of wavelengths to fit in an orbit’s circumference; that is,
$nλ n = 2πr n n = 1, 2, 3 ... , nλ n = 2πr n size 12{nλ rSub { size 8{n} } =2πr rSub { size 8{n} } } {} n = 1, 2, 3 ... , size 12{ left (n=1, 2, 3 "." "." "." right ),} {}$
where $λnλn size 12{λ rSub { size 8{n} } } {}$ is the electron’s de Broglie wavelength.
• Owing to the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits; rather, there are clouds of probability.
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
$ΔE=hf=Ei−Ef,ΔE=hf=Ei−Ef, size 12{ΔE= ital "hf"=E rSub { size 8{i} } - E rSub { size 8{f} } } {}$
where $ΔEΔE size 12{ΔE} {}$ is the change in energy between the initial and final orbits and $hfhf size 12{ ital "hf"} {}$ is the energy of an absorbed or emitted photon.
• It is useful to plot orbit energies on a vertical graph called an energy-level diagram.
• The allowed orbits are circular, Bohr proposed, and must have quantized orbital angular momentum given by
$L = m e vr n = n h 2π n = 1, 2, 3 ... , L = m e vr n = n h 2π size 12{L=m rSub { size 8{e} } ital "vr" rSub { size 8{n} } =n { {h} over {2π} } } {} n = 1, 2, 3 ... , size 12{ left (n=1, 2, 3 "." "." "." right ),} {}$
where $LL size 12{L} {}$ is the angular momentum, $rnrn size 12{r rSub { size 8{n} } } {}$ is the radius of orbit $nn size 12{n rSup { size 8{"th"} } } {}$, and $hh size 12{h} {}$ is Planck’s constant.

### 30.7Patterns in Spectra Reveal More Quantization

• The Zeeman effect—the splitting of lines when a magnetic field is applied—is caused by other quantized entities in atoms.
• Both the magnitude and direction of orbital angular momentum are quantized.
• The same is true for the magnitude and direction of the intrinsic spin of electrons.

### 30.8Quantum Numbers and Rules

• Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number $nn size 12{n} {}$ labels the basic states of a system and is given by
$n=1,2,3,....n=1,2,3,.... size 12{n=1, 2, 3, "." "." "." } {}$
• The magnitude of angular momentum is given by
$L = l l + 1 h 2π l = 0, 1, 2, ... , n − 1 , L = l l + 1 h 2π l = 0, 1, 2, ... , n − 1 ,$
where $ll size 12{l} {}$ is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the $zz size 12{z} {}$-axis is given by
$Lz = ml h2π ml = −l, −l+ 1, ... ,−1, 0, 1, ...l−1,l, Lz = ml h2π ml = −l, −l+ 1, ... ,−1, 0, 1, ...l−1,l,$
$LzLz size 12{L rSub { size 8{z} } } {}$ is the $zz size 12{z} {}$-component of the angular momentum and $mlml size 12{m rSub { size 8{l} } } {}$ is the angular momentum projection quantum number. Similarly, the electron’s intrinsic spin angular momentum $SS size 12{S} {}$ is given by
$S=ss+1h2π(s=1/2 for electrons),S=ss+1h2π( size 12{S= sqrt {s left (s+1 right )} { {h} over {2π} } } {}s=1/2 for electrons), size 12{s=1/2} {}$
$ss size 12{s} {}$ is defined to be the spin quantum number. Finally, the direction of the electron’s spin along the $zz size 12{z} {}$-axis is given by
$Sz=msh2πms=−12,+12,Sz=msh2π size 12{S rSub { size 8{z} } =m rSub { size 8{s} } { {h} over {2π} } } {}ms=−12,+12, size 12{ left (m rSub { size 8{s} } = - { {1} over {2} } , + { {1} over {2} } right )} {}$
where $SzSz size 12{S rSub { size 8{z} } } {}$ is the $zz size 12{z} {}$-component of spin angular momentum and $msms size 12{m rSub { size 8{s} } } {}$ is the spin projection quantum number. Spin projection $ms=+1/2ms=+1/2 size 12{m rSub { size 8{s} } "=+"1/2} {}$ is referred to as spin up, whereas $ms=−1/2ms=−1/2 size 12{m rSub { size 8{s} } = - 1/2} {}$ is called spin down. Table 30.1 summarizes the atomic quantum numbers and their allowed values.

### 30.9The Pauli Exclusion Principle

• The state of a system is completely described by a complete set of quantum numbers. This set is written as $n, l,ml,msn, l,ml,ms$.
• The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers; that is, no two electrons can be in the same state.
• This exclusion limits the number of electrons in atomic shells and subshells. Each value of $nn size 12{n} {}$ corresponds to a shell, and each value of $ll size 12{l} {}$ corresponds to a subshell.
• The maximum number of electrons that can be in a subshell is $22l+122l+1 size 12{2 left (2l+1 right )} {}$.
• The maximum number of electrons that can be in a shell is $2n22n2 size 12{2n rSup { size 8{2} } } {}$.